The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.
The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of onehalf. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize.
In this lecture, Ken Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances. He will conclude with a discussion of recent joint work with mathematicians Michael Griffin of Brigham Young University, Larry Rolen of Georgia Tech, and Don Zagier of the Max Planck Institute, which sheds new light on this famous problem.
The Gyárfás–Sumner conjecture states the following: Let $a,b$ be positive integers. Then there exists a function $f$, such that if $G$ is a graph of clique number at most $a$ and chromatic number at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as induced subgraphs. This conjecture is still open, though for several special cases it is known to be true. We study the oriented version of this conjecture: Does there exist a function $g$, such that if the chromatic number of an oriented graph $G$ (satisfying certain properties) is at least $g(s)$ then $G$ contains all oriented trees on at most $s$ vertices as its induced subgraphs. In general this statement is not true, not even for triangle free graphs. Therefore, we consider the next natural special class – namely the 4cycle free graphs – and prove the above statement for that class. We show that $g(s) \leq 4s^2$ in this case.
We also consider the rainbow (colorful) variant of this conjecture. As a special case of our theorem, we significantly improve an earlier result of Gyárfás and Sarkozy regarding the existence of induced rainbow paths in $C_4$ free graphs of high chromatic number. I will also discuss the recent results of Seymour, Scott (and Chudnovsky) on this topic.
Let $O$ be the ring of integers of a nonArchimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let e be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of JaikinZapirain and StasinskiStevens. However, those methods do not work for p=2. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. JaikinZapirain and AvniKlopschOnnVoll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{r})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are NOT isomorphic for $r \geq 2e+2$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{r}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{r}))]$ are NOT isomorphic for $r\geq4$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
Automorphic and modular forms in one complex variable (on the upper halfplane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We will show both original and known results on Harmonic Analysis for functions defined on the infinitedimensional torus, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the onedimensional torus, with its corresponding Haar measure. Such results will include:
Several open problems and other questions will be considered. Some of the results presented are joint work with Emilio Fernandez (Universidad de La Rioja, Spain).
The study of Leavitt path algebras has two primary sources, the work of W.G. Leavitt in the early 1960’s on the module type of a ring, and the work by Kumjian, Pask, and Raeburn in the 1990’s on CuntzKrieger graph $C^*$algebras. Given a directed graph $\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an $F$algebra essentially built from the directed paths in the graph $\Gamma$. Reasonable necessary and sufficient graphtheoretic conditions for two directed graphs to have isomorphic Leavitt path algebras do not seem to be known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with a directed graph $\Gamma$, that we call the Leavitt inverse semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then $L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$ to have isomorphic Leavitt inverse semigroups.
This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.
We will present some recent studies on ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are matrices whose entries are polynomials of degree at most 1. We will discuss, how a good Groebner basis for these ideals help us compute primary decompositions and gather various other homological informations.
I will discuss a few examples of concepts that have interesting extensions if loops are allowed (but not required). I will include interval graphs, strongly chordal graphs, and other concepts.
Automorphic and modular forms in one complex variable (on the upper halfplane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles (nonabelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions. We will discuss how these results can be applied to obtain an alternate proof of a result of PaulyRamanan. If time permits, we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.
Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.
The Lehmer conjecture and Serre’s lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.
G.C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.
We follow G.C. Rota’s advice. This leads to several new results and improvement of known results. This includes Kostant’s nonvanishing results attached to simple complex Lie algebras, a new nonvanishing zone of the NekrasovOkounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly signchanges results of reciprocals of the cubic root of Klein’s absolute $j$ invariant, and hence the $j$invariant itself. Finally we give an interpretation of the first nonsign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.C. Rota.
Automorphic and modular forms in one complex variable (on the upper halfplane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s wellknown bigconstant/smallconstant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newlyintroduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoytype theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoytype theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
The sphere packing problem asks for the densest packing by congruent nonoverlapping spheres in n dimensions. It is a famously hard problem, though easy to state, and with many connections to different parts of mathematics and physics. In particular, every dimension seems to have its own idiosyncracies, and until recently no proven answers were known beyond dimension 3, with the 3dimensional solution being a tour de force of computeraided mathematics.
Then in 2016, a breakthrough was achieved by Viazovska, solving the sphere packing problem in 8 dimensions. This was followed shortly by joint work of CohnKumarMillerRadchenkoViazovska solving the sphere packing problem in 24 dimensions. The solutions involve linear programming bounds and modular forms. I will attempt to describe the main ideas of the proof.
We describe the leading terms in the asymptotic behavior of the eigenvalues and the eigenfunctions to an elliptic Dirichlet spectral problem in a thin finite cylindrical domain with a periodically oscillating boundary by means of homogenization. Under suitable scaling and structure assumptions, the eigenfunctions show oscillatory behavior, and asymptotically localize with a profile solving a diffusion equation with quadratic potential on the real line. Methods for analysis of spectral asymptotics for heterogeneous media will be briefly discussed.
We will discuss certain rationality results for the critical values of the degree$2n$ $L$functions attached to $GL_1 \times O(n,n)$ over a totally real number field for an even positive integer $n$. We will also discuss some relations for Deligne periods of motives. This is part of a joint work with A. Raghuram.
Let W be a Weyl group and V be the complexification of its natural reflection representation. Let H be the discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) – H) is the Artin group described by the Dynkin diagram of W.
We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) – H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group.
The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes–Wall lattice for which some similar results hold.
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicityone result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially squarefree. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$.
We expect similar applications. We also discuss few results on the squarefree Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, \mathbb{Z})$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau’))$ ($(\tau,z,z,\tau’)$ in Siegel’s upper halfplane of degree 2) to $H \times H$ is its restriction to $z=0$, which we denote by $F_{z=0}$. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$norms of such pullbacks to central values of $L$functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$ $L$functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$ with the norm of its pullback provides a measure of concentration of $F_g$ along $z=0$. We recall certain conjectures pertaining to the size of the’mass’. We use the amplification method to improve the currently known bound for $N(F_g)$.
We will define an invariant for annular links using the combinatorial link Floer complex that gives genus bounds for annular cobordisms. The celebrated sliceBennequin inequality relates slice genus of a knot with its contact geometric invariants. We investigate similar relations in our context. In particular, we will define an invariant of transverse knots that refines the transverse invariant $\theta$ in knot Floer homology.
In this talk, we discuss the “local smoothing” phenomenon for Fourier integral operators with amplitude function belongs to the “symbol class”. We give an overview of the regularity results which have been proven to date. We use harmonic analysis of Hermite functions in the study of Fourier integral operators. Finally, we give an application of the local smoothing estimate to the wave equation and maximal operators. This is a joint work with Prof. P. K. Ratnakumar.
If $T$ is a cnu (completely nonunitary) contraction on a Hilbert space, then its NagyFoias characteristic function is an operator valued analytic function on the unit disc $\mathbb{D}$ which is a complete invariant for the unitary equivalence class of $T$. $T$ is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all elements $\varphi$ of the group $M$ of biholomorphic maps on $\mathbb{D}$. A stronger notion is of an associator. $T$ is an associator if there is a projective unitary representation $\sigma$ of $M$ such that $\varphi(T) = \sigma(\varphi)^* T \sigma(\varphi)$ for all $\varphi$ in $M$. In this talk we shall discuss the following result from a recent work of the speaker with G. Misra and S. Hazra: A cnu contraction is an associator if and only if its characteristic function $\theta$ has the factorization $\theta(z) = \pi_* (\varphi_z) C \pi(\varphi_z), z \in \mathbb{D}$ for two projective unitary representations $\pi, \pi_∗$ of $M$.
This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms. We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries.
In the first part of this talk I shall recall what the Hot spots conjecture is. Putting it in mathematical terms, I shall provide a brief history of the conjecture. If time permits I shall explain a proof of the conjecture for Euclidean triangles.
I will begin by reviewing the relationship between Hitchin’s Integrable System and 4d N=2 Supersymmetric Quantum Field Theories. I will then discuss two classes of deformations of the Hitchin system which correspond, in the physical context, to relevant and marginal deformations of a conformal theory. The study of relevant deformations turns out to be related to the theory of sheets in a complex Lie algebra and their classification leads to a surprising duality between sheets in a Lie algebra and Slodowy slices in the Langlands dual Lie algebra (work done with J. Distler) . If there is time, I will discuss marginal deformations which are related to studying the Hitchin system as a family over the moduli space of curves including over nodal curves (ongoing project with J. Distler and R. Donagi) .
Modular forms are certain functions defined on the upper half plane that transform nicely under $z\to 1/z$ as well as $z\to z+1$. By a modular relation (or a modulartype transformation) for a certain function $F$, we mean that which is governed by only the first map, i.e., $z\to 1/z$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha \beta = 1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha) = F(iw, \beta)$ or other relations of the form $F(z, \alpha) = F(z, \beta)$ etc. The famous RamanujanGuinand formula, equivalent to the functional equation of the nonholomorphic Eisenstein series on ${\rm SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha) = F(z, iw, \beta)$ recently obtained by Kesarwani, Moll and the speaker. This implies that one can superimpose the theta structure on the RamanujanGuinand formula.
The current work arose from answering a similar question  can we superimpose the theta structure on a recent modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ which generalizes an important result of Ramanujan? In the course of answering this question in the affirmative, we were led to a surprising new generalization of $\zeta(z, a)$. This new zeta function, $\zeta_w(z, a)$, satisfies interesting properties, albeit they are much more difficult to derive than the corresponding ones for $\zeta(z, a)$. In this talk, I will briefly discuss the theory of the Riemann zeta function $\zeta(z)$, the Hurwitz zeta function $\zeta(z, a)$ and then describe the theory of $\zeta_w(z, a)$ that we have developed. In order to obtain the generalized modular relation (with the theta structure) satisfied by $\zeta_w(z, a)$, one not only needs this theory but also has to develop the theory of reciprocal functions in a certain kernel involving Bessel functions as well as the theory of a generalized modified Bessel function. (Based on joint work with Rahul Kumar.)
A beautiful $q$series identity found in the unorganized portion of Ramanujan’s second and third notebooks was recently generalized by Maji and I. This identity gives, as a special case, a threeparameter identity which is a rich source of partitiontheoretic information allowing us to prove, for example, Andrews’ famous identity on the smallest parts function $\mathrm{spt}(n)$, a recent identity of Garvan, and identities on divisor generating functions, to name a few. Guo and Zeng recently derived a finite analogue of Uchimura’s identity on the generating function for the divisor function $d(n)$. This motivated us to look for a finite analogue of my generalization of Ramanujan’s aforementioned identity with Maji. Upon obtaining such a finite version, our quest to look for a finite version of Andrews’ $\mathrm{spt}$identity necessitated finding finite analogues of rank, crank and their moments. We could obtain finite versions of rank and crank for vector partitions. We were also able to obtain a finite analogue of a partition identity recently conjectured by George Beck and proven by Shane Chern. I will discuss these and some related results. This is joint work with Pramod Eyyunni, Bibekananda Maji and Garima Sood.
Given a closed orientable surface $S$, a $(G,X)$structure on $S$ is the datum of a maximal atlas whose charts take values on $X$ and transition functions are restrictions of elements in $G$. Any such structure induces a holonomy representation $\rho:\pi_1(\widetilde{S})\to X$ which encodes geometric data of the structure. Conversely, can we recover a geometric structure from a given representation? Is such a structure unique? In this talk we answer these questions by providing old and new results.
We study risksensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risksensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzerosum and zerosum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddlepoint equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risksensitive ergodic control problem for controlled reflecting diffusion processes in the nonnegative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding HamiltonJacobiBellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risksensitive zerosum/nonzerosum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the nonnegative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding HamiltonJacobiIsaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzerosum stochastic differential games with risksensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
In this talk, we shall discuss a combinatorial characterization of the family of secant lines of the 3dimensional projective space $PG(3,q)$ which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with points and planes. This is joint work with Puspendu Pradhan.
Consider a stochastic matrix $P$ for which the Perron–Frobenius eigenvalue has multiplicity larger than 1, and for $\epsilon > 0$, let
where $Q$ is a stochastic matrix for which the Perron–Frobenius eigenvalue has multiplicity 1. Let $\pi^\epsilon$ be the Perron–Frobenius eigenfunction for $P^\epsilon$. We will discuss the behavior of $\pi^\epsilon$ as $\epsilon \to 0$.
This was an important ingredient in showing that if two players repeatedly play Prisoner’s Dilemma, without knowing that they are playing a game, and if they play rationally, they end up cooperating. We will discuss this as well in the second half.
The talk will include the required background on Markov chains.
A famous result of Leonhard Euler says that his socalled “convenient numbers” $N$ have the property that a positive integer $n$ has a unique representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and only if $n$ is a prime, a prime power, twice one of these, or a power of 2. The set of known 65 convenient numbers is ${ 1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 }$, and it is conjectured that these are all of them. So, when we look at this set, we see that 11 is the first “inconvenient” number, and therefore we consider the natural question which positive integers have a representation of the form $n=x^2+11 y^2$ with $\gcd(x,11y)=1$.
Our approach is split into two parts. First we introduce the modular class group $G_{11}$
of level 11 and give a detailed description of its structure. We show that there are four
conjugacy classes of elliptic elements of order 2, we provide concrete matrices representing these
elliptic elements, and we give an explicit representation of $G_{11}$ using them.
Then we conjugate the first of these matrices, namely $t_1=\binom{0, 1}{1, 0}$, by the elements
of $G_{11}$ and get matrices whose top right entry is of the form $x^2+11 y^2$.
Conversely, we construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are
conjugate to one of the generators. Then the matrices conjugate to $t_1$ are the ones
we are interested in, and we find a set of candidate numbers $C$ such that $C=S_1 \cup
S_2$
, where $S_1$
is the set we want to characterise. Thus the task is reduced to distinguishing
between $S_1$ and $S_2$.
This problem is addressed in the second part of the talk using number rings in $K=\mathbb{Q}(\sqrt{11})$. The ring of integers of this number field is $\mathcal{O}_K=\mathbb{Z}[(11+\sqrt{11})/2]$, and the more natural ring $\mathbb{Z}[\sqrt{11}]$ is its order of conductor 2. By realizing the elements of $S_1$ and $S_2$ as norms of elements in $\mathcal{O}_K$, we get some of their basic properties. The main theorem provides a precise description of the primitive representations $n=x^2+11 y^2$ into four classes, where cubic numbers and prime numbers are two classes which admit separate, detailed descriptions. For the prime numbers in $S_1$, we need to use some consequences of ring class field theory for $\mathbb{Z}[\sqrt{11}]$, but all other results are largely selfcontained.
Counting holomorphic curves in a symplectic manifold has been an area of research since Gromov’s work on this subject in the 1980s. Symplectic manifolds naturally allow a ‘cut’ operation. We explore what happens to curve countbased invariants when a collection of cuts is applied to a symplectic manifold. An interesting feature of curves in a multiplycut manifold is that they have an underlying ‘tropical graph’, which is a graph that lives in the polytope associated to the cut.
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod$l$ reduction of the representation $\pi_F$, i.e.,
This conjecture is verified by Ronchetti when $\pi_F$ is a depthzero cuspidal representation using compact induction model. We will explain a proof in the case where $n=2$ and $\pi_F$ has arbitrary depth, using Kirillov model. If time permits, we will discuss the general case by local Rankin–Selberg convolutions.
We discuss some progress in the last decade (and ongoing interest) of modulation spaces from the PDE point of view. We prove some local and global wellposedness for Hartree–Fock equations in modulation spaces. We shall also prove similar results when a harmonic potential is added to these equations. This is a joint work with Manoussos Grillakis and Kasso Okoudjou.
We study the spectrum of a random geometric graph, in a regime where the graph is dense and highly connected. As opposed to other random graph models (e.g. the ErdosRenyi random graph), even when the graph is dense, not all the eigenvalues are concentrated around 1. In the case where the vertices are generated uniformly in a unit ddimensional box, we show that for every $0\le k \le d$ there are $\binom{d}{k}$ eigenvalues at $12^{k}$. The rest of the eigenvalues are indeed close to 1. The spectrum of the graph Laplacian plays a key role in both theory and applications. We also show that the corresponding eigenfunctions are tightly related to the geometric configuration of the points. Aside from the interesting mathematical phenomenon we reveal here, the results of this paper can also be used to analyze the homology of the random VietorisRips complex via spectral methods.
The talk will be based on a joint work with R. Adler, O. Bobrowski, and R. Rosenthal.
Macdonald polynomials are a remarkable family of orthogonal symmetric polynomials in several variables. An enormous amount of combinatorics, group theory, algebraic geometry and representation theory is encoded in these polynomials. It is known that the characters of level one Demazure modules are nonsymmetric Macdonald polynomials specialized at t=0. In this talk, I will define a class of polynomials in terms of symmetric Macdonald polynomials and using representation theory we will see that these polynomials are Schurpositive and are equal to the graded character of level two Demazure modules for affine $\mathfrak{sl}_{n+1}$. As an application we will see how this gives rise to an explicit formula for the graded multiplicities of level two Demazure modules in the excellent filtration of Weyl modules. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.
In 2001, Jeff Kahn showed that a disjoint union of $n/(2d)$ copies of the complete bipartite graph $K_{d,d}$ maximizes the number of independent sets over all $d$regular bipartite graphs on n vertices, using Shearer’s entropy inequality. In this lecture I will mention several extensions and generalizations of this extremal result (to graphs and hypergraphs) and will describe a stability result (in the spectral sense) to Kahn’s result.
The lecture is based on joint works with Emma Cohen, David Galvin, Will Perkins, Michail Sarantis and Hiep Han.
The connection between the multiplicative and additive structures of an arbitrary integer is one of the most intriguing problems in number theory. It is in this context that we explore the problem of identifying those consecutive integers which are divisible by a power of their largest prime factor. For instance, letting $P(n)$ stand for the largest prime factor of $n$, then the number $n=1294298$ is the smallest integer which is such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2$. No one has yet found an integer $n$ such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2,3$. Why is that? In this talk, we will provide an answer to this question and explore similar problems.
We will review the history of solvability of polynomial equations by radicals, concentrating on the two Memoirs of Evariste Galois. We will show how the first Memoir allows us to determine all equations of prime degree which are solvable by radicals, and the second Memoir similarly leads to the determination of all primitive equations which are solvable by radicals. A finite separable extension $L$ of a field $K$ is called primitive if there are no intermediate extensions, and solvable if the Galois group of its Galois closure is a solvable group. Galois himself proved in his Second Memoir that if $L$ is both primitive and solvable over $K$, then the degree $[L:K]$ has to be the power of a prime. We parametrise the set of all primitive solvable extensions in terms of other more computable things attached to $K$. Thus, when $K$ is a local field with finite residue field of characteristic $p$, we can explicitly write down all primitive extensions! This involves the determination of all irreducible $\mathbb{F}_p$representations of the absolute Galois group of $K$.
In his seminal paper in 2001, Henri Darmon proposed a systematic construction of padic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of padic cohomology classes/cycles in the Harris–Soudry–Taylor representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato–Beilinson conjecture as well as Gross–Zagier type results. This is based on a joint work with Chris Williams (Imperial College London).
The historical development of the study of operator algebras is intimately tied with the quest to develop a mathematical formalism for quantum mechanics. This is what motivated von Neumann to study ‘rings of operators’ in his seminal series of papers with Murray. He was dissatisfied with his original Hilbert space formalism involving the socalled type $I_{\infty}$ algebras and envisaged the type $II_1$ factors as providing the appropriate description of the logic of quantum systems. In this talk, we will briefly trace the history of the field before turning our attention to the question of representing the Heisenberg commutation relation, $QP  PQ = i\hbar I$, using operators on a Hilbert space. In the type $II_1$ case, this obstinately leads towards nonselfadjoint operators and nonselfadjoint operator algebras which have so far been secondclass citizens in comparison to their selfadjoint counterparts (namely, C*algebras, von Neumann algebras). We will conduct some mock drills in the world of matrices before moving our discussion to the algebra of affiliated operators corresponding to a $II_1$ factor (otherwise known as Murrayvon Neumann algebras).
A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.
Computing the determinant using the Schur complement of an invertible minor is wellknown to undergraduates. Perhaps less wellknown is why this works even when the minor is not invertible. Using this and the Cayley–Hamilton theorem as illustrative examples, I will gently explain one “practical” usefulness of Zariski density outside commutative algebra.
This talk will be a continuation of my presentation on ‘Factorization of Contractions’, which was held at the inhouse symposium this year.
We will start by proving the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries which states that: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exist a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\Phi}, M_{\Psi})$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where
for $z \in \mathbb{D}$ (the unit disc in $\mathbb{C}$).
We shall then proceed to derive a similar characterization for pure contractions. In particular, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z_{\mathcal{Q}}$ be the Sz.Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$, where $\mathcal{D}$ is a closed subspace of $\mathcal{H}$. We will show that $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exist $\mathcal{B}(\mathcal{D})$valued polynomials $\varphi$ and $\psi$ of degree $\leq 1$ such that $\mathcal{Q}$ is a joint ($M^*_{\varphi}, M^*_{\psi}$)invariant subspace and
Moreover, there exist a Hilbert space $\mathcal{E}$ and an isometry $V \in \mathcal{B}(\mathcal{D}; \mathcal{E})$ such that
where the pair $(\Phi, \Psi)$, as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on $H^2_{\mathcal{E}}(\mathbb{D})$. As an application, we shall obtain a sharper von Neumann inequality for commuting pairs of contractions.
If time permits, then we will look at some of the recent work by other authors in this direction and also try to figure out the challenges that lie in extending this result to a tuple of contractions.
We prove an explicit central value formula for a family of complex Lseries of degree 6 for GL2 × GL3 which arise as factors of certain Garret–Rankin triple product Lseries associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito–Kurokawa lifts, and as an application, we prove Deligne’s conjecture about the algebraicity of the central values of the considered Lseries up to the relevant periods. I would also include some other arithmetic applications towards the subconvexity problem, construction of associated padic L function, etc.
This is joint work with Carlos de Vera Piquero.
A diagonalizable matrix has linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every matrix is the limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based on regularizing the pseudospectrum with a complex Gaussian perturbation.
Joint work with J. Banks, A. Kulkarni, S. Mukherjee.
Problems associated with the algorithmic counting of combinatorial structures arise naturally in many applications and also constitute an interesting subfield of computational complexity theory. In this talk, we consider one of the most studied of such problems: that of counting proper colourings of bounded degree graphs. Here, a constant maximum degree $\Delta$ and a set of $q$ colours are fixed. The input is a graph $G$ of maximum degree at most $\Delta$, and a parameter $\varepsilon > 0$. The problem is to output “efficiently”, i.e. in time that grows polynomially in $1/\varepsilon$ and the size $n$ of the graph, and up to a multiplicative error of at most $(1+\varepsilon)$, the number of proper colourings of $G$ with the given set of $q$ colours.
The problem has been attacked using randomized Markov chain methods, and a delightfully simple analysis using the path coupling method gives an efficient randomized algorithm when $q \geq 2\Delta + 1$. The best currently known Markov chain analysis, due to Vigoda, requires $q \geq 11 \Delta / 6$. However, the condition required for efficient deterministic algorithms (i.e., those that do not use any randomness and do not have a probability of error) was until recently $q \geq 2.58 \Delta$, which lagged behind even the simple path coupling analysis. In this work, we improve this to $q \geq 2 \Delta$, thus finally matching at least the path coupling bound for deterministic algorithms. Our method, based on a paradigm proposed by Alexander Barvinok, uses information on the complex zeros of the associated Potts model partition function. Perhaps surprisingly, the information we need about the complex zeros of this partition function is obtained using methods inspired from previous analyses of phenomena related to Markov chain algorithms for the problem.
Part of the talk will also be a general survey of Barvinok’s paradigm and the growing body of work connecting it to more probabilistic notions.
Joint work with Singcheng Liu and Alistair Sinclair.
We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let ${X_1,X_2,\dots,X_N}$ be a set of $N$ sample points sampled randomly from an unknown Riemannian manifold $M$. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+\eta_{jk}$, where $j,k\in {1,2,\dots,N}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $\eta_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{\eta_{jk}}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.
This is joint work with Charles Fefferman, Sergei Ivanov and Matti Lassas.
Hida once described his theory of families of ordinary padic modular eigenforms as obtained from cutting “the clear surface out of the pitchdark well too deep to see through” of the space of all elliptic modular forms. In this colloquiumstyle talk, we shall peer into the well of Drinfeld modular forms instead of classical modular forms. More precisely, we shall explain how to construct families of finite slope Drinfeld modular forms over Drinfeld modular varieties of any dimension. In the ordinary case (the “clear surface”), we show that the weight may vary padically in families of Drinfeld modular forms (a direct analogue of Hida’s Vertical Control Theorem). In the deeper & murkier waters of positive slope, the situation is more subtle: the weight may indeed vary continuously, but not analytically, thereby contrasting markedly with Coleman’s wellknown padic theory.
Joint work with G. Rosso (Cambridge University / Concordia University (Montréal)).
Let $\{ M_k \}_{k=1}^{\infty}$ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold $(N^{n+1},g), n+1 \geq 3$. Suppose, the volumes of $M_k$ are uniformly bounded from above and the $p$th Jacobi eigenvalues of $M_k$ are uniformly bounded from below. Then, there exists a closed, singular, minimal hypersurface $M$ in $N$ with the above mentioned volume and eigenvalue bounds such that possibly after passing to a subsequence, $M_k$ weakly converges (in the sense of varifolds) to $M$, possibly with multiplicities. Moreover, the convergence is smooth and graphical over the compact subsets of $reg(M) \setminus Y$ where $Y$ is a finite subset of $reg(M)$ with $Y\leq p1$. This result generalizes the previous results of Sharp and AmbrozioCarlottoSharp in higher dimensions.
To any convex integral polygon $N$ is associated a cluster integrable system that arises from the dimer model on certain bipartite graphs on a torus. The large scale statistical mechanical properties of the dimer model are largely determined by an algebraic curve, the spectral curve $C$ of its Kasteleyn operator $K(x,y)$. The vanishing locus of the determinant of $K(x,y)$ defines the curve $C$ and coker $K(x,y)$ defines a line bundle on $C$. We show that this spectral data provides a birational isomorphism of the dimer integrable system with the Beauville integrable system related to the toric surface constructed from $N$.
This is joint work with Alexander Goncharov and Richard Kenyon.
A three dimensional acoustic medium (operator is $\partial_t^2 {} \Delta_x + q(x)$) is probed by plane waves coming from infinity and the medium response is measured at infinity. One aims to recover the acoustic property (the function $q(x)$) from this type of measurement. Two such longstanding open problems are the “fixed angle” scattering problem and the “backscattering” problem. Both these problems involve studying the injectivity and the inversion of some nonlinear map from compactly supported smooth functions on $\mathbb{R}^3$ to distributions on $\mathbb{R}^3$. These maps are defined through nonexplicit solutions of the perturbed wave equation  the existence and uniqueness of these solutions is well understood.
We will state these two problems, describe what was known and then state our (with Mikko Salo of University of Jyvaskyla, Finland) injectivity result for the “fixed angle scattering” problem. If time permits we will describe some of the ideas used to prove our result for the fixed angle scattering problem – Carleman estimates for the perturbed wave equation play a big role.
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a onetoone correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$filtration.
Introduced in 1957 by Broadbent and Hammersley as a simple probabilistic model for movement of fluid through porous media, percolation theory has emerged as one of the most active and richest areas of research in modern probability theory. As one the most wellknown models demonstrating an orderdisorder phase transition, the techniques developed in the context of percolation theory have been widely successful in rigorous understanding of models in classical statistical physics, e.g. Ising and Potts. Furthermore in two dimensions it enjoys a deep connection with conformal field theory. In this talk I will attempt to provide a very brief tour of some horizons of classical percolation theory including some very recent results. I will then move onto percolation models in dependent media which pose new challenges to overcome. In this context I will present some recent results about a model related to the geometry of level sets of a canonical random Gaussian function (Gaussian free field) on lattice graphs. I will also briefly discuss some directions for future research. Some of the recent results are based on joint works with Hugo DuminilCopin, Aran Raoufi, PierreFrancois Rodriguez, Franco Severo and Ariel Yadin. No knowledge about percolation theory is assumed on the part of the audience.
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three planar random graph models, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(\mid xy \mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid\cdot\mid$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_{\lambda}$ are endowed with weights that are nonnegative independent random variables $W$, where $P(W>w)=w^{\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of vertices of degree $j$ converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
Let $O$ be the ring of integers of a nonArchimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let e be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of JaikinZapirain and StasinskiStevens. However, those methods do not work for p=2. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. JaikinZapirain and AvniKlopschOnnVoll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{2m})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{2m}))]$ are NOT isomorphic for $m > e$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{2m}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{2m}))]$ are NOT isomorphic for $m>1$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $\rho^{12s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
This talk would have two parts. In the first part, we will discuss some topics which can be classified as ‘Linniktype’ problems (the motivation being his original question about locating the first prime in an arithmetic progression) in the context of Hecke eigenvalues of modular forms on various groups, and then talk about the distribution of their signs. In the second part we will discuss differential operators on modular forms, and then talk about their applications to questions about Jacobi forms.
It is wellknown that the sequence of Hecke eigenvalues mentioned above are often real, and has infinitely many sign changes. First part of the talk would discuss the problem of estimating the location of the first such sign change in the context of Hecke eigenvalues of Yoshida lifts (a certain subspace of the Siegel modular forms) and Fourier coefficients of Hilbert modular forms. We show how to improve the previously best known results on this topic significantly.
The crucial inputs behind these would be to establish a nontrivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level for treating Yoshida lifts; and exploiting Hecke relations along with generalising related results due to K. Soundararajan, K. Matomaki et al. for the case of Hilbert modular forms. In both cases we measure the location of the eigenvalues or Fourier coefficients in terms of an analytic object called the ‘analytic conductor’, which would be introduced during the talk. Moreover in the case of Hilbert modular forms, we will also discuss quantitative results about distribution of positive and negative Hecke eigenvalues. The proof depends on establishing a certain result on a particular types of multiplicative functions on the set of integral ideals of a totally real number field.
In the second part of the talk, we will introduce the space of Jacobi forms and certain results due to J. Kramer and, briefly, a conjecture due to Hashimoto on theta series attached to quaternion algebras to motivate the results to follow. The (partial) solution of this conjecture by Arakawa and B"ocherer transfers the question to one about differential operators on Jacobi forms, and we would report on previously known and new results on this topic.
The heart of the second part of the talk would focus on the question about the differential operators on Jacobi forms. It is well known that certain differential operators ${D_{v}}_{0}^{2m}$ map the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the sum of the differential operators $D_{v}$ for $v={1,2,…2m}$ map $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. The question alluded to above boils down to investigate whether one can omit certain differential operators from the list above, maintaining the injective property. In this regard, we would discuss results of Arakawa–B"ocherer, Das–Ramakrishnan, and finally our results. The main point would be to establish automorphy of the Wronskian of a certain tuple of congruent theta series of weight 3/2.
Integer partitions have been an interesting combinatorial object for centuries but we still have a lot left to understand. In this talk we will see an uncommon way to view partitions using ‘Abaci’ and use this to illustrate partition division (defining tcores and tquotients of a given integer partition). I am planning to go in more detail with a special kind of partition called tcore (partitions) and discuss different ways to enumerate related objects.
Trace is a kind of “special noncommutative integration” and trace formulae attempts to relate traces of certain expressions, involving operators with associated geometric/topological quantities. A simple example, involving two orthogonal projections in a Hilbert space, will be discussed, in which some of these features appear naturally.
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s wellknown bigconstant/smallconstant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newlyintroduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoytype theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoytype theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
The study of multiplicative B(H)semigroups with selfadjointideals property (called SI semigroups, in short) is related to the solvability of a certain operator equation. This introduces a new unitary invariant property for B(H)semigroups. We characterized singly generated SI semigroups for a normal operator and for a rank one operator. The SI property is closely related to the simplicity of B(H)semigroups.
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms,having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicityone result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$
have the same eigenvalues of the $p$th Hecke operator $T_p$
for almost all primes $p$
, then $f_1=f_2$
.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square–free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$. We expect similar applications. We also discuss few results on the square–free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, Z)$ then we consider its Saito–Kurokawa lift $F_g$
and a certain ‘restricted’ $L^2$
norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau'))$
($(\tau,z,z,\tau')$
in Siegel’s upper halfplane of degree 2) to $H \times H$
is its restriction to $z=0$
, which we denote by $F\_{z=0}$
. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$
norms of such pullbacks to central values of $L$functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$
), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$
$L$
functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$
with the norm of its pullback provides a measure of concentration of $F_g$
along $z=0$. We recall certain conjectures pertaining to the size of the ‘mass’. We use the amplification method to improve the currently known bound for $N(F_g)$
.
An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchell, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an Hcategory will denote an “Hmodule algebra with several objects” and a coHcategory will denote an “Hcomodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. We study the cohomology in such module categories. In particular, we consider Hequivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the Hlocally finite cohomologies of these objects. We also introduce relative (D,H)Hopf modules over a Hopf comodule category D. These generalize relative (A,H)Hopf modules over an Hcomodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. We will develop these cohomology theories in a manner similar to the “Hfinite cohomology” obtained by Guedenon and the cohomology of relative Hopf modules studied by Caenepeel and Guedenon respectively. This is one of the two thesis problems which we plan to discuss in detail.
If time permits, we will also give a brief presentation of the other thesis project. In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” ($\mathbb{F}_1$) has been developed. It is in this context that monoids became topologically and geometrically relevant objects of study. In our work, we abstract out the topological characteristics of the prime spectrum of a commutative monoid, endowed with the Zariski topology is homeomorphic to the spectrum of a ring i.e., it is a spectral space. Spectral spaces, introduced by Hochster, are widely studied in the literature. We use ideals and modules over monoids to present many such spectral spaces. We introduce closure operations on monoids and obtain natural classes of spectral spaces using finite type closure operations. In the process, various closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids. We study their persistence and localization properties in detail. Next, we make a study of valuation on monoids and prove that the collection of all valuation monoids having the same group completion forms a spectral space. We also prove that the valuation spectrum of any monoid gives a spectral space. Finally, we prove that the collection of continuous valuations on a topological monoid whose topology is determined by any finitely generated ideal also gives a spectral space.
We construct certain representations of affine Hecke algebras and Weyl groups, which depend on several auxiliary parameters. We refer to these as “metaplectic” representations, and as a direct consequence we obtain a family of “metaplectic” polynomials, which generalizes the wellknown Macdonald polynomials.
Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the KazhdanPatterson action on metaplectic forms for GL(n); more generally it recovers the ChintaGunnells action on pparts of Weyl group multiple Dirichlet series.
This is joint work with Jasper Stokman and Vidya Venkateswaran.
We study risksensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risksensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzerosum and zerosum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddlepoint equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risksensitive ergodic control problem for controlled reflecting diffusion processes in the nonnegative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding HamiltonJacobiBellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risksensitive zerosum/nonzerosum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the nonnegative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding HamiltonJacobiIsaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzerosum stochastic differential games with risksensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
This talk consists of the finite element analysis of a distributed optimal control problem governed by the von Kármán equation that describe the deflection of very thin plates defined on a polygonal domain of R2 with box constraints on the control variable. In this talk we discuss a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower order norms for the state and adjoint variables are derived. The lower order estimates for the adjoint variable and a postprocessing of control leads to an improved error estimate for the control variable. Finally we discuss several numerical results to illustrate our theoretical results.
Schur classified polynomial representations of GL(n) using the commuting actions of the dth symmetric group and GL(n) on the dfold tensor power of ndimensional space. We set out to investigate what happens when the symmetric group in this picture is shrunk to the alternating group. This led to a simple interpretation and clearer understanding of the Koszul duality functor on the category of polynomial representations of GL(n).
This is based on joint work with T. Geetha and Shraddha Srivastava (https://arxiv.org/abs/1902.02465).
Consider the tensor product of two irreducible finite dimensional representations of a simple Lie algebra. The submodules generated by a tensor product of extremal vectors of the two components are called KostantKumar submodules. These are parametrized by a double coset space of the Weyl group.
Littelmann’s path model is a very general combinatorial model for representations, which encompasses many classical constructs such as Young tableaux and LakshmibaiSeshadri chains. The path model for the full tensor product is simply the set of concatenations of paths of the individual components. We describe a way to associate a Weyl group element (rather, a double coset) to each such concatenated path and thereby obtain a path model for KostantKumar submodules.
Finally, we recall the many descriptions of Demazure modules, which may be viewed as the analog of the above picture for single paths (rather than concatenations). In this case, the Weyl group element associated to a path admits different descriptions in different path models in terms of statistics such as initial direction, minimal standard lifts and Kogan faces of GelfandTsetlin polytopes. Along the way, we mention some relations to jeudetaquin and the Schutzenberger involution.
This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan.
Each finite dimensional irreducible representation V of a simple Lie algebra L admits a filtration induced by a principal nilpotent element of L. This, socalled, Brylinski or BrylinskiKostant filtration, can be restricted to the dominant weight spaces of V, and the resulting Hilbert series is very interesting qanalogs of weight multiplicity, first defined by Lusztig.
This picture can be extended to certain infinitedimensional Lie algebras L and to irreducible highest weight, integrable representations V. We focus on the level 1 vacuum modules of special linear affine Lie algebras. In this case, we show how to produce a basis of the dominant weight spaces that is compatible with the Brylinski filtration. Our construction uses the socalled Walgebra, a natural vertex algebra associated to L.
This is joint work with Sachin Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).
In this talk, we will introduce the notion of an embedding of a quadratic space (in an associative algebra). Familiar examples of embeddings are given by Complex Numbers, Quaternions, Octonions, Clifford Algebras, and Suslin Matrices.
When there are two embeddings of the same quadratic space, then we can gather information about one embedding using the other. This is the main theme of the talk. To illustrate this, we will see the connection between Suslin Matrices and Clifford Algebras. In one direction, we are able to give a simple description of Clifford Algebras using matrices. Conversely, we can now explain some (seemingly) accidental properties of Suslin matrices in a conceptual way.
The systematic study of determinantal processes began with the work of Macchi (1975), and since then they have appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, nonintersecting paths, measures on Young diagrams), and physics (fermions). A particularly interesting and wellknown example of a discrete determinantal process is the Uniform spanning tree (UST) on finite graphs. We shall describe UST on complete graphs and complete bipartite graphs—in these cases it is possible to make explicit computations that yield some special cases of Aldous’ result on CRT.
The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a Hilbert space of functions on a given set. Let $H$ and $K$ are two finite dimensional subspaces of a Hilbert space, and let $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalue of Wishart matrix ensembles $W(N,N)$ and $W(N1,N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M,N)$ has the same distribution as the directed lastpassage time $G(M,N)$ on $Z^2$ with i.i.d. exponential weights. We, thus, obtain stochastic domination between $G(N,N)$ and $G(N1,N+1)$  answering a question of Riddhipratim Basu. Similar connections are also known between the largest eigenvalue of Meixner ensemble and directed lastpassage time on $Z^2$ with i.i.d. geometric weights. We prove a stochastic domination result which combined with the Lyons’ result gives the stochastic domination between Meixner ensemble $M(N,N)$ and $M(N1,N+1)$.
We shall present the circle of ideas concerning the coarse geometric approach to the BaumConnes conjecture, via the analytic surgery sequence of N. Higson and J. Roe. In joint work with M.T. Benameur, we elicit its relations with some secondary invariants of Dirac operators on cocompact coverings, which generalize classical deep results originally due to N. Keswani. We have now partially extended this program further to the context of noncompact, complete spin foliations, which provide new obstructions to the existence of leafwise metrics of (uniformly) positive scalar curvature via the coarse index, generalizing the results of Connes for smooth foliations of compact manifolds. If time permits, we shall also outline the connections of this framework with the work of GromovLawson.
Modular forms are characterized by certain transformation laws. In general, these properties are not preserved by (holomorphic) differentiation. I will discuss several ways to overcome this trouble: Analytically one can use nonholomorphic differential operators (‘‘MaaßShimura operators’’) or use bilinear holomorphic differential operators (‘‘Rankin brackets’’). More arithmetically, one can show that derivatives of modular forms are still congruent to modular forms (‘‘Ramanujan’s $\theta$operators’’). I will describe the connection between the analytic and the arithmetic aspects with focus on some recent results on $\theta$operators.
Representations of Symmetric groups $S_n$ can be considered as homomorphisms to the orthogonal group $\mathrm{O}(d,\mathbb{R})$, where $d$ is the degree of the representation. If the determinant of the representation is trivial, we call it achiral. In this case, its image lies in the special orthogonal group $\mathrm{SO}(V)$. It is called chiral otherwise. The group $\mathrm{O}(V)$ has a nontrivial topological double cover $\mathrm{Pin}(V)$. We say the representation is spinorial if it lifts to $\mathrm{Pin}(V)$. We obtained a criterion for whether the representation is spinorial in terms of its character. We found similar criteria for orthogonal representations of Alternating groups and products of symmetric groups. One can use these results to count the number of spinorial irreducible representations of $S_n$, which are parametrized by partitions of $n$. We say a partition is spinorial if the corresponding irreducible representation of $S_n$ is spinorial. In this talk, we shall present a summary of these results and count for the number of odddimensional, irreducible, achiral, spinorial partitions of $S_n$. We shall also prove that almost all the irreducible representations of $S_n$ are achiral and spinorial. This is joint work with my supervisor Dr. Steven Spallone.
The Pick–Nevanlinna problem refers to the problem of – given two connected open sets in complex Euclidean spaces and finite sets of distinct points in each – characterizing (in terms of the given point data) the existence of a holomorphic map between the two sets that interpolates the given points. The problem gets its name from Pick – who provided a beautiful characterization for the existence of an interpolant when the domain and the codomain are the unit disc – and from Nevanlinna, who rediscovered Pick’s result. This characterization is in terms of matrix positivity. I shall begin by presenting an easy argument by Sarason, which he says is already implicit in Nevanlinna’s work, for the necessity of the Pick–Nevanlinna condition. How does one prove the sufficiency of the latter condition? Sarason’s ideas have provided the framework for a long chain of complicated characterizations for the more general problem. The last word on this is still to be written. But in the original setup of Pick, geometry provides the answer. Surprisingly, Pick did not notice that his approach provides a very clean solution to the interpolation problem where the codomain is any Euclidean ball. We shall see a proof of the latter. This proof uses a general observation about conditional positivity (which also made an appearance in the previous talk), which is attributed to Schur. If time permits, we shall see what can be said when the codomain is a bounded symmetric domain.
I will present a historical account of some work of Schoenberg in metric geometry: from his metric space embeddings into Euclidean space and into spheres (Ann. of Math. 1935), to his characterization of positive definite functions on spheres (Duke Math. J. 1942). It turns out these results can be viewed alternately in terms of matrix positivity: from appearances of (conditionally) positive matrices in analysis, to the classification of entrywise positivity preservers in all dimensions.
In this talk I consider crossed product $C^*$algebras of higher dimensional noncommutative tori with actions of cyclic groups. I will discuss $K$theory of those $C^*$algebras and some applications.
A theorem of J. Jones relates cohomology of free loop space of a simply connected manifold to Hochschild homology of its singular cochain algebra. In this talk I will discuss a potential Hochschild chain model for a product on cohomology of free loop space known as the GoreskyHingston product.
In this talk we prove that the solution of a general linear recurrence with constant coefficients can be interpreted as the determinant of some suitable matrix using a purely combinatorial method. As a consequence of our approach, we give combinatorial proofs of some recent identities due to Sury and McLaughlin in a unified way.
In this talk, we consider tilings on surfaces (2dimensional manifolds without boundary). If the facecycle at all the vertices in a tiling are of same type then the tiling is called semiregular. In general, semiregular tilings on a surface M form a bigger class than vertextransitive and regular tilings on M. This gives us little freedom to work on and construct semiregular tilings. We present a combinatorial criterion for tilings to decide whether a tiling is elliptic, flat or hyperbolic. We also present classifications of semiregular tilings on simplyconnected surfaces (i.e., on 2sphere and plane) in the first two cases. At the beginning, we present some examples and brief history on semiregular tilings.
‘Holomorphic eta quotients’ are certain explicit classical modular forms on suitable Hecke subgroups of the full modular group. We call a holomorphic eta quotient $f$ ‘reducible’ if for some holomorphic eta quotient $g$ (other than $1$ and $f$), the eta quotient $f/g$ is holomorphic. An eta quotient or a modular form in general has two parameters: weight and level. We shall show that for any positive integer $N$, there are only finitely many irreducible holomorphic eta quotients of level $N$. In particular, the weights of such eta quotients are bounded above by a function of $N$. We shall provide such an explicit upper bound. This is an analog of a conjecture of Zagier which says that for any positive integer $k$, there are only finitely many irreducible holomorphic eta quotients of weight $k/2$ which are not integral rescalings of some other eta quotients. This conjecture was established in 1991 by Mersmann. We shall sketch a short proof of Mersmann’s theorem and we shall show that these results have their applications in factorizing holomorphic eta quotient. In particular, due to Zagier and Mersmann’s work, holomorphic eta quotients of weight $1/2$ have been completely classified. We shall see some applications of this classification and we shall discuss a few seemingly accessible yet longstanding open problems about eta quotients.
This talk will be suitable also for nonexperts: We shall define all the relevant terms and we shall clearly state the classical results which we use.
The study of projective modules along the lines initiated by J.P. Serre, H. Bass is outlined; the impetus given by A. Suslin in that direction is described, and recent progress made along the vision of A. Suslin on it by Fasel–Rao–Swan is shared briefly.
I will continue my discussion on the strong unique continuation theorem for zero order perturbations of the fractional heat equation based on a monotonicity property of a generalized Almgren frequency in weighted Gauss space. This is joint with Nicola Garofalo. Then I will briefly describe how similar considerations are useful while studying the regularity of the free boundary in the obstacle problem for the fractional heat operator where some other monotonicity formulas are also required. This is a more recent work with Donatella Danielli, Nicola Garofalo and Arshak Petrosyan.
We describe how a completion of the factorial row by Suslin led to showing some ideals in a polynomial ring are settheoretic complete intersections.
Consider Riemannian functionals defined by L^2norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature. I will talk about rigidity, stability and local minimizing properties of Einstein metrics and their products as critical metrics of these quadratic functionals. We prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for certain quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small. We also prove the stability of L^{n/2}norm of Weyl curvature at compact quotients of Sn × Hm.
The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems strongly related to the moments of some classical orthogonal polynomials (Hermite, Laguerre, Askey–Wilson). The partition function has been interpreted with various combinatorial objects such as permutations, alternative and treelike tableaux, etc. We introduce a new one called “Laguerre heaps of segments”, which seems to play a central role in the network of bijections relating all these interpretations.
I will discuss a strong unique continuation theorem for zero order perturbations of the fractional heat equation
based on a monotonicity property of a generalized Almgren frequency in weighted Gauss space. This is a joint work
with Nicola Garofalo. If time permits, I will also talk on how such a truncated variant of the Almgren type
monotonicity formula combined with new Weiss and Monneau type monotonicity formulas allows to study the structure
of the singular set on the free boundary in the obstacle problem for the fractional heat operator.
This part of the talk ( if at all I get time !) is joint with Donatella Danielli, Nicola Garofalo and Arshak Petrosyan.
The theory of orthogonal polynomials started with analytic continued fractions going back to Euler, Gauss, Jacobi, Stieltjes… The combinatorial interpretations started in the late 70’s and is an active research domain. I will give the basis of the theory interpreting the moments of general (formal) orthogonal polynomials, Jacobi continued fractions and Hankel determinants with some families of weighted paths. In a second part I will give some examples of interpretations of classical orthogonal polynomials and of their moments (Hermite, Laguerre, Jacobi, …) with their connection to theoretical physics.
According to Wikipedia, a continued fraction is “an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.” It has connections to various areas of mathematics including analysis, number theory, dynamical systems and probability theory.
Continued fractions of numbers in the open interval (0,1) naturally gives rise to a dynamical system that dates back to Gauss and raises many interesting questions. We will concentrate on the extreme value theoretic aspects of this dynamical system. The first work in this direction was carried out by the famous FrenchGerman mathematician Doeblin (1940), who rightly observed that exceedances of this dynamical system have Poissonian asymptotics. However, his proof had a subtle error, which was corrected much later by Iosifescu (1977). Meanwhile, Galambos (1972) had established that the scaled maxima of this dynamical system converges to the Frechet distribution.
After a detailed review of these results, we will discuss a powerful SteinChen method (due to Arratia, Goldstein and Gordon (1989)) of establishing Poisson approximation for dependent Bernoulli random variables. The final part of the talk will be about the application of this SteinChen technique to give upper bounds on the rate of convergence in the DoeblinIosifescu asymptotics. We will also discuss consequences of our result and its connections to other important dynamical systems. This portion of the talk will be based on a joint work with Maxim Sølund Kirsebom and Anish Ghosh.
Familiarity with standard (discrete and absolutely continuous) probability distributions will be sufficient for this talk.
The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the twodimensional unit square have raised many questions over the past century. The nodal domain theorems of Courant (1924), Stern (1926) and Pleijel (1956), give deterministic (upper, liminf, and limsup resp.) bounds for the number of nodal domains which can be exhibited by an nth eigenfunction, assuming the eigenvalues are ordered increasingly and with multiplicities listed. Prominent amongst contemporary research is the closely related question of the number of ‘nodal components’ (connected components of the zero set) of a ‘typical’ eigenfunction.
The spectral degeneracy for the square means that the nodal count of an nth eigenfunction could take a wide range of values, and to understand the ‘typical’ behaviour of this number we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundaryadapted arithmetic random waves’. The number of nodal components —now a random variable— can then be studied, and this talk will draw together tools from various areas of mathematics (random fields, integral geometry, number theory, mathematical physics, …) in order to say a few things about its asymptotic properties.
We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ε. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ε tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is nondegenerate at this point.
It is open question if the standard round metric on $S^4$ is the unique positive Einstein metric, up to isometry and scaling. In this talk, I will discuss a compactness theorem which rules out nonstandard unitvolume Einstein metrics whose scalar curvatures lie within a certain range.
An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchel, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an Hcategory will denote an “Hmodule algebra with several objects” and a coHcategory will denote an “Hcomodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. In this talk, we present a study of cohomology in such module categories.
In particular, we consider Hequivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the Hlocally finite cohomologies of these objects. We also introduce relative (D,H)Hopf modules over a Hopf comodule category D. These generalize relative (A,H)Hopf modules over an Hcomodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants.
(Joint work with Abhishek Banerjee and Mamta Balodi.)
Motivated by mirror symmetry considerations (the deformed HermitianYangMills equation due to JacobsYau) and a desire to study stability conditions involving higher Chern forms, a vector bundle version of the usual complex MongeAmpere equation (studied by Calabi, Aubin, Yau, etc) will be discussed. I shall also discuss a KobayashiHitchin type correspondence for a special case (a dimensional reduction to Riemann surfaces).
Mathematical analysts seem to have an obsession with inequalities and they search for them even within equalities! Then they ask for conditions under which equality holds. They even go to the extent of quantifying the inequalities and ask for sharp constants. Strange as it may seem, we demonstrate that this is the case, with several examples.
We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$cycles of the form $(i,n1,n)$ and $(i,n,n1)$. We call this the transpose top$2$ with random shuffle. We find the spectrum of the transition matrix for this shuffle. We mainly use the representation theory of alternating group. We show that the mixing time is of order $\left(n\frac{3}{2}\right)\log n$ and prove that there is a total variation cutoff for this shuffle.
Valiant (1979) showed that unless P = N P, there is no polynomialtime algorithm to compute the number of perfect matchings of a given graph – even if the input graph is bipartite. Earlier, the physicist Kasteleyn (1963) introduced the notion of a special type of orientation of a graph, and we refer to graphs that admit such an orientation as Pfaffian graphs. Kasteleyn showed that the number of perfect matchings is easy to compute if the input graph is Pfaffian, and he also proved that every planar graph is Pfaffian. The complete bipartite graph $K_{3,3}$ is the smallest graph that is not Pfaffian. In general, the problem of deciding whether a given graph is Pfaffian is not known to be in N P.
Special types of minors, known as conformal minors, play a key role in the theory of Pfaffian orientations. In particular, a graph is Pfaffian if and only if each of its conformal minors is Pfaffian. It was shown by Little (1975) that a bipartite graph $G$ is Pfaffian if and only if $G$ does not contain $K_{3,3}$ as a conformal minor (or, in other words, if and only if $G$ is $K_{3,3}$free); this places the problem of deciding whether a bipartite graph is Pfaffian in co – N P. Several years later, a structural characterization of $K_{3,3}$free bipartite graphs was obtained by Robertson, Seymour and Thomas (1999), and independently by McCuaig (2004), and this led to a polynomialtime algorithm for deciding whether a given bipartite graph is Pfaffian.
Norine and Thomas (2008) showed that, unlike the bipartite case, it is not possible to characterize all Pfaffian graphs by excluding a finite number of graphs as conformal minors. In light of everything that has been done so far, it would be interesting to even identify rich classes of Pfaffian graphs (that are nonplanar and nonbipartite).
Inspired by a theorem of Lovasz (1983), we took on the task of characterizing graphs that do not contain $K_4$ as a conformal minor – that is, $K_4$free graphs. In a joint work with U. S. R. Murty (2016), we provided a structural characterization of planar $K_4$free graphs. The problem of characterizing nonplanar $K_4$free graphs is much harder, and we have evidence to believe that it is related to the problem of recognizing Pfaffian graphs. In particular, we conjecture that every graph that is $K_4$free and $K_{3,3}$free is also Pfaffian. The talk will be mostly selfcontained. I will assume only basic knowledge of graph theory. For more details, see: https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.21882.
The main aim of this thesis is to explain the of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the AumannCarathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show the convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the AumannCaratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasihyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasihyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
A topological dynamical system is a pair $(X,T)$ where $T$ is a homeomorphism of a compact space $X$. A measure preserving action is a triple $(Y, \mu, S)$ where $Y$ is a standard Borel space, $\mu$ is a probability measure on $X$ and $S$ is a measurable automorphism of $Y$ which preserves the measure $\mu$. We say that $(X,T)$ is universal if it can embed any measure preserving action (under some suitable restrictions).
Krieger’s generator theorem shows that if $X$ is $A^{\mathbb{Z}}$ (biinfinite sequences in elements of $A$) and $T$ is the transformation on $X$ which shifts its elements one unit to the left then $(X,T)$ is universal. Along with Tom Meyerovitch, we establish very general conditions under which $\mathbb{Z}^d$ (where now we have $d$ commuting transformations on $X$)dynamical systems are universal. These conditions are general enough to prove that the following models are universal:
A selfhomeomorphism with non uniform specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet).
A generic (in the sense of dense $G_\delta$) selfhomeomorphism of the 2torus preserving Lebesgue measure (extending result by Lind and Thouvenot to infinite entropy).
Proper colourings of the $\mathbb{Z}^d$ lattice with more than two colours and the domino tilings of the $\mathbb{Z}^2$ lattice (answering a question by Şahin and Robinson).
Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson. The talk will not assume background in ergodic theory and dynamical systems.
Given a group G, we will define conjugacy invariant norm and discuss the boundedness of such norms. We will relate this problem with the existence of homogeneous quasimorphisms on groups. If time permits, we will discuss bounded cohomology and its relation to the above. This talk will be largely selfcontained. No background of any of the mentioned objects will be required.
We say a unital ring $R$ has the Invariant Basis Number (IBN) property in case, for each pair of positive integers $i,j$ if the left $R$modules $R^i$ and $R^j$ are isomorphic, then $i=j$. The first examples of non IBN rings were studied by William Leavitt in the 1950s and he defined (what are now known as) Leavitt algebras which are ‘universal’ with non IBN property. In 2004 the algebraic structures arising from directed (multi)graphs known as Leavitt path algebras (LPA for short) were initially developed as algebraic analogues of graph $C^*$ algebras. LPAs generalize a particular class of Leavitt algebras.
During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in $C^*$algebras, and symbolic dynamicists as well. The goal of this talk is to introduce the notion of Leavitt path algebras and to present some results on LPAs arising from weighted Cayley graphs of finite cyclic groups.
As a natural generalization of the classical uniformization theorem, one could ask if every conformal class of Riemannian metrics contains a metric with constant scalar curvature. In 1960, Yamabe attempted to solve the problem, and gave an incorrect “proof” of an affirmative answer to the above question. Yamabe’s idea was to to use techniques from calculus of variations and partial differential equations to minimize a certain functional, now named after him. Building on some work of Trudger, Aubin solved the Yamabe problem under an additional analytic condition, namely that the Yamabe invariant of the manifold is strictly smaller than the corresponding for the ndimensional sphere. Aubin was able to verify this analytic condition for all manifolds of dimension strictly greater than 5 that are not locally conformally flat. Most remarkably, the problem was finally solved by Schoen in 1984 using the newly proven positive mass theorem of Schoen and Yau, which in turn has origins in general theory of relativity. A curious feature of Schoen’s proof is that it works precisely in the cases left out by Aubin’s result.
In the first part of the talk, I will provide some background in Riemannian geometry, and then describe the developments surrounding the Yamabe problem until Aubin’s work in 1976. A recurring theme will be the role played by the ndimensional sphere and its conformal transformations, in all aspects of the problem. In the second part of the talk, I will describe the positive mass theorem, and outline Schoen’s proof of the Yamabe problem in low dimensions and for locally conformally flat manifolds. Most of the Riemannian geometric quantities (including scalar curvature, gradients, Laplacians etc.) will be be defined in the talk, and so the hope is that both the talks will be accessible to everyone.
Given a hypergraph $\mathcal{H}$ with vertex set $[n]:={1,\ldots,n}$, a bisecting family is a family $\mathcal{A}\subseteq\mathcal{P}([n])$ such that for every $B\in\mathcal{H}$, there exists $A\in\mathcal{A}$ with the property $A\cap BA\cap\overline{B}\in{1,0,1}$. Similarly, for a family of bicolorings $\mathcal{B}\subseteq {1,1}^{[n]}$ of $[n]$ a family $\mathcal{A}\subseteq\mathcal{P}([n])$ is called a System of Unbiased Representatives for $\mathcal{B}$ if for every $b\in\mathcal{B}$ there exists $A\in\mathcal{A}$ such that $\sum_{x\in A} b(x) =0$.
The problem of optimal families of bisections and bicolorings for hypergraphs originates from what is referred to as the problem of Balancing Sets of vectors, and has been the source for a few interesting extremal problems in combinatorial set theory for about 3 decades now. We shall consider certain natural extremal functions that arise from the study of bisections and bicolorings and bounds for these extremal functions. Many of the proofs involve the use of polynomial methods (not the Polynomial Method, though!).
(Joint work with Rogers Mathew, Tapas Mishra, and Sudebkumar Prasant Pal.)
Since Dvir proved the Finite Kakeya Conjecture in 2008, the Polynomial Method has become a new and powerful tool and a new paradigm for approaching extremal questions in combinatorics (and other areas too). We shall take a look at the main philosophical principle that underlies this method via two recent (2017) theorems. One is the upper bound for Capsets by Ellenberg–Gijswijt, and the other, a function field analogue of a theorem of Sarkozy, due to Ben Green.
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In the first part of my talk, we discuss the quasiconformal equivalence for general open Riemann surfaces and give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. In the second part, we consider the quasiconformal equivalence of Riemann surfaces which are the complements of Cantor sets.
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In the first part of my talk, we discuss the quasiconformal equivalence for general open Riemann surfaces and give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. In the second part, we consider the quasiconformal equivalence of Riemann surfaces which are the complements of Cantor sets.
Spectral networks are certain decorated graphs drawn on a Riemann surface. I will describe a conjectural picture in which spectral networks can be viewed as analogues of HermitianEinstein metrics on vector bundles, and in which holomorphic differentials on the Riemann surface arise as stability conditions on certain Fukayatype categories. This talk is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, and Carlos Simpson.
While the enumeration of linear extensions of a given poset is a wellstudied question, its cyclic counterpart (enumerating extensions to total cyclic orders of a given partial cyclic order) has been subject to very little investigation. In this talk I will introduce some classes of partial cyclic orders for which this enumeration problem is tractable. Some cases require the use of a multidimensional version of the classical boustrophedon construction (a.k.a. SeidelEntringerArnold triangle). The integers arising from these enumerative questions also appear as the normalized volumes of certain polytopes.
This is partly joint work with Arvind Ayyer (Indian Institute of Science) and Matthieu JosuatVerges (Laboratoire d’Informatique Gaspard Monge / CNRS).
When HarishChandra died in 1983, he left behind a voluminous pile of handwritten manuscripts on harmonic analysis on semisimple Lie groups over real/complex and padic fields. The manuscripts were turned over to the archives of the Institute for Advanced Study at Princeton, and are archived there.
Robert Langlands is the Trustee of the HarishChandra archive, and has always been interested in finding a way of salvaging whatever might be valuable in these manuscripts. Some years ago, at a conference in UCLA, he encouraged V. S. Varadarajan and me to look at these manuscripts.
The results of our efforts have resulted in the publication of the Volume 5 (Posthumous) of the Collected works of HarishChandra by Springer Verlag, which appeared in July this year. This volume covers only those manuscripts that deal with Real or Complex groups. The manuscripts dealing with padic groups remain in the archive.
My talk will be devoted to an outline of the results in this volume, without much detail. However, I will try to describe the main ingredient of the methods that cut across most of this work.
Let $X=(x_1, … ,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix with $(i,j)$ entry equal to gcd$(x_i,x_j)$, the greatest common divisor of $x_i$ and $x_j$, is called the GCD matrix on $X$. By a surprising result of Beslin and Ligh (1989), all GCD matrices are positive definite. In this talk, we will discuss new characterizations of the GCD matrices satisfying the stronger property of being totally nonnegative (i.e., all their minors are nonnegative).
Joint work with Lucas Wu (U. Delaware).
Consider the following three properties of a general group $G$:
Algebra: $G$ is abelian and torsionfree.
Analysis: $G$ is a metric space that admits a “norm”, namely, a translationinvariant metric $d(.,.)$ satisfying: $d(1,g^n) = n d(1,g)$ for all $g \in G$ and integers $n$.
Geometry: $G$ admits a length function with “saturated” subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g \in G$.
While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a nonabelian group with a “norm”.
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and if time permits, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint  as D.H.J. PolyMath  with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)
A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps. As dynamical systems, Schwarz reflection maps produce various instances of “matings” of rational maps and groups.
We will introduce this new class of dynamical systems, and illustrate the abovementioned mating phenomenon with a few concrete examples. We will then describe a specific oneparameter family of Schwarz reflection maps such that “typical” maps in this family arise as unique conformal matings of a quadratic antiholomorphic polynomial and the ideal triangle group.
Time permitting, we will also mention how Schwarz reflection maps provide us with a framework for constructing correspondences on the Riemann sphere that are “matings” of rational maps and groups.
Two groups are considered ‘elementarily equivalent’, if they have the same ‘elementary’ theory. Classification of various families of groups based on elementary equivalence, has been of long standing interest to both group theorists and model theorists, the most celebrated example of which was the elementary equivalence in free groups posed by Tarski. By studying examples of groups with different elementary theories, we can gain insight into the nature of expressibility of properties of groups. In this talk, I shall elementary equivalence in Artin groups of finite type, which forms a generalization of braid groups and are of interest in geometric group theory. This was a part of joint work with Arpan Kabiraj and Rishi Vyas. The talk should be largely selfcontained. No background in logic, model theory or braid groups shall be assumed.
Let $\mathfrak{O}$ be the ring of integers of a nonArchimedean local field such that the residue field has characteristic $p$. Then the abscissa of convergence of representation zeta function of Special Linear group $\mathrm{SL}_2(\mathfrak{O})$ is $1.$ The case $p\neq 2$ is already known in the literature. For $p=2$ we need more tools to prove the result. In this talk I will discuss the difference between those cases and give an outline of the proof for $p=2.$
Let $\mathfrak{p}$ be the maximal ideal of $\mathfrak{O}$ and $\mathfrak{O}/\mathfrak{p}=q.$ It is already shown in literature that for $r \geq 1,$ the group algebras $\mathbb C[\mathrm{GL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{GL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. Also for $2\nmid q,$ the group algebras $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. In this talk I will also show that if $2\mid q$ and $\mathrm{char}(\mathfrak{O})=0$ then, the group algebras, $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{2\ell})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2\ell}))]$ are not isomorphic for $\ell > \mathrm{e}$, where $\mathrm{e}$ is the ramification index of $\mathfrak{O}.$
Inspired by the recent developments in differential geometry and calculus of variations, there have been several approaches to identifying a suitable notion of local (Ricci) curvature on nonsmooth spaces, such as graphs and Markov chains. I will describe some of these approaches and review a few recent developments in this topic on discrete curvature. Some of the consequences include a tight Cheeger inequality in abelian Cayley graphs, and diameter bounds on the spectral gap of the graph Laplacian. Several open questions remain.
This will be an elementary introduction to the Calculus of Variations, a mathematical theory that provides the basis for understanding physical phenomena. Starting with Euclid, the talk will attempt to describe some aspects of this theory, with examples.
Let $\mathfrak{gl}(n)$ denote the Lie algebra of the general linear group $GL(n)$. Given two finite dimensional irreducible representations $L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor product decomposition $L(\lambda) \otimes L(\mu)$ is given by the LittlewoodRichardson rule.
The situation becomes much more complicated when one replaces $\mathfrak{gl}(n)$ by the general linear Lie superalgebra $\mathfrak{gl}(mn)$. The analogous decomposition $L(\lambda) \otimes L(\mu)$ is largely unknown. Indeed many aspects of the representation theory of $\mathfrak{gl}(mn)$ are more akin to the study of Lie algebras and their representations in prime characteristic or to the BGG category $\mathcal{O}$. I will give a survey talk about this problem and explain why some approaches don’t work and what can be done about it. This will give me the chance to speak about a) the character formula for an irreducible representation $L(\lambda)$, b) Deligne’s interpolating category $Rep(GL_t)$ for $t \in \mathbb{C}$ and c) the process of semisimplification of a tensor category.
The noncommutative generalisation of the notion of stopping time was introduced by Hudson. Such quantum stopping times may be used to stop the CCR flow and its isometric cocycles, i.e., left operator Markovian cocycles on Boson Fock space. This stopping of the CCR flow yields a homomorphism from the semigroup of stopping times into the semigroup of unital endomorphisms of the algebra of bounded operators on the ambient Fock space, and this homomorphism is a generalised version of the shift semigroup. Moreover, stopped cocycles satisfy a nondeterministic version of the cocycle relation, which is a significant generalisation of a result of Applebaum. (Joint work with K. B. Sinha)
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesquianalytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are nonnegative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a nonnegative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the pushforward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the pushforward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued nonnegative kernel $K$ is quasiinvariant, then $K^{(1,1)}$ is also a quasiinvariant kernel.
In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” has been developed. One of the simplest approaches to this is via the theory of monoid schemes. The concept of a monoid scheme itself goes back to Kato and was further developed by Deitmar and by Connes, Consani and Marcolli. The idea is to replace prime spectra of commutative rings, which are the building blocks of ordinary schemes, by prime spectra of commutative pointed monoids. In our work, we focus mostly on abstracting out the topological characteristics of the prime spectrum of a commutative pointed monoid. This helps to obtain several classes of topological spaces which are homeomorphic to the the prime spectrum of a monoid. Such spaces are widely studied and are called spectral spaces. They were introduced by M. Hochster. We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations like integral, saturation, Frobenius and tight closures on monoids. We prove that the collection of all continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.
In this talk we shall take an eclectic tour through three centuries since the inception of this function by Euler in 1729. The function has been the object of intense study by every great mathematician of the 19th century and continues to tantalize the modern mathematicians.
We shall dwell upon some of the more important identities, their modern ramifications and some new proofs of old results. We shall also discuss some recent developments that have occured in the last decade.
Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigidmotion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less wellstudied. The BetkeKneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for latticeinvariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. In this talk, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.
The wandering subspace problem for an analytic (normincreasing) misometry T on a Hilbert space H asks whether every Tinvariant subspace of H can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 is ascribed to BeurlingLaxHalmos, while that for m = 2 is due to Richter. In this talk, we discuss present status of this problem including some partial solutions in case m > 2.
Professor H. Upmeier of the Marburg University would be visiting the Department of Mathematics, IISc as the InfoSys Visiting Professor during the period Dec 4  Feb 28. He will give a series of lectures on the broad theme of “Geometric Quantization in Complex and Harmonic Analysis”. The first set of lectures will take place according to the following schedule
All lectures will take place in LH1, Department of Mathematics.The first lecture will be at 3:00 pm. All subsequent lectures will be at 4:00 pm.
In this series, the speaker will discuss some basic material (connexions, curvature etc) and then cover, with full proofs, the BorelWeilBott theorem and the Kodaira embedding theorem.
A second set of lectures will be announced subsequently.
The main aim of coding theory is to construct codes that are easier to encode and decode, can correct or at least detect many errors, and contain a sufficiently large number of codewords. To study errordetecting and errorcorrecting properties of a code with respect to various communication channels, several metrics (e.g. Hamming metric, Lee metric, RosenbloomTsfasman (RT) metric, symbolpair metric, etc.) have been introduced and studied in coding theory.
In this talk, we will establish algebraic structures of all repeatedroot constacyclic codes of prime power lengths over finite commutative chain rings. Using their algebraic structures, we will determine Hamming distances, bsymbol distances, RT distances, and RT weight distributions of these codes. As an application of these results, we will identify MDS (maximumdistance separable) Hamming, MDS bsymbol and MDS RT codes within this particular class of constacyclic codes. We will also present an algorithm to decode these codes with respect to the Hamming, symbolpair and RT metrics.
A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a (possibly infinite) family of polynomial inequalities. In this talk, we will discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to such convexity constraints. These embeddings arise naturally in connection with higherdimensional analogues of the fact that the algebra of continuous functions on a circle can be generated by two smooth functions. We will expand on this connection, discuss the local and global challenges in constructing such embeddings, and discuss two distinct cases in which new bounds have been obtained. This is joint work with R. Shafikov.
An ordinary ring may be seen as a preadditive category with just one object. This leads to the powerful analogy, first formulated explicitly by Mitchell in 1975, that a small preadditive category should be seen as a “ring with several objects”. We will trace the history and development of the category of modules over a preadditive category.
We study operators of multiplication by $z^k$ in Dirichlettype spaces $D_\alpha$. We establish the existence of $k$ and $\alpha$ for which some $z^k$invariant subspaces of $D_\alpha$ do not satisfy the wandering property. As a consequence of the proof, any Dirichlettype space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by $z^k$ for any $k \geq 6$.
Symmetric polynomials are interesting not just in their own right. They appear naturally in various contexts, e.g., as characters in representation theory and as cohomology classes of Grassmannians and other homogeneous spaces in geometry. There are many different bases for the space of symmetric polynomials, perhaps the most interesting of which is the one formed by Schur polynomials. The LittlewoodRichardson (LR) rule expresses the product of two given Schur polynomials as a linear combination of Schur polynomials. The first half of the talk will be a tour of these topics that should be accessible even to undergraduate students.
The second half will be an exposition (which hopefully will continue to be widely accessible!) of recent joint research work with Mrigendra Singh Kushwaha and Sankaran Viswanath, both of IMSc. The LR rule above has a natural interpretation as giving the decomposition as a direct sum of irreducibles of the tensor product of two irreducible representations of the unitary (or general linear) group. A generalised version of it (due to Littelmann, still called the LR rule) gives the analogous decomposition for any reductive group. On the tensor product of two irreducible representations, there is the natural “KostantKumar” filtration indexed by the Weyl group. This consists of the cyclic submodules generated by the highest weight vector tensor an extremal weight vector. We obtain a refined LR rule that gives the decomposition as a direct sum of irreducibles of the KostantKumar submodules (of the tensor product). As an application, we obtain alternative proofs of refinements of “PRV type” results proved by Kumar and others. (PRV = Parthasarathy, Ranga Rao, Varadarajan)
The counts of algebraic curves in projective space (and other toric varieties) has been intensely studied for over a century. The subject saw a major advance in the 1990s, due to groundbreaking work of Kontsevich in the 1990s. Shortly after, considerations from high energy physics led to an entirely combinatorial approach to these curve counts, via piecewise linear embeddings of graphs, pioneered by Mikahlkin. I will give an introduction to the surrounding ideas, outlining new results and new proofs that the theory enables. Time permitting I will discuss generalizations, difficulties, and future directions for the subject.
Tropical geometry studies combinatorial structures that arise as “shadows” or “skeletons” of algebraic varieties. These skeletons were motivated in part by the mirror symmetry conjectures in mathematical string theory, but have now grown to function broadly as a tool for the study of algebraic varieties. Much of the work in the subject in the past decade has focused on the geometry of moduli spaces of abstract and parameterized algebraic curves, their tropical analogues, and the relationship between the two. This has led to new results on the topology of $M_{g,n}$, the geometry of spaces of elliptic curves, and to classical questions about the geometry of Hurwitz spaces. I will give an introduction to these ideas and recent advances.
Characters of classical groups appear in the enumeration of many interesting combinatorial problems. We show that, for a wide class of partitions, and for an even number of variables of which half are reciprocals of the other half, Schur functions (i.e., characters of the general linear group) factorize into a product of two characters of other classical groups. Time permitting, we will present similar results involving sums of two Schur functions. All the proofs will involve elementary applications of ideas from linear algebra.
This is joint work with Roger Behrend.
Formality is a topological property, defined in terms of Sullivan’s model for a space. In the simplyconnected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring.
In 1975, Deligne, Griffiths, Morgan and Sullivan proved that any compact Kaehler manifold is formal. We study the analogue for some contact manifolds. Such spaces are obtained as the total space of some circle and sphere bundles over symplectic manifolds. These include some Sasakian manifolds.
We will describe Gelfand’s criterion for the commutativity of associative algebras and discuss some of its applications towards the multiplicity one theorems for the representations of finite groups.
We’ll discuss two applications of forcing to analysis. (1) For every partition of a set of reals into countable sets, there is a transversal of the same Lebesgue outer measure. (2) The existence of a continuumsized family of entire functions which take fewer than continuum values at each complex number is undecidable in ZFC plus the negation of the continuum hypothesis. Both results are joint work with Saharon Shelah.
We will study recurrence patterns in decimal expansions of rational numbers (in any integer base for this talk). After making some initial observations, we will compute the length of the repeating part of any fraction. We conclude by explaining this result over a Euclidean domain.
Lag windows are commonly used in the time series, steady state simulation, and Markov chain Monte Carlo (MCMC) literature to estimate the long range variances of ergodic averages. We propose a new lugsail lag window specifically designed for improved finite sample performance. We use this lag window for batch means and spectral variance estimators in MCMC simulations to obtain strongly consistent estimators that are biased from above in finite samples and asymptotically unbiased. This quality is particularly useful when calculating effective sample size and using sequential stopping rules where they help avoid premature termination. Further, we calculate the bias and variance of lugsail estimators and demonstrate that there is little loss compared to other estimators. We also show mean square consistency of these estimators under weak conditions. Our results hold for processes that satisfy a strong invariance principle, providing a wide range of practical applications of the lag windows outside of MCMC. Finally, we study the finite sample properties of lugsail estimators in various examples.
In the first part of this talk we introduce a classical family of symmetric polynomials called Schur polynomials and discuss some of their properties, and a problem that arises from their study, namely the combinatorial interpretation of Littlewood–Richardson coefficients.
In the second part, we explain how the work of Robinson, Schensted, Knuth, Lascoux, and Schuetzenberger on words in an ordered alphabet led to a solution of this combinatorial problem. We will then mention some relatively recent developments in this subject.
The aim of this talk is to give a highlevel overview of the theory of expander graphs and introduce motivations and possible approaches to generalizing it to higher dimensions. I shall begin with three perspectives on expansion in graphs discrepancy, isoperimetry and mixing time, and show a qualitative equivalence of these notions in defining expansion for graphs. Next I shall briefly discuss upper and lower bounds on expansion, and sketch the LubotzkyPhillipsSarnak construction of Ramanujan graphs. Finally, I hope to motivate highdimensional expanders using two interesting topics the overlapping problem, and the threshold problem.
I will start by recalling some classical formulae that one usually encounters in a first course in Calculus. For example, Euler proved in the 1730’s that the sum of reciprocals of squares of positive integers is onesixth of the square of \pi. Such formulae are the prototypical examples of an entire of research in modern number theory called special values of Lfunctions. The idea of an Lfunction is crucial in the work of Andrew Wiles in his proof of Fermat’s Last Theorem. The aim of this lecture will be to give an appreciation for Lfunctions and to convey the grandeur of this subject that draws upon several different areas of mathematics such as representation theory, algebraic and differential geometry, and harmonic analysis. Towards the end of the talk, I will present some of my own recent results on the special values of certain automorphic Lfunctions.
We shall discuss the difficulty in solving and numerically integrating differentialalgebraic equation systems of the dx/dt = f(x,u), g(x) = 0 where x is in R^n and u is in R^m and m <= n. In this context we shall introduce a horizontal lift and its exponentiation toward construction of a solution. Especially, the solution and behavior of the algebraic variable is of interest. Cases where u can be rough (belong to fractional Hoelder space) are of interest. A numerical approximation that can produce useful results in computer simulations will be discussed.
Start with a system of particles with possibly different masses, and consider a process where the particles merge, as time passes, according to some random mechanism. At some point of time the identity of the most massive particle–the leader–becomes fixed. We study the fixation time of the identity of the leader in the general setting of Aldous’s multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of nearcritical coalescent processes, including the classical ErdosRenyi process. In particular, this generalizes a result of Luczak. Based on joint work with Louigi AddarioBerry and Shankar Bhamidi.
Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$ is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$ modulo the relations $[u, v]=0$ if there is no edge between the vertices $u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$ is naturally graded and the graded dimensions of the Lie algebra $\mathfrak{g}$ of $G$ have some deep connections with the vertex colorings of $G$. In this talk, I will explain how to get the generalized chromatic polynomials of $G$ in terms of graded dimensions of the Lie algebra of $G$. We will use this connection to give a Lie theoretic proof of of Stanley’s reciprocity theorem of chromatic polynomials.
I will give a gentle introduction to the Diamond Lemma. This is a useful technique to prove that certain “PBWtype” bases exist of algebras given by generators and relations. In particular, we will see the PBW theorem for usual Lie algebras.
This is joint work with N. Prabhu (IISER Pune). We derive new bounds for moments of the error in the SatoTate law over families of elliptic curves. As applications, we deduce new almostall results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method builds on recent work by N. Prabhu and K. Sinha who derived a Central Limit Theorem on the distribution of the errors in the SatoTate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch’s identities connect moments of coefficients of HasseWeil Lfunctions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak’s identity is combinatorial in nature.
The five Platonic solids are some of the oldest known objects of mathematical study. In this talk (in joint work with David Aulicino and Pat Hooper) we discuss how understanding the flat geometry of Platonic solids leads to very interesting 20th and 21st century mathematics. In particular, we use the geometry of Riemann surfaces to construct a closed geodesic on the dodecahedron that passes through exactly one vertex, solving a longstanding open question.
A grove is a spanning forest of a triangular portion of the triangular lattice with a prescribed boundary connectivity. A large random grove exhibits a limit shape i.e. there is a nonrandom algebraic curve outside which the grove is “frozen”. TK Petersen and D Speyer proved that for the uniform measure on groves, the curve is the inscribed circle. I will talk about extensions of their results to probability measures on groves that are periodic in appropriate coordinates. These measures give interesting algebraic curves with higher genus and cusp singularities as limit shapes, as well as new “gaseous” phases.
We consider the primary Brownian loop soup (BLS) layering vertex fields and show the existence of the fields in smooth bounded domains for a suitable range of parameters $\beta$’s. To show this at a fixed cutoff, we use Kahane’s theory of Gaussian multiplicative chaos. On the other hand, when the cutoff is removed, we use WeinerIto chaos expansion to establish that the $\lambda\beta^2$ limit as the intensity $\lambda$ of the BLS diverges and $\beta$ goes to 0 such that $\lambda\beta^2$ is constant, is a complex Gaussian multiplicative Chaos. Based on joint work with F. Camia, A. Gandolfi and G. Peccati.
We consider the collection of near maxima of the discrete logcorrelated Gaussian field in the interior of a box. We provide a rough description of the geometry of the set of near maxima. We show that two near maxima can other either simultaneously either at microscopic or at macroscopic level, but not at mesoscopic level.
Weyl’s law gives an asymptotic formula for the number of eigenfunctions of the LaplaceBeltrami operator on a Riemannian manifold. In 2005, Elon Lindenstrauss and Akshay Venkatesh gave a proof of this law for quite general quotients of semisimple Lie groups. The proof crucially uses the fact that solutions of the corresponding wave equation propagate with finite speed. I will try to explain what they did in the simplest setting of the upper half plane. I will also try to explain why such eigenfunctions, also known as automorphic forms, are of central importance in number theory. The first 45 minutes of the talk should be accessible to students who have a knowledge of some basic complex analysis and calculus.
Proofs of many famous problems in Number Theory (including Fermat’s Last Theorem) rely on understanding some properties about Elliptic Curves, which makes this topic inevitable and very interesting . One of the seven ‘Millennium Prize Problems’ stated by the Clay Mathematics Institute is the Birch SwinnertonDyer Conjecture, which is a statement regarding Elliptic Curves.
In this talk I will briefly describe the idea of a proof of the MordellWeil Theorem and introduce the nSelmer group and TateShafarevitch group associated to Elliptic Curves. I will define the Lfunction and some other arithmetic invariants attached to the Elliptic Curves and state the celebrated BirchSwinnertonDyer Conjecture.
Elliptic curves are important objects of study in various areas of research in modern mathematics. In this talk I will develop some algebraic and geometric tools to understand the group structures on elliptic curves and their Isogenies (certain kind of homomorphisms). I will specialise the general study of Elliptic Curves over finite fields, define zeta functions of associated Elliptic Curve and state the Weil Conjectures.
The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the AumannCarathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the AumannCaratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasihyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. It’s construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasihyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
Let G be a finitedimensional complex simple Lie algebra and G[t] be its current algebra. The degree grading on the polynomial ring gives a natural grading on G[t] and makes it a graded Lie algebra. Local Weyl modules introduced by Chari and Pressley are interesting finitedimensional graded G[t]modules. Corresponding to a dominant integral weight x of G there is a local Weyl module denoted by W(x). The zeroth graded piece of W(x) is the irreducible Gmodule V(x). In this talk, we discuss how to obtain a basis for W(x) from the basis of V(x) given by GelfandTsetlin patterns, when G is of type A and C.
In this talk we shall see three very different areas of applications of combinatorics in mathematics and computer science, illustrating four different flavours of combinatorial reasoning.
The first problem is on the decomposition, into irreducible representations, of the Weil representation of the full symplectic group associated to a finite module of odd order over a Dedekind domain. We shall discuss how a poset structure defined on the orbits of finite abelian pgroups under automorphisms can be used to show the decomposition of the Weil representation is multiplicityfree, as well as parametrize the irreducible subrepresentations, compute their dimensions in terms of p, etc. Joint works with Amritanshu Prasad (IMSc, Chennai).
Next, we consider lower bounds on the maximum size of an independent set, as well as the number of independent sets, in kuniform hypergraphs, together with an extension to the maximum size of a subgraph of bounded degeneracy in a hypergraph. Joint works with C. R. Subramanian (IMSc, Chennai), Dhruv Mubayi (UIC, Chicago) and Jeff Cooper (UIC, Chicago) and Arijit Ghosh.
Finally, we shall look at Haussler’s Packing Lemma from Computational Geometry and Machine Learning, for set systems of bounded VC dimension. We shall go through its generalisation to the Shallow Packing Lemma for systems of shallow cell complexity, and see how it can be used to prove the existence of small representations of set systems, such as epsilon nets, Mnets, etc. Joint works with Arijit Ghosh (IMSc, Chennai), Nabil Mustafa (ESIEE Paris), Bruno Jartoux (ESIEE Paris) and Esther Ezra (Georgia Inst. Tech., Atlanta).
Let G be a central product of two groups H and K. In this talk, I shall discuss about the second cohomology group of G, having coefficients in a divisible abelian group D with trivial Gaction, in terms of the second cohomology groups of certain quotients of H and K.
For an elliptic curve $E$ over $\mathbb{Q}$, the distribution of the number of points on $E$ mod $p$ has been wellstudied over the last few decades. A relatively recent study is that of extremal primes for a given curve $E$. These are the primes $p$ of good reduction for which the number of points on $E$ mod $p$ is either maximal or minimal. If $E$ is a curve with CM, an asymptotic for the number of extremal primes was determined by James and Pollack. The talk will discuss the nonCM case and focus on obtaining upper bounds. This is joint work with C. David, A. Gafni, A. Malik and C. TurnageButterbaugh.
The aim of this talk is to answer the Nielsen Realisation problem: Can every finite subgroup of the mapping class group can be realised as a subgroup of the isometry group of some hyperbolic surface? In other words, does every finite subgroup fix a point in the Teichmüller space of the surface?
The usual FenchelNielsen coordinates can be thought of as fixing a pants decomposition and keeping track of the length of the boundary of these together with the amount on ‘twist’ while glueing. Shear coordinates on the other hand, instead of using pair of pants, use ideal triangles as the basic pieces. As ideal triangles are unique up to isometry, only the gluing data needs to be tracked in this case. We shall see a convexity result concerning the length of simple closed curves with respect to these coordinates. This result leads to a positive answer for the Nielsen Realisation problem. I’ll be mainly following the paper by Bestvina, Bromberg, Fujiwara, and Souto (AMJ, 2013). Some technical results will be assumed. Familiarity with FenchelNielsen coordinates will be helpful.
Since its introduction, the class of entanglement breaking maps played a crucial role in the study of quantum information science and also in the theory of completely positive maps. In this talk, I will present a certain class of linear maps on matrix algebras that have the property that they become entanglement breaking after composing finite or infinite number of times with themselves. These maps are called eventually entanglement breaking maps. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. It turns out that the set of eventually entanglement breaking maps forms a rich class within the set of all unital completely positive maps. I will relate these maps with irreducible positive linear maps which have been studied a lot in the noncommutative PerronFrobenius theory. Various spectral properties of a ucp map on finite dimensional C*algebras will be discussed. The motivation of this work is the “PPTsquared conjecture” made by M. Christandl that says that every PPT channel, when composed with itself, becomes entanglement breaking. In this work, it is proved that every unital PPTchannel becomes entanglement breaking after finite number of iterations. This is a joint work with Sam Jaques and Vern Paulsen.
A hyperplane arrangement cuts up a vector space into several pieces. The combinatorics and topology of this subdivision is encoded in the associated abelian category of perverse sheaves. This category has an alternate algebraic description due to Kapranov and Schechtman, in terms of representations of a quiver with relations. I will first explain the background and setup. I will then focus on gluing, or “recollement”, which is a recipe to reconstruct the category of perverse sheaves on a space from an open subset and its complement. The aim of the talk is to describe how recollement on the above category of perverse sheaves translates to the category of quiver representations.
A Riemann surface appears in many different guises in mathematics, for example, as a branched cover of the Riemann sphere, an algebraic subset of a projective space, or a complex analytic 1manifold. What is the relationship between various representations of the same Riemann surface? In the first part of my talk, I will describe a conjectural answer to one aspect of this question, due to Mark Green. In the second part, I will talk about ribbons. Ribbons are a particular kind of nonreduced schemes—spaces that carry “infinitesimal functions.” I will explain how studying these seemingly strange objects helps us understand properties of regular Riemann surfaces relevant for Green’s conjecture.
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesquianalytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are nonnegative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a nonnegative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the pushforward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the pushforward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued nonnegative kernel $K$ is quasiinvariant, then $K^{(1,1)}$ is also a quasiinvariant kernel.
In this talk I will outline a proof of the classical GaussBonnet theorem. The proof uses ChernWeil theory (which is standard) but more interestingly, Morse theory. The proof has appeared in a recent paper of mine in the Journal of Pure and Applied Mathematics of INSA.
Algebraic identities play a pivotal role in the study of many mathematical structures although once understood, they are subconsciously regarded as being obvious or even tautological. For instance, polarization identity in convexity results in Hilbert space theory, Sylvester’s determinant identity in the study of determinantal processes, rank identities in the proof of Cochran’s theorem, etc. In this talk, the main goal is to discuss a systematic approach towards developing a theory of rank identities and determinant identities. This makes contact with Cohn’s work on free ideal rings, particularly free associative algebras over a field. By taking a universal approach, we will see how these methods translate to the world of finite von Neumann algebras (specifically II1 factors) where there is a natural notion of centervalued rank which measures the degree of nondegeneracy of an operator, and a notion of determinant known as the FugledeKadison determinant. We will also see some applications to the (nonselfadjoint) algebraic structure of finite von Neumann algebras and to certain operator inequalities.
In this talk we will describe connections between second order partial differential equations and Markov processes associated with them. This connection had been an active area of research for several decades. The talk is aimed at Analysts and does not assume familiarity with probability theory.
We consider a finite version of the onedimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0,n_1)system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)system and (n_0, 1)system, using an enriched twodimensional model.
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function f: O → E, where O is an open subset of the complex plane and E a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For Ω_1 a subset of the Riemann sphere, we consider the space of locally holomorphic maps of Ω_1 into E vanishing at infinity if infinity belongs to Ω_1, denoted by P(Ω_1,E). For two complementary subsets Ω_1 and Ω_2 of the Riemann sphere we prove that given two locally convex topological vector spaces E and F in separating duality, under some general conditions, we can define a separating duality between P(Ω _1,E) and P(Ω_2,F).
We will survey recent progress of birational geometry in positive characteristic fields. As well, we will introduce subadditiviy of Kodaira dimension and canonical bundle formula in positive characteristics. These are joint work with Caucher Birkar, Lei Zhang and Yi Gu.
We will consider the following question:
Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a finite field F, what is the maximum number of common solutions they can have in the corresponding protective space over F?
The case of a single homogeneous polynomial (i.e., hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. We will outline these developments and report on some recent progress.
An attempt will be made to keep the prerequisites at a minimum. If there is time and interest, connections to coding theory or to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension will also be outlined.
Kazhdan’s theory loosely states that the complex representation theory of the group G($F$), where G is a split connected reductive group over $\mathbb{Z}$ and $F$ is a nonarchimedean local field of characterstic $p$, can be viewed as a “limit” of the complex representation theories of the groups G($F’$), where $F’$ varies over nonarchimedean local fields of characteristic 0 with residue characteristic $p$. A similar theory for representations of the Galois group Gal($F_s/F$) is due to Deligne. In this talk we will review this theory, discuss some applications of this theory to the local Langlands correspondence, and some ingredients in generalizing the work of Kazhdan and some variants of it to nonsplit groups.
We give examples of two inequivalent smooth structures on the complex projective 9space such that one admits a metric of nonnegative scalar curvature and the other does not. Following this example and the work of Thomas Farrell and Lowell Jones, we also construct examples of closed negatively curved Riemannian 18manifolds, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds. (Joint work with Samik Basu.)
In this talk, we present our new results on the numerical analysis of nonlocal fracture models. We begin by giving a brief introduction to the Peridynamic theory and the nonlocal potentials considered in our work. We consider a force interaction characterized by a double well potential. Here, one well, near zero strain, corresponds to the linear response of a material, and the other well, for large strain, corresponds to the softening of a material. We show the existence of a regularized model with evolving displacement field in either Hölder space or Sobolev space. Assuming exact solutions in Hölder space, we obtain apriori error estimates due to finite difference approximation. We show that the error converges to zero, uniformly in time, in the mean square norm. The rate depends on the nonlocal length scale and is proportional to 𝐶(Δ𝑡+ℎ𝛾/𝜖2). Here $ℎ$ is the size of mesh, $\epsilon$ is the nonlocal length scale, $\Delta t$ is the size of time step, and $\gamma \in (0,1]$ is the Hölder exponent. $C$ is the constant independent of mesh size and size of time step and may depend on nonlocal length scale through the norm of the exact solution. We also study the finite element approximation and show that the error uniformly converges to zero at the rate C(Δt+h2/ϵ2). We consider piecewise linear continuous elements. Theoretical claims are supported by numerical results. This is a joint work with Dr. Robert Lipton and is funded by the US Army Research Office under grant/award number W911NF1610456.
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is called \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, given any metric graph, its metric can be rescaled so that it can be essentially and isometrically embedded on a closed hyperbolic surface. The essential genus $g_e(G)$ of a metric graph $(G, d)$ is the lowest genus of a surface on which such an embedding of the graph is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ can be essentially and isometrically embedded (possibly after a rescaling the metric $d$) on a surface of genus $g$.
Next, we study minimal embeddings, where each complementary region has Euler characteristic $1$. The maximum essential genus $g_e^{\max}(G)$ a graph $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, nonsparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such nonsparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in manyvariable systems. Using simulated data from nonlinear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply nonparametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed pointprocess and real timeseries data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, nonsparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such nonsparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in manyvariable systems. Using simulated data from nonlinear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply nonparametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed pointprocess and real timeseries data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
Classical geometric notion of the YangMills functional has been generalized to the noncommutative context by A. Connes. In this talk we will see a suitable formulation of “subadditivity” and “additivity” of this action functional under a natural hypothesis on spectral triples, and show that in general the YangMills functional is always subadditive. An instance of additivity will be discussed for the case of noncommutative torus.
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain. Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priori error estimates to show the optimal convergence of the method with respect to the regularity, for this we have enriched the finite element space with elementwise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem. Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results here are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a postprocessed solution ˜uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and redrefinement of a tetrahedron. We present conclusions and possible extensions for the future works.
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
A geodesic conjugacy between two closed manifolds is a homeomorphism between their unit tangent bundles that takes geodesic flow orbits of one to that of the other in a timepreserving manner. One of the central problems in Riemannian geometry is to understand the extent to which a geodesic conjugacy determines a closed Riemannian manifold itself. While an answer to the question in this generality has yet remained elusive, we give an overeview of results on closed surfaces – the most important illustrative case where a complete picture about questions of geodesic conjugacy rigidity is available.
We begin with a introduction to the notion of a resolution of a module over a Noetherian ring, leading to Betti numbers over local or graded rings, and some problems related to them. Most of the talk will focus on the graded case. One of the recent developments in this area is the resolution of the BoijSoderberg conjectures by EisenbudSchreyer (2009). We discuss the motivation behind the conjectures, with a quick word on the techniques used in their resolution. If time permits, we will see other scenarios where parts of the BoijSoderberg conjectures hold, and discuss obstacles in extending the EisenbudSchreyer techniques in general. This last part is joint work with Rajiv Kumar.
Let G be a group and H a subgroup of G. Let $\pi_1$ and $\pi_2$ be irreducible representations of $G$ and $H$ respectively. By “branching laws” one refers to the rules of describing the vector space $Hom_{H} (\pi_1, \pi_2)$. The well known Langlands’ conjectures predict connections between the representation theory of reductive groups (over local and global fields) and the study of Galois representations. In the nineties, B. Gross and D. Prasad started a systemic investigations into the study of branching laws for the groups of interest to Langlands program, and their predictions are known as GrossPrasad conjectures. We discuss two basic examples of these predictions. A covering group of a reductive groups is a certain central extensions. We discuss branching laws involving covering groups (namely, a two fold central extension of padic $GL_2$) which may be seen as a variation of GrossPrasad conjectures for covering groups.
In a topological space, a point x is said to be a specialization of another point y if x is in the closure of y. Specialization closed subsets occur naturally when considering the notion of support in the Zariski topology in algebra/algebraic geometry. We will define them and show their classical use in classifying certain subcategories. This will allow us to give a characterization of CohenMacaulay local rings. Time permitting, we will also discuss some reductions of Ktheoretic invariants (of derived categories with support).
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
In this talk I will give a brief introduction to Liouville firstpassage percolation (LFPP) which is a model for random metric on a finite planar grid graph. It was studied primarily as a way to understand the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. In short the Liouville quantum gravity is a (conjectured) one parameter family of ``canonical’’ random metrics on a Riemann surface. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will highlight the apparent disagreement of these estimates with a prediction made in the physics literature about the LQG metric. I will also mention some (of many) future problems in this program. Based on a joint work with Jian Ding.
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function f: O → E, where O is an open subset of the complex plane and E a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For Ω_1 a subset of the Riemann sphere, we consider the space of locally holomorphic maps of Ω_1 into E vanishing at infinity if infinity belongs to Ω_1, denoted by P(Ω_1,E). For two complementary subsets Ω_1 and Ω_2 of the Riemann sphere we prove that given two locally convex topological vector spaces E and F in separating duality, under some general conditions, we can define a separating duality between P(Ω_1,E) and P(Ω_2,F).
We consider a finite version of the onedimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0, n_1)system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)system and (n_0, 1)system, using an enriched twodimensional model.
We will begin by providing a brief introduction to selfnormalizing concentration inequalities for scalar and finitedimensional martingales, which are very useful in measuring the size of a stochastic process in terms of another, growing, process (the 2009 book of de la Pena et al is a good reference). We will then present a selfnormalizing concentration inequality for martingales that live in the (potentially infinitedimensional) Reproducing Kernel Hilbert Space (RKHS) of a p.s.d. kernel. We will conclude by illustrating applications to online kernel leastsquares regression and multiarmed bandits with infinite action spaces, a.k.a. sequential noisy function optimization [Joint work with Sayak Ray Chowdhury (IISc)].
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
The main aim of this talk is to construct a canonical Fisocrystal $H(A)_K$ for an abelian scheme A over a padic complete discrete valuation ring of perfect residue field K. This Fisocrystal $H(A)_K$ comes with a filtration and admits a natural map to the usual Hodge sequence of A. Even though $H(A)_K$ admits a map to the crystalline cohomology of A, the Fstructure on $H(A)_K$ is fundamentally distinct from the one on the crystalline cohomology. When A is an elliptic curve, we further show that $H(A)$ itself is an Fcrystal and that implies a strengthened version of Buium’s result on differential characters. The weak admissibility of $H(A)$ depends on a modular parameter over the points of the moduli of elliptic curves. Hence the Fontaine functor associates a new padic Galois representation to every such weakly admissible Fcrystal $H(A)$. This is joint work with Jim Borger.
Let $\Omega\subset\mathbb{R}^d$ be a unbounded domain. A positive harmonic function u in $\Omega$ that vanishes on the boundary $\partial\Omega$ is called a Martin function on $\Omega$. In this talk, we will discuss various analytic and geometric aspects of Martin functions, namely how fast they grow at infinity, maximum on a slice, and convexity properties of their level lines. If time permits, we will also present a inverse balayage problem from Potential theory.
In recent years, the theory of complex valued analytic functions defined on multiply connected domains has been recognized to have several applications in applied mathematics. In this talk, we will review the theory of SchottkyKlein prime functions and other allied special functions defined on multiply connected circular domains, and discuss the numerical computation of these special functions. We will also briefly present applications to selected problems in fluid dynamics through conformal mapping methods.
In 1980, Gross conjectured a formula for the expected leading term at s=0 of the padic Lfunction associated to characters of totally real number fields. The conjecture states a precise relation between this leading term and padic regulator of punits in an abelian extension. In the talk I will present a precise formulation of the conjecture and describe its relevance for Hilbert’s 12th problem. I will then sketch proof of this conjecture of Gross. This is a joint work with Samit Dasgupta and Kevin Ventullo.
I will present work done with students and colleagues on the collective behaviour of motile organisms, viewed as interacting particles with an autonomous velocity and noise. The talk will include a bit of stochastic processes, some statistical mechanics, and some hydrodynamics. I will discuss experiments, analytical theory and some computation.
Edelman and Greene constructed a bijective correspondence between the reduced words of the reverse permutation (n, n  1, …, 2, 1) and standard Young tableaux of the staircase shape (n  1, …, 1). Recently, motivated by random sorting networks, we studied this bijection and discovered some new properties in joint work with Svante Linusson. In this talk, I will discuss them and, if time permits, also a related project with Linusson and Robin Sulzgruber on random sorting networks where the intermediate permutations avoid the pattern 132.
The classification of homogeneous scalar weighted shifts is known. Recently, Koranyi obtained a large class of inequivalent irreducible homogeneous bilateral 2by2 block shifts. We construct two distinct classes of examples not in the list of Koranyi. It is then shown that these new examples of irreducible homogeneous bilateral 2by2 block shifts, together with the ones found earlier by Koranyi, account for every unitarily inequivalent irreducible homogeneous bilateral 2by2 block shift.
In this talk we will discuss an analytic model theory for pure hypercontractions (introduced by J. Agler) which is analogous to Sz.NagyFoias model theory for contractions. We then proceed to study analytic model theory for doubly commuting ntuples of operators and analyze the structure of joint shift coinvariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces H_k on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable H_kinner functions. We also prove that H_kinner functions are contractive multipliers and deduce a result on the multiplier norm of quasihomogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
Given the recent successfully concluded Polymath project – and the next Polymath that has already started over the weekend – I will present the origins of Polymath and discuss its workings, using slides of myself and of a Polymath collaborator.
Mirror symmetry is a phenomenon predicted by string theory. It allows one to translate questions in symplectic geometry to questions in complex geometry, and vice versa. The homological mirror symmetry program interprets mirror symmetry within the unifying categorical framework of derived noncommutative geometry. After introducing these ideas, I will describe an approach to a theory of Kähler metrics in derived noncommutative geometry. We will see how this leads to (i) a nonArchimedean categorical analogue of the DonaldsonUhlenbeckYau theorem, inspired by symplectic geometry, and (ii) the discovery of a refinement of the HarderNarasimhan filtration which controls the asymptotic behavior of certain geometric flows. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.
Let X1 , X2 , X3, … Xn be iid random variables. Laws of large numbers roughly state that the average of these variables converges to the expectation value of each of them when n is large. Various forms of these laws have many applications. The strong and weak laws along with the following three applications will be discussed: a) Cointossing. b) The Weierstrass approximation theorem. c) The Glivenko–Cantelli theorem.
In the second half of this talk, a law of large numbers is proven for spaces with infinite “volume” (measure) as opposed to the above version for probability measures (“volume” =1).
In joint work with Steve Lalley and Jenya Sapir, we study the tessellation of a compact, hyperbolic surface induced by a typical long geodesic segment. We show, that when properly scaled, the local behavior of a typical geodesic is that of a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.
Determinantal point processes, which first appeared in Dyson’s work on random matrices, arise in diverse problems of asymptotic combinatorics, ergodic theory, representation theory. They have strong chaotic properties: for example, the sineprocess has the Kolmogorov property and satisfies the Central Limit Theorem. A Functional Limit Theorem for the sineprocess has been established in joint work with A. Dymov.
A delicate aspect of the behaviour of a determinantal point process is that particles interact at infinite radius. For instance, Ghosh and Peres showed that, under the sineprocess, the number of particles in a bounded interval is determined by the configuration in the outside of the interval. For determinantal point processes with socalled integrable kernels, an explicit description is given of conditional measures of the process in a bounded interval with respect to the fixed exterior. These conditional measures are given by orthogonal polynomial ensembles with explicitly found weights. A key step in the argument is that projections inducing our processes satisfy a weaker analogue of the division axiom of de Branges: in fact, this weak division property, as shown in joint work with Roman Romanov, characterizes integrable kernels. Similar results for determinantal point processes governed by orthogonal projections onto Hilbert spaces of holomorphic functions are obtained in joint work with Y. Qiu. The talk is based on the preprints
https://arxiv.org/abs/1707.03463, (joint with R. Romanov), https://arxiv.org/abs/1701.00111 (joint with A. Dymov), and the paper
Alexander I. Bufetov, Yanqi Qiu, “Conditional measures of generalized Ginibre point processes”, J. Funct. Anal., 272:11 (2017), 4671–4708.
Terence Tao posted on his blog a question of Apoorva Khare, asking whether the free group on two generators has a length function $l: F_2 \to\mathbb{R}$ (i.e., satisfying the triangle inequality) which is homogeneous, i.e., such that $l(g^n) = nl(g)$. A week later, the problem was solved by an active collaboration of several mathematicians (with a little help from a computer) through Tao’s blog. In fact a more general result was obtained, namely that any homogeneous length function on a group $G$ factors through its abelianization $G/[G, G]$.
I will discuss the proof of this result and also the process of discovery (in which I had a minor role).
Gorenstein rings are very common and significant in many areas of mathematics. The following are two important and widely open problems in commutative algebra and algebraic geometry:
Recently, in a joint work with M.E. Rossi, we obtained partial results to these problems in some cases (Kalgebras of socle degree 4). In this talk, we will discuss these new developments.
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain.
Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priorierror estimates to show the optimal convergence of the method with respect to the regularity, forthis we have enriched the finite element space with elementwise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem.Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximationof the elliptic obstacle problem. The results here are two fold. Firstly, we address the influenceof the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a postprocessed solution uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and redrefinement of a tetrahedron. We present conclusions and possible extensions for the future works.
This talk will focus on the rooted GaltonWatson (GW) tree. The offspring distribution we consider is Poisson(\lambda), but our results extend to more general distributions. First order properties on rooted trees capture the local, finite structures inside the tree. We analyze the probabilities of first order properties under the GW measure, and obtain these probabilities as fixed points of contracting distributional maps. Moreover, we come up with nice functions that express these probabilities conditioned on survival of the GW tree. This is joint work with Joel Spencer.
A version of the uniformization theorem states that any compact Riemann surface admits a metric of constant curvature. A deep and important problem in complex geometry is to characterize Kahler manifolds admitting constant scalar curvature Kahler (cscK) metrics or extremal Kahler metrics. Even in the special case of KahlerEinstein metrics, starting with the work of Yau and Aubin in the 1970’s, a complete solution was obtained only very recently by ChenDonaldsonSun (and Tian). Their main results says that a Fano manifold admits a KahlerEinstein metric if and only if it is Kstable. I will survey some of these recent developments, and then focus on a refinement obtained in collaboration with Gabor Szekelyhidi. This has led to the discovery of new KahlerEinstein manifolds. If time permits, I will also talk about some open problems on constructing cscK and extremal metrics on blowups of extremal manifolds, and mention some recent progress.
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractaltype constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a timedependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a nonlinear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a nonlinear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s wellknown theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operatortheoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractlydefined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$to$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hardtocheck) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
Derived category is an important tool in homological algebra invented by Grothendieck and Verdier. In these days derived categories play important roles in many areas of algebraic geometry. In this talk, I will discuss derived categories and their applications to study Ulrich bundles on some Fano manifolds.
In this talk, I will tell you about the BoreldeSibenthal theorem which gives the classification of all maximal closed subroot systems of finite crystallographic root systems. I will start my talk by introducing the notion of finite root systems and it’s closed subroot systems.
The concept of root system is very fundamental in the theory of Lie groups and Lie algebras. Especially they play a vital role in the classification of finite dimensional semisimple Lie algebras. Closed subroot systems of finite root systems naturally appear in the BoreldeSibenthal theory which describes the closed connected subgroups of a compact Lie group that have maximal rank. The classification of closed subroot systems is essential in the classification of semisimple subalgebras of semisimple Lie algebras.
Through out this talk, we will try to stay within the theory of root systems and reflection groups. No knowledge of Lie algebras or Lie groups will be assumed. If time permits I will discuss about my joint work with R. Venkatesh which gives explicit descriptions of the maximal closed subroot systems of affine root systems.
We will present results obtained in collaboration with J. Burgos and M. Sombra. These extend the well known dictionary between the geometric properties of toric varieties and convex geometry. In particular, we give combinatorial descriptions of classical invariants of arithmetic geometry, such as metric, height or essential minimum.
In this talk, we prove the unique factorization property of Schur functions. This fundamental property of Schur functions was first observed and proved by C. S. Rajan in 2004. I give a different proof of this beautiful fact which I jointly obtained with my adviser S. Viswanath. I begin my talk with introducing the Schur functions and their connections with representation theory of general linear groups. Basic knowledge of elementary algebra will be assumed like group theory and linear algebra. If time permits, I will tell you about the possible generalizations of this result.
It is a wellknown result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.
We consider the computation of nvariate polynomials over a field F via a sequence of arithmetic operations such as additions, subtractions, multiplications, divisions, etc. It has been known for at five decades now that a random nvariate polynomial of degree n is hard to compute. Yet not a single explicit polynomial is provably known to be hard to compute (although we have a lot of good candidates). In this talk we will first describe this problem and its relationship to the P vs NP problem. We will then describe several partial results on this problem, both old and new, along with a more general approach/framework that ties together most of these partial results.
A partition of integer $n$ is a sequence $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_k, \cdots)$ of non negative integers such that $\lambda_i \ge \lambda_{i+1}$ and $\sum_i \lambda_i = n$. It follows that there are finitely many nonzero $\lambda_i$’s. One can restrict the number of them and the largest value of $\lambda_i$ and observe that the set of such partitions form a poset under a suitable relation. Several natural questions arise in this setting. Some of these questions have been answered by Proctor, Stanley and Kathy O’Hara among others. We take a look at some techniques as given by Stanley and ask if it is possible to extend it to higher dimensions.
I will describe our work that establishes (akin to) von Neumann’s conjecture on condition number, the ratio of the largest and the smallest singular values, for sparse random matrices. Nonasymptotic bounds on the extreme singular values of large matrices have numerous uses in the geometric functional analysis, compressed sensing, and numerical linear algebra. The condition number often serves as a measure of stability for matrix algorithms. Based on simulations von Neumann and his collaborators conjectured that the condition number of a random square matrix of dimension $n$ is $O(n)$. During the last decade, this conjecture was proved for dense random matrices.
Sparse matrices are abundant in statistics, neural networks, financial modeling, electrical engineering, and wireless communications. Results for sparse random matrices have been unknown and requires completely new ideas due to the presence of a large number of zeros. We consider a sparse random matrix with entries of the form $\xi_{i,j} \delta_{i,j}, \, i,j=1,\ldots,n$, such that $\xi_{i,j}$ are i.i.d. with zero mean and unit variance and $\delta_{i,j}$ are i.i.d. Ber$(p_n)$, where $p_n \downarrow 0$ as $n \to \infty$. For $p_n < \frac{\log n}{n}$, this matrix becomes noninvertible, and hence its condition number equals infinity, with probability tending to one. In this talk, I will describe our work showing that the condition number of such sparse matrices (under certain assumptions on the moments of $\{\xi_{i,j}\}$) is $O(n^{1+o(1)})$ for all $p_n > \frac{\log n}{n}$, with probability tending to one, thereby establishing the optimal analogous version of the von Neumann’s conjecture on condition number for sparse random matrices.
This talk is based on a sequence of joint works with Mark Rudelson.
We shall discuss a theorem of Bernstein published in 1975 about the number of common solutions of n complex polynomials in n variables in terms of the mixed volumes of their Newton polytopes. This is a far reaching generalisation of the Fundamental Theorem of Algebra and Bezout’s Theorem about intersections of plane algebraic curves. If time permits, we shall sketch a proof of Bernstein’s theorem using Hilbert functions of monomial ideals in polynomial rings.
I will give a gentle historical (and ongoing) account of matrix positivity and of operations that preserve it. This is a classical question studied for much of the past century, including by Schur, PolyaSzego, Schoenberg, Kahane, Loewner, and Rudin. It continues to be pursued actively, for both theoretical reasons as well as applications to highdimensional covariance estimation. I will end with some recent joint work with Terence Tao (UCLA).
The entire talk should be accessible given a basic understanding of linear algebra/matrices and onevariable calculus. That said, I will occasionally insert technical details for the more advanced audience. For example: this journey connects many seemingly distant mathematical topics, from Schur (products and complements), to spheres and Gram matrices, to Toeplitz and Hankel matrices, to rank one updates and Rayleigh quotients, to CauchyBinet and JacobiTrudi identities, back full circle to Schur (polynomials).
In this talk we discuss a formulation of Quantum Theory of Dark matter and discuss some operators on Hilbert spaces of singular measures.
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s wellknown theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operatortheoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractlydefined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$to$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hardtocheck) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
In this talk we will discuss an analytic model theory for pure hypercontractions (introduced by J. Agler) which is analogous to Sz.NagyFoias model theory for contractions. We then proceed to study analytic model theory for doubly commuting ntuples of operators and analyze the structure of joint shift coinvariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces H_k on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable H_kinner functions. We also prove that H_kinner functions are contractive multipliers and deduce a result on the multiplier norm of quasihomogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form
where, $\sigma_{L}$
and $\sigma_{R}$
are projective representation of the
Mobius group Mob with a common multiplier. We give another proof
of the ``product formula’’.
Also, we prove that the projective representations $\sigma_L$
and
$\sigma_R$
for a class of multiplication operators, the two
representations $\sigma_{R}$
and $\sigma_{L}$
are unitarily equivalent to
certain known pair of representations $\sigma_{\lambda + 1}$ and
$\sigma_{\lambda  1},$` respectively. These are described explicitly.
Let $G$ be either (i) the direct product of $n$copies of the
biholomorphic automorphism group of the disc or (ii) the biholomorphic
automorphism group of the polydisc $\mathbb D^n.$
A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots
,T_n)$
is said to be $G$homogeneous if the joint spectrum of $\mathsf{T}$
lies in $\widebar{\mathbb{D}}^n$
and $\varphi(\mathsf{T}),$
defined using
the usual functional calculus, is unitarily equivalent with $\mathsf{T}$
for all $\varphi \in G.$
We show that a commuting tuple $\mathsf{T}$
in the CowenDouglas class of
rank $1$ is $G$  homogeneous if and only if it is unitarily equivalent
to the tuple of the multiplication operators on either the reproducing
kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n}
\frac{1}{(1  z_{i}\overline{w}_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n}
\frac{1}{(1  z_{i}\overline{w}_{i})^{\lambda}},$
where $\lambda,$
$\lambda_i$
, $1 \leq i \leq n,$
are positive real numbers, according as
$G$ is as in (i) or (ii).
Let $\mathsf T:=(T_1, \ldots ,T_{n1})$
be a $G$homogeneous $(n1)$tuple
of rank $1$ CowenDouglas class, where $G$ is the the direct product of
$n1$copies of the biholomorphic automorphism group of the disc. Let
$\hat{T}$
be an irreducible homogeneous (with respect to the
biholomorphic group of automorphisms of the disc) operator in the
CowenDouglas class on the disc of rank $2$. We show that every
irreducible $G$  homogeneous operator, $G$ as in (i), of rank $2$ must be
of the form
We also show that if $G$ is chosen to be the group as in (ii), then there are no irreducible $G$ homogeneous operators of rank $2.$
In this talk the interplay between the combinatorial structures of finite simple graphs and various homological invariants like regularity, depth etc. of related algebraic objects shall be discussed. Some open problems, recent developments and ongoing projects shall be discussed. In particular some new techniques developed in my thesis to study CastelnuovoMumford regularity of algebraic objects related to graphs shall be discussed in some details.
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractaltype constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a timedependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a nonlinear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a nonlinear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random discrete structures: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using nonrigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on $n$ vertices and degree exponent $\tau>3$, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{\frac{\tau\wedge 4 3}{\tau\wedge 4 1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including (i) components under critical percolation, (ii) the vacant set left by a random walk, and (iii) the minimal spanning tree, constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the GromovHausdorff sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed at a complete resolution of these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90’s, the proof of which relies on a variation of Stein’s method and a quantification of a classical argument in percolation theory.
Based on joint work with Louigi AddarioBerry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.
The notion of a weakly proregular sequence in a commutative ring was first formally introduced by AlonsoJeremiasLipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel and PortaShaulYekutieli: a precise definition of this notion will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. Every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the MatlisGreenleesMay (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically Itorsion (i.e. complexes with Itorsion cohomology) and cohomologically Icomplete complexes in the derived category of A.
In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.
Dualizing complexes were first introduced in commutative algebra and algebraic geometry by Grothendieck and play a fundamental role in SerreGrothendieck duality theory for schemes. The notion of a dualizing complex was extended to noncommutative ring theory by Yekutieli. There are existence theorems for dualizing complexes in the noncommutative context, due to Van den Bergh, Wu, Zhang, and Yekutieli amongst others.
Most considerations of dualizing complexes over noncommutative rings are for algebras defined over fields. There are technical difficulties involved in extending this theory to algebras defined over more general commutative base rings. In this talk, we will describe these challenges and how to get around them. Time permitting, we will end by presenting an existence theorem for dualizing complexes in this more general setting.
The material described in this talk is work in progress, carried out jointly with Amnon Yekutieli.
The field of geodynamics deals with the large scale forces shaping the Earth. Computational geodynamics, which uses numerical modeling, is one of the most important tools to understand the mechanisms within the deep Earth. With the help of these numerical models we can address some of the outstanding questions regarding the processes operating within the Earth’s interior and their control on shaping the surface of the planet. Much of Earth’s surface observations such as gravity anomalies, plate motions, dynamic topography, lithosphere stress field, owe their origin to convection within the Earth’s mantle. While we understand the basic nature of such flow in the mantle, a lot remains unexplained, including the complex rheology of the deep mantle and how this density driven convective flow couples with the shallow surface. In this talk I will discuss how my group is using numerical modeling to understand the influence of the deep mantle on surface observations.
For a finite abelian group $G$ with $G = n$, the Davenport Constant $DA(G)$ is defined to be the least integer $k$ such that any sequence $S$ with length $k$ of elements in $G$ has a nonempty $A$ weighted zerosum subsequence. For certain sets $A$, we already know the precise value of constant corresponding to the cyclic group $\mathbb{Z} / n \mathbb{Z}$. But for different group $G$ and $A$, the precise value of it is still an open question. We try to find out bounds for these combinatorial invariant for random set $A$. We got few results in this connection. In this talk I would like to present those results and discuss about an extremal problem related to this combinatorial invariant.
The study of weighted inequalities in Classical Harmonic Analysis started in 70’s, when B. Muckenhoupt characterised in 1972 the weights $w$ for which the Hardy–Littlewood maximal function is bounded in $L^p(w)$. At that time the question about how the operator depended on the constant associated with $w$, which we denote by $[w]_{A_p}$, was not considered (i.e., quantitative estimates) were not investigated.
From the beginning of 2000’s, a great activity has been carried out in order to obtain the sharp dependence for singular integral operators, reaching the solution of the socalled $A_2$ conjecture by T. P. H\“ytonen.
In this talk we consider operators with homogeneous singular kernels, on which we assume smoothness conditions that are weaker than the standard ones (this is why they are called rough). The first qualitative weighted estimates are due to J. Duoandikoetxea and J. L. Rubio de Francia. For the norm of these operators in the space $L^2(w)$ we obtain a quantitative estimate which is quadratic in the constant $[w]_{A_2}$.
The results are based on a classical decomposition of the rough operators as a sum of other operators with a smoother kernel, for which a quantitative reelaboration of a dyadic decomposition proposed by M. T. Lacey is applied.
We will overview as well the most recent advances, mainly associated with quantitative estimates for these rough singular integrals. In particular, CoifmanFefferman type inequalities (which are new even in their qualitative version), weighted $A_p$$A_{\infty}$ inequalities and a quantitative version of weak $(1,1)$ estimates will be shown.
Conformal blocks are refined invariants of tensor product of representations of a Lie algebra that give a special class of vector bundles on the moduli space of curves. In this talk, I will introduce conformal blocks and explore connections to questions in algebraic geometry and representation theory. I will also focus on some ``strange” dualities in representation theory and how they give equalities of divisor classes on the moduli space of curves.
We shall consider sampling procedures to construct subgraphs to infer properties of a network. Specifically, we shall consider sampling procedures in the context of dense and sparse graph limits. We will explore open questions and interesting explorations at the undergraduate level. The talk will be accessible to a general audience.
Given a metric space (X, d), there are several notions of it being negatively curved. In this talk, we single out certain consequences of negative curvature – which are themselves weaker than (X, d) being negatively curved – that turn out to be very useful in proving results about holomorphic maps. We shall illustrate what this means by giving a proof of the WolffDenjoy theorem. (This theorem says that given a holomorphic selfmap f of the open unit disc, either f has a fixed point in the open unit disc or there exists a point p on the unit circle such that ALL orbits under the successive iterates of f approach p.) For most of this talk, we shall focus on metric spaces or on geometry in one complex variable. Towards the end, we shall briefly point out what can be proved in domains in higher dimensions that have the aforementioned negativecurvaturetype properties. This part of the talk is joint work with Andrew Zimmer.
The RiemannRoch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the RiemannRoch theorem for graphs. In fact, this theorem is not a mere analogue but has concrete relations with its algebrogeometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on my dissertation and joint work with i. Bernd Sturmfels, ii. FrankOlaf Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.
The Hadamard product of two matrices is formed by multiplying corresponding entries, and the Schur product theorem states that this operation preserves positive semidefiniteness.
It follows immediately that every analytic function with nonnegative Maclaurin coefficients, when applied entrywise, preserves positive semidefiniteness for matrices of any order. The converse is due to Schoenberg: a function which preserves positive semidefiniteness for matrices of arbitrary order is necessarily analytic and has nonnegative Maclaurin coefficients.
For matrices of fixed order, the situation is more interesting. This talk will present recent work which shows the existence of polynomials with negative leading term which preserve positive semidefiniteness, and characterises precisely how large this term may be. (Joint work with D. Guillot, A. Khare and M. Putinar.)
Commutators of singular integral operators with BMO functions were introduced in the seventies by CoifmanRochberg and Weiss. These operators are very interesting for many reasons, one of them being the fact that they are more singular than CalderonZygmund operators. In this lecture we plan to give several reasons showing the “bad” behavior of these operators.
I will introduce the concepts of backward, forward, and covariant Lyapunov vectors (that are basically “eigenfunctions” of certain operators) for a dynamical system and discuss their relevance in studying various stability results, both for the system itself and for nonlinear filtering problem, which will also be introduced along the way. The “relevance” to nonlinear filters is in the form of open questions whose answers (as a set of conjectures) may be gleaned from numerical results and linear filtering theory.
Given n random points on a manifold embedded in a Euclidean space, we wish to understand what topological features of the manifold can be inferred from the geometry of these points. One procedure is to consider union of Euclidean balls of radius r around the points and study the topology of this random set (i.e., the union of balls). A more robust method (known as persistent homology) of inferring the topology of the underlying manifold is to study the evolution of topology of the random set as r varies. What topological information (for example, Betti numbers and some generalizations) about the underlying manifold can we glean for different choices of r ? This question along with some partial answers in the recent years will be the focus of the talk. I shall try to keep the talk mostly selfcontained assuming only knowledge of basic probability and pointset topology.
The goal of this talk is to present an algorithm which takes a compact square complex belonging to a special class as input and decides whether its fundamental group splits as a free product. The special class is built by attaching tubes to finite graphs in such a way that they satisfy a nonpositive curvature condition. This construction gives rise to a rich class of complexes, including, but not limited to, closed surfaces of positive genus. The algorithm can be used to deduce the celebrated Stallings theorem for this special class, as also the well known Grushko decomposition theorem.
A group is cyclic iff its subgroup lattice is distributive. Ore’s generalized one direction of this result. We will discuss a dual version of Ore’s result, for any boolean interval of finite groups under the assumption that the dual Euler totient of the interval is nonzero. We conjecture that the dual Euler totient is always nonzero for boolean intervals. We will discuss some techniques which may be helpful in proving it. We first see that dual Euler totient of an interval of finite groups is the Mobius invariant (upto a sign) of its coset poset P. Next in the boolean group complemented case, we prove that P is CohenMacaulay, using the existence of an explicit ELlabeling. We then see that nontrivial betti number of the order complex is nonzero, and so is the dual Euler totient.
Quasialgebras were introduced as algebras in a monoidal category. Since the associativity constraints in these categories are allowed to be nontrivial, the class of quasialgebras contains various important examples of nonassociative algebras like the octonions and other Cayley algebras. The diamond lemma is a reduction method used in algebra. The original diamond lemma was stated in graph theory by Newman which was later generalized to associative algebras by Bergman. In this talk, we will see the analog of this lemma for the group graded quasialgebras with some interesting examples like octonion algebra and generalized octonions
Vortex streets are a common feature of fluid flows at high Reynolds numbers and their study is now well developed for incompressible fluids. Much less is known, however, about compressible vortex streets. A fundamental reason appears to be the inapplicability of the point vortex model to compressible flows. In this talk, we discuss point vortices in the context of weakly compressible flows and elaborate on the problems involved. We then adopt the hollow vortex model where each vortex is modelled as a finitearea constant pressure region with nonzero circulation. For weakly compressible flows steady hollow vortex solutions are well known to be candidates for the leading order solution in a perturbative RayleighJansen expansion of a compressible flow. Here we give details of that expansion based on the vortex street solutions of Crowdy & Green (2012). Physical properties of the compressible vortex streets are described. Our approach uses the ImaiLamla method coupled with analytic function theory and conformal mapping. (Joint work with Darren Crowdy)
The formalism of an “abelian category” is meant to axiomatize the operations of linear algebra. From there, the notion of “derived category” as the category of complexes “upto quasiisomorphisms” is natural, motivated in part by topology. The formalism of tstructures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example). In this talk we want to discuss a notion of punctual (=”pointwise”) gluing of tstructures which is possible in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to useful constructions in the motivic world.
In this talk, I would continue dealing with Sabra shell model of Turbulence and study one of the important questions for fluid flow problems namely, finding controls which are capable of preserving the invariant quantities of the flow. Controls are designed in the feedback form such that resultant controlled flow will preserve certain physical properties of the state such as enstrophy, helicity. We use the theory of nonlinear semigroups and represent the feedback control as a multivalued feedback term which lies in the normal cone of the convex constraint space under consideration.
Graphpartitioning problems are a central topic of research in the study of algorithms and complexity theory. They are of interest to theoreticians with connections to error correcting codes, sampling algorithms, metric embeddings, among others, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications. One of the central problems studied in this field is the sparsest cut problem, where we want to compute the cut which has the least ratio of number of edges cut to size of smaller side of the cut. This ratio is known as the expansion of the cut. In this talk, I will talk about higher order variants of expansion (i.e. notions of expansion corresponding to partitioning the graph into more than two pieces, etc.), and how they relate to the graph’s eigenvalues. The proofs will also show how to use the graph’s eigenvectors to compute partitions satisfying these bounds. Based on joint works with Prasad Raghavendra, Prasad Tetali and Santosh Vempala.
In this lecture I am going to present control problems associated with shell models of Turbulence. Shell models of turbulence are simplified caricatures of equations of fluid mechanics in wavevector representation. They exhibit anomalous scaling and local nonlinear interactions in wave number space. We would like to study control problem related to one such widely accepted shell model of turbulence known as sabra shell model. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the ow near a priori known state. We derive the optimal controls in terms of the solution of adjoint equation for corresponding linearised problems.
Understanding the origins of intermittency in turbulence remains one of the most fundamental challenges in applied mathematics. In recent years there has been a fresh attempt to understand this problem through the development of the method of Fourier decimation. In this talk, we will review these recent results and analyse the role of precise numerical simulations in understanding the mathematics of the NavierStokes and Euler equations.
It is generally well known that there is an innate notion of things like isomorphisms or epimorphisms. This allows us to talk about isomorphisms or epimorphisms of various objects: groups, rings, algebras, etc. In other words, “isomorphism” is really a categorical notion. However, it is not so well known that finiteness itself is alsocategorical. In this talk, we will discuss how finiteness applies to various categories. This will allow usto see finite sets, finite dimensional vector spaces, finitely generated algebras and compact sets as manifestations of the same basic idea.
Consider the infinite Ginibre ensemble (the distributional limit of the eigenvalues of nxn random matrices with i.i.d. standard complex Gaussian entries) in the complex plane. For a bounded set $U$, let $H_r(U)$ denote the probability (hole probability) that no points of infinite Ginibre ensemble fall in the region $rU$. We study the asymptotic behavior of $H_r(U)$ as $r \to \infty$. Under certain conditions on $U$ we show that $\log H_r(U)=C_U \cdot r^4 (1+o(1))$ as $r \to \infty$. Using potential theory, we give an explicit formula for $C_U$ in terms of the minimum logarithmic energy of the set with a quadratic external field. We calculate $C_U$ explicitly for some special sets such as the annulus, cardioid, ellipse, equilateral triangle and half disk.
Moreover, we generalize the above hole probability results for a class of determinantal point processes in the complex plane.
In algebraic geometry the concept of height pairing (a particular example of linking numbers) of algebraic cycles lies at the confluence of arithmetic, Hodge theory and topology. In a series of two talks, I will explain the notion of Beilinson’s height pairing for cycles homologous to zero. This will bring into picture the notion of Arakelov/arithmetic intersection theory. I will give sufficient background of this theory and provide examples. Finally, I will talk about my recent work with Dr. Jose Ignacio Burgos, about a generalization of Beilinson’s height pairing for higher algebraic cycles.
In this talk, I shall consider an abstract Cauchy problem for a class of impulsive subdiffusion equation. Existence and regularity of solution of the problem shall be established via eigenfunction expansion. Further, I shall establish the approximate controllability of the problem by applying unique continuation property via internal control acts on a subdomain.
It was shown by Basu, Sidoravicius and Sly that a TASEP starting with the step initial condition, i.e., with one particle each at every nonpositive site of $\mathbb{Z}$ and no particle at positive sites, with a slow bond at the origin where a particle jumping from the origin jumps at a smaller rate $r < 1$, has an asympototic current which is strictly less than 1/4. Here we study the limiting measure of the TASEP with a slow bond. The distribution of regular TASEP started with the step initial condition converges to the invariant product Bernoulli measure with density 1/2. The slowdown due to the slow bond implies that there is a long range effect near the origin where the region to the right of origin is sparser and there is a traffic jam to the left of the slow bond with particle density higher than a half. However, the distribution becomes close to a product Bernoulli measure as one moves far away from the origin, albeit with a different density ? < 1/2 to the right of the origin and ?’ > 1/2 to the left of the origin. This answers a question due to Liggett. The proof uses the correspondence between TASEP and directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times, and the geometric properties of the maximal paths there.
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed at the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.
In this talk, diffusion and reaction of several mobile chemical species are considered in the pore space of a heterogeneous porous medium. The reactions amongst the species are modelled via mass action kinetics and the modelling leads to a system of multispecies diffusion; reaction equations (coupled semilinear partial differential equations) at the micro scale where the highly nonlinear reaction rate terms are present at the right hand sides of the system of PDEs, cf. [2]. The existence of a unique positive global weak solution is shown with the help of a Lyapunov functional, Schaefer’s fixed point theorem and maximal Lpregularity, cf. [2, 3]. Finally, with the help of periodic homogenization and twoscale convergence we upscale the model from the micro scale to the macro scale, e.g. [1, 3]. Some numerical simulations will also be shown in this talk, however for the purpose of illustration, we restrict ourselves to some relatively simple 2 dimensional situations.
As an extension to the previous model, we consider the mixture of two fluids. For such models, a system of StokesCahnHilliard equations will be considered at the micro scale in a perforated porous medium. We first explain the periodic setting of the model and the existence results. At the end homogenization of the model will be shown using some extension theorems on Sobolev spaces, twoscale convergence and periodic unfolding.
As big data sets have become more common, there has been significant interest in finding and understanding patterns in them. One example of a simple pattern is the distance between data points, which can be thought of as a 2point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3point configurations. In this talk I will present recent Falconer type theorems, established by myself and my collaborators, for a wide class of finite point configurations in any dimension. The techniques we used come from analysis and geometric measure theory, and the key step was to obtain bounds on multilinear analogues of generalized Radon transforms.
Let H denote a connected component of a stratum of translation surfaces. We show that the SiegelVeech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the MasurVeech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.
We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture. This is an attempt to prove the McKennaWalter conjecture which is open for the last two decades.
I will provide a glimpse at the recent extension of free probability to systems with left and right variables based on a notion of bifreeness. This will include the simplest cases of nonlinear convolution operations on noncommutative distributions snd the analogue of extreme values in this setting.
Given any finite simple graph G one can naturally associate two ideals, namely the edge ideal I(G) and the binomial edge ideal J_G in suitable polynomial rings. In this talk we shall discuss the interplay between combinatorics of the graph and depth and regularity of I(G), J_G and their powers. Some recent progress and some open problems will be discussed.
Differentiable manifolds whose Ricci curvature is proportional to the metric are called Einstein manifolds. Such manifolds have been central objects of study in differential geometry and Einstein’s theory for general relativity, with some strong recent results. In this talk, we shall focus on positively curved 3+1 Lorentzian Einstein manifolds with one spacelike rotational isometry. After performing the dimensional reduction to a 2+1 dimensional Einstein’s equations coupled to ‘shifted’ wave maps, we shall prove two explicit positive mass theorems:
Abstract: Given a proper coarse structure on a locally compact Hausdorff space $X$, one can construct the Higson compactification for the coarse structure. In the opposite direction given a compactification of $X$, one can construct a coarse structure. We use unitizations of a nonunital C$^*$\\nobreakdash algebra $A$ to define a noncommutative coarse structure on $A$. We also set up a framework to abstract coarse maps to this noncommutative setting. The original motivation for this work comes from Physics where quantum phenomenon when probed at large scales give classical results. We show equivalence of the canonical coarse structure on the classical plane $\\mathbb{R}^{2n}$ with a certain noncommutative coarse structure on the Moyal plane which models the hypothetical phase space of Quantum physics. If time permits we shall also discuss other examples of noncommutative coarse equivalences. This is a joint work with Prof. Ralf Meyer.
What functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, and has also recently received renewed attention due to its applications in highdimensional statistics. However, effective characterizations of entrywise functions preserving positivity in a fixed dimension remain elusive to date.
I will present recent progress on this question, obtained by: (a) imposing rank and sparsity constraints, (b) restricting to structured matrices, and (c) restricting the class of functions to special families such as polynomials or power functions. These constraints arise in theory as well as applications, and provide natural ways to relax the elusive original problem. Moreover, novel connections to symmetric function theory, matrix analysis, and combinatorics emerge out of these refinements.
Quasilinear symmetric and symmetrizable hyperbolic system has a wide range of applications in engineering and physics including unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, nonNewtonian fluid dynamics, and Einstein field equations of general relativity. In the past, the Cauchy problem of smooth solutions for these systems has been studied by several mathematicians using semigroup approach and fixed point arguments. In a recent work of M. T. Mohan and S. S. Sritharan, the local solvability of symmetric hyperbolic system is established using two different methods, viz. local monotonicity method and a frequency truncation method. The local existence and uniqueness of solutions of symmetrizable hyperbolic system is also proved by them using a frequency truncation method. Later they established the local solvability of the stochastic quasilinear symmetric hyperbolic system perturbed by Levy noise using a stochastic generalization of the localized MintyBrowder technique. Under a smallness assumption on the initial data, a global solvability for the multiplicative noise case is also proved. The essence of this talk is to give an overview of these new local solvability methods and their applications.
We give a relatively simple proof of the famous theorem of Narasimhan and Seshadri on vector bundles on a compact Riemann surface. The theorem relates the algebraic geometric notion of stability of vector bundles on a compact Riemann surface with a transcendental construct  unitary representations of a suitable Fuchsian group associated to the Riemann surface.
The DeligneHitchin moduli space is a partial compactification of the moduli space of $\lambda$connections. It includes as closed subvarieties the moduli spaces of Hitchin bundles ($\lambda=0$) and of holomorphic connections ($\lambda=1$), exhibiting the later as a deformation of the former. We show a Torelli theorem for a parabolic version of this moduli space (joint work with David Alfaya). I will try to make the talk accessible to a wide mathematical audience.
The PerronFrobenius theorem is a powerful and useful result about the eigenvalues and eigenvectors of a nonnegative matrix. I will not be proving the theorem but will instead focus on its applications. In particular, I will discuss how it can be used to understand the behaviour of a certain class of dynamical systems, namely certain systems of firstorder ordinary differential equations where the couplings between variables are specified by a graph. Some familiarity with graph theory will be useful, but I will try to recapitulate all the basic concepts needed for this talk.
In this talk I will present algorithmic applications of an approximate version of Caratheodory’s theorem. The theorem states that given a set of $d$dimensional vectors $X$, for every vector in the convex hull of $X$ there exists an epsilonclose (under the $p$norm, for $2 \leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on epsilon and the norm $p$, and is independent of the ambient dimension $d$. This theorem can be obtained by instantiating Maurey’s lemma (c.f. Pisier 1980/81 and Carl 1985).
I will describe how this approximate version of Caratheodory’s theorem leads novel additive approximation algorithms for finding (i) Nash equilibria in twoplayer games (ii) dense subgraphs.
A classical theorem in Riemannian geometry asserts that products of compact manifolds cannot admit Riemannian metrics with negative sectional curvature. A fibration (or a fibre bundle) is a natural generalization of a product and hence one can ask if a fibration can admit such a metric. This question is still open and I will discuss it in the context of Kahler manifolds. This leads to the study of graphs of holomorphic motions, which originally arose in complex dynamics. I will sketch a proof that the graph cannot be biholomorphic to a ball or more generally, a strongly pseudoconvex domain.
The enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field is the same as the classifi cation of commuting tuples of matrices over a finite field up to simultaneous similarity. Let C_{n,k}(q) denote the number of isomorphism classes of ndimensional Fq[x1,…,xk]modules. The generating function in k of the C_{n,k}(q) is a rational function. The computation of this was done explicitly for n <= 4. I shall give a summary of my recently published work on this study of the C_{n,k}(q)s for n <= 2.
In this talk, we’ll report on progress towards a conjecture of B.Krtz about the holomorphic extensions of nonzero Kfinite vectors of irreducible admissible Banach representations of simple real Lie groups and the relation to a distinguished domain  the socalled crown domain. We’ll explain some of the main ideas  the CasselmanWallach smooth globalisation, vanishing of matrix coefficients at infinity etc. Indeed we prove the conjecture with some additional growth conditions on the Banach globalisations. This is joint work with Gang Liu, Uni. Metz.
This talk is about computing (approximately) the “best” map between two polygons taking vertices to vertices. It arises out of a reallife problem, namely, surface registration. Our notion of “best” is extremal quasiconformality (least angledistortion). I will try to keep the talk as selfcontained as possible. It is based on a jointwork with M. Goswami, G. Telang, and X. Gu.
In this talk I will describe the interrelationship between Margulis spacetimes and Anosov representations. Moreover, I will define the pressure metric on the moduli space of Margulis spacetimes and sketch some of it’s properties.
We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study. This talk will be accessible to the general mathematical audience.
We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study. This talk will be accessible to the general mathematical audience.
Multiobjective optimisation involves optimising several quantities, such as time and money, simultaneously. The result is a polyhedral frontier of best possible solutions, which cannot improve one quantity without a tradeoff against another. For linear programming, this frontier can be generated using Benson’s outer approximation algorithm, which uses a sequence of scalarisations (singleobjective optimisations), combined with classical algorithms from polytope theory.
We consider a class of wave propagation models with aleatoric and epistemic uncertainties. Using mathematical analysisbased, shapeindependent, a priori parameter estimates, we develop offline/online strategies to compute statistical moments of a key quantity of interest in such models. We present an efficient reduced order model (ROM) and high performance computing (HPC) framework with analysis for quantifying aleatoric and epistemic uncertainties in the propagation of waves through a stochastic media comprising a large number of three dimensional particles. Simulation even for a single deterministic three dimensional configuration is inherently difficult because of the large number of particles. The aleatoric uncertainty in the model leads to a larger dimensional system involving three spatial variables and additional stochastic variables. Accounting for epistemic uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity. Our hybrid ROM and HPC framework can be used in conjunction with any computational method to simulate a single particle deterministic wave propagation model.
Let $G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $P = MN$ be a Levi decomposition of a maximal parabolic subgroup of $G$, and $\sigma$ an irreducible unitary supercuspidal representation of $M(F)$. One can then consider the representation Ind$_{P(F)}^{G(F)}\sigma$ (normalized parabolic induction). This induced representation is known to be either irreducible or of length two. The question of when it is irreducible turns out to be (conjecturally) related to local $L$functions, and also to poles of a family of so called intertwining operators.
Let $\Delta$ be the Laplacian on a Riemannian symmetric space $X=G/K$ of the noncompact type and let $\sigma(\Delta)\subseteq \mathbb{C}$ denote its spectrum. The resolvent $(\Deltaz)^{1}$ is a holomorphic function on $\mathbb{C} \setminus \sigma(\Delta)$, with values in the space of bounded operators on $L^2(X)$. If we view it as a function with values in Hom$(C_c^\infty(X), C_c^\infty(X)^*)$, then it often admits a meromorphic continuation beyond $\mathbb{C} \setminus \sigma(\Delta)$. We study this meromorphic continuation as a map defined on a Riemann surface above $\mathbb{C} \setminus \sigma(\Delta)$. The poles of the meromorphically extended resolvent are called resonances. The image of the residue operator at a resonance is a $G$module. The main problems are the existence and the localization of the resonances as well as the study of the (spherical) representations of $G$ so obtained. In this talk, based on joint works with Joachim Hilgert and Tomasz Przebinda, we will describe a variety of different situations occurring in the rank two case.
Kardar, Parisi and Zhang introduced a universality class (the socalled KPZ universality class) in 1986 which is believed to explain the universal behaviour in a large class of two dimensional random growth models including first and last passage percolation. A number of breakthroughs has led to an explosion of mathematically rigorous results in this field in recent years. However, these have mostly been restricted to the class of exactly solvable models, where exact formulae are available using powerful tools of random matrices, algebraic combinatorics and representation theory; beyond this class the understanding remains rather limited. I shall talk about a geometric approach to these problems based on studying the geometry of geodesics (optimal paths), and describe some recent progress along these lines.
The ndimensional matrix representations of a group or an associative algebra A form a space (algebraic variety) Rep(A,n) called the nth representation variety of A. This is a classical geometric invariant that plays a role in many areas of mathematics. The construction of Rep(A,n) is natural (functorial) in A, but it is not `exact’ in the sense of homological algebra. In this talk, we will explain how to refine Rep(A,n) by constructing a derived representation variety DRep(A,n), which is an example of a derived moduli space in algebraic geometry. For an application, we will look at the classical varieties of commuting matrices, and present a series of combinatorial conjectures extending the famous Macdonald conjectures in representation theory.
The ndimensional matrix representations of a group or an associative algebra A form a space (algebraic variety) Rep(A,n) called the nth representation variety of A. This is a classical geometric invariant that plays a role in many areas of mathematics. The construction of Rep(A,n) is natural (functorial) in A, but it is not `exact’ in the sense of homological algebra. In this talk, we will explain how to refine Rep(A,n) by constructing a derived representation variety DRep(A,n), which is an example of a derived moduli space in algebraic geometry. For an application, we will look at the classical varieties of commuting matrices, and present a series of combinatorial conjectures extending the famous Macdonald conjectures in representation theory.
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Ranklevel duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra.
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Ranklevel duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra.
The FarrellJones Isomorphism conjecture gives a single statement to understand many standard conjectures in Topology and Algebra= . We will discuss understanding K and Ltheory of groups acting on trees from the vertex stabilizers of the action, in the context of Isomorphism conjecture.
This is a topic in classical algebraic KTheory. I will recall definitions of elementary linear group, elementary symplectic group, linear transvection group, and symplectic transvection group. These group= s have natural action on the set of unimodular elements. I will briefly discuss how bijections between orbit spaces of unimodular elements under different group actions are established. Finally, I will talk about an application of these results, namely improving injective stability bound for the K1 group.
One of the earliest instance of integral geometric discussion can be traced back to Buffon through the “needle problem”. The solution, as we know now, is based on a platform of theory of measures on geometrical spaces which are invariant under some group operations. In this talk, we shall walk through this subject by following a specific line of problems/results called “kinematic fundamental formulae” by studying some specific examples.
Inverse spectral theory in one dimension is to recover a selfadjoint operator given its spectrum and some spectral measure. In general there are lots of selfadjoint operators (of a given form) this collection is called an isospectral set. We will give a brief introduction to the questions in this area and also some connections to other areas of mathematics.
Suppose V is a finite dimensional representation of a complex finite dimensional simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the nontrivial irreducible factors that occur in the tensor product factorization of V are uniquely determined, up to reordering, by the isomorphism class of V. I will present an elementary proof of Rajan’s theorem. This is a joint work with S.Viswanath.
We study asymptotic analysis (homogenization) of secondorder partial differential equations(PDEs) posed on an oscillating domain. In general, the motivation for studying problems defined on oscillating domains, come from the need to understand flow in channels with rough boundary, heat transmission in winglets, jet engins and so on. There are various methods developed to study homogenization problems namely; multiscale expansion, oscillating test function method, compensated compactness, twoscale convergence, blockwave method, method of unfolding etc.
In this thesis, we consider a two dimensional oscillating domain (comb shape
type) $\Omega_{\epsilon}$
consists of a fixed bottom region $\Omega^$
and an oscillatory
(rugose) upper region $\Omega_{\epsilon}^{+}$. We introduce an optimal control problems in
$\Omega_{\epsilon}$
for the Laplacian operator. There are mainly two types of optimal
control problems; namely distributed control andboundary control. For distributed control
problems in the oscillatingdomain, one can put control on the oscillating part or on the fixed
part and similarly for boundary control problem (control on the oscillatingboundary or on the
fixed part the boundary). Considering controls on theoscillating part is more interesting and
challenging than putting control on fixed part of the domain. Our main aim is to characterize
the controlsand study the limiting analysis (as $\epsilon \to 0$
) of the optimalsolution.
In the thesis, we consider all the four cases, namely distributed and boundary controls both
on the oscilalting part and away from the oscillating part. Since, controls on the oscillating
part is more exciting, in this talk, we present the details of two sections. First we consider
distributed optimal control problem, where the control is supported on the oscillating part
$Omega_{\epsilon}^{+}$
with periodic controls and with Neumann condition on the oscillating
boundary $\gamma_{\epsilon}$
. Secondly, we introduce boundary optimal control
problem, control applied through Neumann boundary condition on the oscillating boundary
$\gamma_{\epsilon}$
with suitable scaling parameters. We characterize the optimal control
using unfolding and boundary unfolding operators and study limiting analysis. In the limit, we
obtain two limit problems according to the scaling parameters and we observe that limit
optimal control problem has three control namely; a distributed control, a boundary control
and an interface control.
Given a simple graph G, the KacMoody Lie algebra of G is the KacMoody algebra whose simply laced Dynkin diagram is G. We give a new interpretation of the chromatic polynomial of G in terms of the KacMoody Lie algebra of G. We show that the chromatic polynomial is essentially th= e qKostant partition function of the associated KacMoody algebra evaluate= d on the sum of the simple roots. As an application, we construct basis of some of the root spaces of the KacMoody algebra of G. This is a joint work with Sankaran Viswanath.
We shall introduce the definition of a kmode Gaussian state and a chain of such states which determine a C* probability space. We present examples of such states exhibiting properties like exchangeability and stationarity. Stationary chains are determined by block Toeplitz matrices. Using the KacMurdochSzego theorems on asymptotic spectral distributions of Toeplitz matrices we compute the entropy rates of some of these chains. This leaves many natural problems open.
In this talk we discuss how the notion of usual convergence is extended using the notion of ideals and the importance of Pideals. We then show how ideals can be generated and in particular how the P ideals can be generated by matrices other than regular summability matrix
One of the most useful analogies in mathematics is the fundamental group functor (also known as the Galois Correspondence) which sends a topological space to its fundamental group while at the same time sending continuous maps between spaces to corresponding homomorphisms of groups in such a way that compositions of maps are preserved.
I will introduce minimal surfaces and explain the WeierstrassEnneper representation of a minimal surface using hodographic coordinates. I will mention an interesting link between minimal surfaces and BornInfeld solitons. If time permits, I will explain my work (with P. Kumar and R.K. Singh) on interpolation of two real analytic curves by a minimal surface of the BjorlingSchwartz type.
Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a very important problem. Theoretically, unconstrained SFM is polynomial time solvable, however, these algorithms are not practical. In 1976, Wolfe proposed a heuristic for projection onto a polytope, and in 1980, Fujishige showed how Wolfe’s heuristic can be used for SFM. For general submodular functions, this FujishigeWolfe heuristic seems to have the best empirical performance. Despite its good practical performance, very little is known about Wolfe’s projection algorithm theoretically.
In this talk, I will describe the first analysis which proves that the heuristic is polynomial time for bounded submodular functions. Our work involves the first convergence analysis of Wolfe’s projection heuristic, and proving a robust version of Fujishige’s reduction theorem.
Having been unclear how to widely define strong (or strict) pseudoconvexity in the infinitedimensional context, we compared the concept in the smoothboundary case with strict convexity. As a result, w= e accomplished establishing definitions of local uniform pseudoconvexity, uniform pseudoconvexity and strict pseudoconvexity for open and bounded subsets of a Banach space. We will see examples of Banach spaces with uniformly pseudoconvex unit ball, as well as examples of Banach spaces whose unit ball is not even strictly pseudoconvex. As an application of the techniques developed, we show that in finite dimension the concept of strict plurisubharmonicity coincides with strict plurisubharmonicity in distribution.
A pair of commuting bounded operators $(S,P)$ acting on a
Hilbert space, is
called a $\Gamma$
contraction, if it has the symmetrised bidisc as a
spectral set. For
every $\Gamma$contraction $(S,P)$
, the operator equation
has a
unique solution $F$ with numerical radius, $w(F)$ no greater than one,
where $D_P$ is the
positive square root of $(IP^*P)$
. This unique operator is called the
fundamental operator of
$(S,P)$. This thesis constructs an explicit normal boundary dilation for a
$\Gamma$contraction. A triple of commuting bounded operators $(A,B,P)$
acting on a
Hilbert space with the closure of the tetrablock
as a spectral set, is called a tetrablock contraction. Every tetrablock
contraction
possesses two fundamental operators and these are the unique solutions of Moreover, $w(F_1)$ and
$w(F_2)$ are no greater than one. This thesisalso constructs an explicit
normal boundary
dilation for a tetrablock contraction. In these constructions, the
fundamental operators play a
pivotal role. Both the dilations in the symmetrized bidisc and in the
tetrablock are proved to
be minimal. But unlike the one variable case, uniqueness of minimal
dilations usually does
not hold good in several variables, e.g., Ando’s dilation is not unique.
However, we show that
the dilations are unique under a certain natural condition. In view of the
abundance of
operators and their complicated structure, a basic problem in operator
theory is to find nice
functional models and complete sets of unitary invariants. We develop a
functional model
theory for a special class of triples of commuting bounded operators
associated with the
tetrablock. We also find a set of complete sort of unitary invariants for
this special class.
Along the way, we find a BeurlingLaxHalmos type of result for a triple
of multiplication
operators acting on vectorvaluedHardy spaces. In both the model theory
and unitary
invariance,fundamental operators play a fundamental role. This thesis
answers the question
when two operators $F$ and $G$ with $w(F)$ and $w(G)$ no greater than one,
are
admissible as fundamental operators, in other words, when there exists a
$\Gamma$contraction $(S,P)$ such that $F$ is the fundamental operator of
$(S,P)$ and
$G$ is the fundamental operator of $(S^*,P^*)$
. This thesis also answers a
similar question
in the tetrablock setting.
In the 1980s, Goldman introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F. This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra.
In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.
We use these results to identify the quotient of the Goldman Lie algebra of a nonclosed surface by its center as a subalgebra of the first Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas, namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in an= y hyperbolic surface F of finite type. Our construction shows that given a selfintersecting geodesic x of F and any selfintersection point P of x, we get a sequence of such pairs.
I will present the Funk and Hilbert metrics on convex sets in the setting of Euclidean, nonEuclidean and timelike geometries. I will explain the motivation for studying these metrics and highlight some of their main properties, concerning geodesics, infinitesima structures and isometries.
There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. In the first half of the talk I will describe some often used methods and discuss their pros and cons (it’s all in the rounding).
One easy method is called Hamilton’s method (also known as the method of largest reminder or Hare’s method). It suffers from a drawback called the Alabama paradox, which e.g. made USA abandon it for the distribution of seats in the house of representatives between states. It is still in use in many other countries including Sweden.
In the second half of the talk I will describe a joint work with Svante Janson (Uppsala Univ.) where we study the probability that the Alabama paradox will happen. We give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector $p_1,\dots,p_m$ of relative sizes of the states.
Eigenvalues and eigenvectors appear in many physical and engineering problems, beginning with the noted study by Euler in 1751 of the kinematics of rigid bodies. From a mathematical point of view, for an operator of a suitable class, acting in a vector space, its point spectrum and associated subspaces refer to its “eigenvalues and eigenvectors”, while the subspaces associated with the continuous spectrum of the operator is said to consist of “eigenfunctions”. The basic ideas will be discussed mostly through examples, in some of which a natural connection with the representation of appropriate groups lurks behind.
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$
be i.i.d matrices of size nxn whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$
, where each
$Y_i = X_i or (X_i)^{1}$
. We show that the eigenvalues form a determinantal
point process. The case where k=2, $Y_1=X_1,Y_2=X_2^{1}$
was derived
earlier by Krishnapur. The case where $Y_i =X_i$
for all $i=1,2,...,n$
, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Let X be a closed, simply connected and orientable manifold of dimension m and LX the space of free loops on X. We use Rational Homotopy Theory to construct a model for the loop space homology. We further define a BV structure which is equivalent, in some cases, to the ChasSullivan BV operator.
In algebraic geometry the concept of height pairing (a particular example
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multi‐scale medium
We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture. This is an attempt to prove the McKennaWalter conjecture which is open for the last two decades.
There is a folklore conjecture that there are infinitely many primes p such that p+2 is also prime. This conjecture is still open. However, in the last two years, spectacular progress has been made to show that there are infinitely many primes p such that p+h is also prime with 1< h < 247. We will discuss the history of this problem and explain the new advances in sieve theory that have led to these remarkable results. We will also highlight what we may expect in the future.
Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an inhomogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. We plan to start with an illustrative example of limiting analysis in 1D for a second order elliptic partial differential equation. We will also see some classical results in the case of periodic composite materials and oscillating boundary domain. The emphasis will be on the computational importance of homogenization in numerics by the introduction of correctors. In the second part of the talk, we will see a study on optimal control problems posed in a domain with highly oscillating boundary. We will consider periodic controls in the oscillating part of the domain with a model problem of Laplacian and try to understand their optimality and asymptotic behavior.
Voevodsky’s conjecture states that numerical and smash equivalence coincide for algebraic cycles. I shall explain the conjecture in more detail and talk about some of the examples for which this conjecture is known.
Enumerative geometry is a branch of mathematics that deals with the following question: How many geometric objects are there that satisfy certain constraints? The simplest example of such a question is How many lines pass through two points?. A more interesting question is How many lines are there in three dimensional space that intersect four generic lines?. An extremely important class of enumerative question is to ask How many rational (genus 0) degree d curves are there in CP^2 that pass through 3d1 generic points? Although this question was investigated in the nineteenth century, a complete solution to this problem was unknown until the early 90’s, when KontsevichManin and RuanTian announced a formula. In this talk we will discuss some natural generalizations of the above question; in particular we will be looking at rational curves on delPezzo surfaces that have a cuspidal singularity. We will describe a topological method to approach such questions. If time permits, we will also explain the idea of how to enumerate genus one curves with a fixed complex structure by comparing it with the Symplectic Invariant of a manifold (which are essentially the number of curves that are solutions to the perturbed dbar equation).
Interplay between arithmetic and analytic objects are some of the beautiful aspects of number theory. In this talk, we will discuss several examples of this.
We consider an nplayer strategic game with finite action sets. The payoffs of each player are random variables. We assume that each player uses a satisficing payoff criterion defined by a chanceconstraint, i.e., players face a chanceconstrained game. We consider the cases where payoffs follow normal and elliptically symmetric distributions. For both cases we show that there always exists a mixed strategy Nash equilibrium of corresponding chanceconstrained game.
This talk shall be an introduction to some aspects of Teichmüller theory, which is concerned with the parameter space of marked Riemann surfaces. I shall describe Wolf’s parametrization of Teichmüller space using harmonic maps, and discuss how it extends to the Thurston compactification. If time permits, I will describe some recent joint work with Wolf.
We get useful insight about various algebraic structures by studying their automorphisms and endomorphisms. Here we give a brief introduction to the theory of semigroups of endomorphisms of the algebra of all bounded operators on a Hilbert space.
Given an irreducible polynomial f(x) with integer coefficients and a prime number p, one wishes to determine whether f(x) is a product of distinct linear factors modulo p. When f(x) is a solvable polynomial, this question is satisfactorily answered by the Class Field Theory. Attempts to find a nonabelian Class Field Theory lead to the development of an area of mathematics called the Langlands program. The Langlands program, roughly speaking, predicts a natural correspondence between the finite dimensional complex representations of the Galois group of a local or a number field and the infinite dimensional representations of real, padic and adelic reductive groups. I will give an outline of the statement of the local Langlands correspondence. I will then briefly talk about two of the main approaches towards the Langlands program  the type theoretic approach relying on the theory of types developed by BushnellKutzko and others; and the endoscopic approach relying on the trace formula and endoscopy. I will then state a couple of my results involving these two approaches.
The TuraevViro invariants are a powerful family of topological invariants for distinguishing between different 3manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. I will discuss this family of invariants, and present an explicit fixedparameter tractable algorithm for arbitrary r which is practical—and indeed preferable—to the prior state of the art for real computation.
Exact computation with knots and 3manifolds is challenging  many fundamental problems are decidable but enormously complex, and many major algorithms have never been implemented. Even simple problems, such as unknot recognition (testing whether a loop of string is knotted), or 3sphere recognition (testing whether a triangulated 3manifold is topologically trivial), have bestknown algorithms that are worstcase exponential time.
In the seventies of last century, Magnus Landstad characterised the coefficient Calgebra inside the multiplier algebra of a given crossed product Calgebra subject to an action of an Abelian locally compact group G. In 2005, Stefaan Vaes extended the Landstad theory for regular locally compact quantum groups. He gave strong indications that this is not possible for nonregular groups. In this talk I shall explain how LandstadVaes theory extends for nonregular groups. To this end we have to consider not necessary continuous, but measurable, actions of locally compact quantum groups. For regular locally compact quantum groups any measurable action is continuous, so our theory contains that of LandstadVaes. This is a joint work with Stanislaw Lech Woronowicz.
I will present a general approach for constructing a Markov process that describes the dynamics of a nonequilibrium process when one or more observables of this process are observed to fluctuate in time away from their typical values.
We study the local behavior of (extreme) quantiles of the sum of heavytailed random variables, to infer on risk concentration. Looking at the literature, asymptotic (for high threshold) results have been obtained when assuming (asymptotic) independence and second order regularly varying conditions on the variables. Other asymptotic results have been obtained in the dependent case when considering specific copula structures. Our contribution is to investigate on one hand, the nonasymptotic case (i.e. for any threshold), providing analytical results on the risk concentration for copula models that are used in practice, and comparing them with results obtained via MonteCarlo method. On the other hand, when looking at extreme quantiles, we assume a multivariate second order regular variation condition on the vectors and provide asymptotic risk concentration results. We show that many models used in practice come under the purview of such an assumption and provide a few examples. Moreover this ties up related results available in the literature under a broad umbrella. This presentation is based on two joint works, one with M. Dacorogna and L. Elbahtouri (SCOR), the other with B. Das (SUTD).
The usual foundations of mathematics based on Set theory and Predicate calculus (and extended by category theory), while successful in many ways, are so far removed from everyday mathematics that the possibility of translation of theorems to their formal counterparts is generally purely a matter of faith. Homotopy type theory gives alternative foundations for mathematics. These are based on an extension of type theory (from logic and computer science) using an unexpectedly deep connection of Types with Spaces discovered by Voevodsky and AwodeyWarren. As a consequence of this relation we also obtain a synthetic view of homotopy theory. In this lecture, I will give a brief introduction to this young subject.
It is well known that finding an explicit solution for partial differential equations is almost impossible in most of the cases. In this talk we will give a brief introduction to the analysis of certain nonlinear PDEs using various tools from analysis. We will discuss some concrete examples and some open problems (depending on time). The talk should be accessible to senior undergraduate students.
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he Lfunction of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the Lfunction. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Independent component analysis (ICA) is a basic problem that arises in several areas including signal processing, statistics, and machine learning. In this problem, we are given linear superpositions of signals. E.g., we could be receiving signals from several sensors but the receivers only get the weighted sums of these signals. The problem is to recover the original signals from the superposed data. In some situations this turns out to be possible: the main assumption being that the signals at different sensors are independent random variables. While independent component analysis is a wellstudied problem, one version of it was not wellunderstood, namely when the original signals are allowed to be heavytailed, such as those with a Pareto distribution. Such signals do arise in some applications. In this talk, I will first discuss the previously known algorithms for ICA and then a new algorithm that applies also to for the heavytailed case. The techniques used are basic linear algebra and probability.
We consider integral lattices $L$ in an Euclidean space $V = \mathbb{R}^m$, i.e. $\mathbb{Z}$submodules of full rank in $V$ such that all vectors in $L$ have integral length. It is impossible to classify such lattices up to isometry, there are just too many of them in general, even if we ﬁx additional invariants such as the discriminant. Therefore one looks for interesting subclasses of lattices, in particular “extremal lattices”, characterized by the property that the smallest length of a nonzero vector in $L$ is “as large as possible”. There are several ways to make this more precise, we will focus on analytic extremality, where modular forms come in. In particular, we will consider extremality for maximal lattices.
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he Lfunction of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the Lfunction. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he Lfunction of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the Lfunction. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
There is a dichotomy of contact structures– tight vs overtwisted. Classification of overtwisted contact structures up to isotopy, due to Eliashberg, is well understood. However, there have been few results towards classification of tight contact structures. In general, given a contact structure it is difficult to know whether it is tight or not.
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he Lfunction of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the Lfunction. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Define $S_R^\alpha f := \int_{\mathbb{R}^d} (1\frac{\xi^2}{R^2})_+^\alpha$ $\hat{f} (\xi) e^{i2\pi (x.\xi)} d\xi$, the Bochner Riesz means of order $\alpha \geq 0$. Let $f\in L^2(\mathbb{R}^d)$ and $f \neq 0$ in an open, bounded set $B.$ It is known that $S_R^\alpha f$ goes to 0 a.e. in $B$ as $R\rightarrow\infty.$ We study the pointwise convergence of Bochner Riesz means $S_{R}^\alpha f, \alpha>0$ as $R \rightarrow \infty$ on sets of positive Hausdorff measure in $\mathbb{R}^d$ by making use of the decay of the spherical means of Fourier Transform of fractal measures. We get an improvement in the range of the Hausdorff dimension of the sets on which it converges. When $0<\alpha<\frac{d1}{4},$ we get the best possible result in $\mathbb{R}^2$ and in higher dimensions we improve the result by L.Colzani, G. Gigante and A. Vargas. Steins interpolation theorem also gives us the corresponding result for $f\in L^p(\mathbb{R}^d), 1<p<2.$
In this talk we prove a version of the Gohberg lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators on compact Lie groups. As a consequence, we prove several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrixvalued symbols of operators. (This is a joint work with Professor Michael Ruzhansky.)
Koszul duality theory is a homological method aiming at constructing an explicit quasifree resolution for quadratic algebras. We introduce the concepts of bar (and cobar) construction for a quadratic algebra (and coalgebra) and provide a quasifree resolution for quadratic algebras, under certain assumptions.
A reductive dual pair in the group Sp(W) of isometries of a symplectic space W, over a local
In the first part of this paper we give a solution for the onedimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson random measure. The reflecting process is right continuous with left limits (RCLL for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the general result we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove existence and uniqueness of the solution of the BSDE.
An operad is an algebraic device which encodes a type of algebra. The classical types of algebras, ie. associative, commutative and Lie algebras give the first examples of algebraic operads. Operadic point of view has several advantages. Firstly, many results known for classical algebras, when written out in operadic language, can be applied to other types of algebras. Secondly, operadic language simplifies the statements and proofs. Thirdly, even for classical cases, operad theory has provided new results. We start with several equivalent definitions, together with examples, and few basic properties.
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$
be i.i.d matrices of size nxn whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$
, where each
$Y_i = X_i or (X_i)^{1}$
. We show that the eigenvalues form a determinantal
point process. The case where k=2, $Y_1=X_1,Y_2=X_2^{1}$
was derived
earlier by Krishnapur. The case where $Y_i =X_i$
for all $i=1,2,...,n$
, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
A problem of interest in geometric measure theory both in discrete and continuous settings is the identification of algebraic and geometric patterns in thin sets. The first two lectures will be a survey of the literature on pattern recognition in sparse sets, with a greater emphasis on continuum problems. The second two contain an exposition of recent work, joint in part with Vincent Chan, Kevin Henriot and Izabella Laba on the existence of linear and polynomial configurations in multidimensional Lebesguenull sets satisfying appropriate Hausdorff and Fourier dimensionality conditions.
This talk will focus on two aspects of vector bundles. One is the calculation and application of certain characteristic and secondary characteristic forms (i.e. Chern, ChernSimons, and BottChern forms). This part is joint work with Leon Takhtajan and Indranil Biswas. The second is to study certain fully nonlinear PDE akin to the MongeAmp\\‘ere equation arising from these differential geometric objects. An existence result or two will be presented along with the difficulties involved in the PDE. Moreover, other areas of geometry and physics from which the same PDE arise will be pointed out.
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3manifolds
$\mathbb{R P}^3$
, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$
,
$S^{\hspace{.2mm}2} \times S^1$
, $\TPSS$
and $S^{\hspace{.2mm}3}/Q_8$
,
where $Q_8$
is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq1,q)$ with $4(q+k1)$ vertices for $q \geq 3$
, $k \geq 2$
and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$manifolds from a given presentation $\langle S \mid R \rangle$
with
$\#S=\#R=2$
. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$
with $4(m+n+k3)+ 2\delta_n^2 + 2
\delta_k^2$
vertices for $m\geq 3$
and $m \geq n \geq k \geq 2$
, where
$\delta_i^j$
is the Kronecker delta.
If $n=2$ or $k\geq 3$
and $m \geq 4$
then these crystallizations
are vertexminimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$
, $S^2 \times S^2$
, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4manifolds which are
homeomorphic
but not plhomeomorphic.
Largedense matrices arise in numerous applications: boundary integral formulation for elliptic partial differential equations, covariance matrices in statistics, inverse problems, radial basis function interpolation, multi frontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements, scaling as O(N^2), and extensive computational time to perform matrix algebra, scaling as O(N^2) or O(N^3), make computations impractical. I will discuss some novel methods for handling these computationally intense problems. In the first half of the talk, I will discuss my contributions to some of the new developments in handling large dense covariance matrices in the context of computational statistics and Bayesian data assimilation. More specifically, I will be discussing how fast dense linear algebra (O(N) algorithms for inversion, determinant computation, symmetric factorisation, etc.) enables us to handle large scale Gaussian processes, thereby providing an attractive approach for big data applications. In the second half of the talk, I will focus on a new algorithm termed Inverse Fast Multipole Method, which permits solving singular integral equations arising out of elliptic PDE’s at a computational cost of O(N).
Let P be the equilibrium potential of a compact set K in R^n. An electrostatic skeleton of K is a positive measure such that the closed support S of has connected complement and empty interior, and the Newtonian (or logarithmic, when n = 2) potential of is equal to P near infinity. We prove the existence and uniqueness of an electrostatic skeleton for any simplex. – This message has been scanned for viruses and dangerous content by MailScanner, and is believed to be clean.
Abstract : In this work, we study certain stability results for Ball Separationproperties in Banach Spaces leading to a discussion in the context of operator spaces. In this work, we study certain stability results for Small Combination of Slices Property (SCSP) leading to a discussion on SCSP in the context of operator spaces. SCSpoints were first introduced as a slice generalisation of the PC (i.e. point ofcontinuity points for which the identity mapping from weak topology to normtopology is continuous.) It is known that X is strongly regular respectively Xis wstrongly regular) if and only if every non empty bounded convex set K in X ( respectively K in X) is contained in the norm closure ( respectively w closure)of SCS(K)( respectively wSCS(K)) i.e. the SCS points ( w SCS points) of K. Later, it was proved that a Banach space has Radon Nikodym Property (RNP) if and only if it is strongly regular and it has the KreinMilamn Property(KMP). Subsequently, the concepts of SCS points was used to investigate the structure of nondentable closed bounded convex sets in Banach spaces. The point version of the result was also shown to be true .
This talk is about random matrix theory. Linear Algebra and maybe a little probability are the only prerequisites. Random matrix theory is now finding many applications. Many more applications remain to be found. It is truly matrix statistics, when traditional statistics has been primarily scalar and vector statistics. The math is so much richer, and the applications to computational finance, HIV research, the Riemann Zeta Function, and crystal growth, to name a few, show how important this area is. I will show some of these applications, and invite you to find some of your own.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface  we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal nonadmissible fat graphs. Another major result of this thesis is that there are infinitely many minimal nonadmissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal nonplanar graphs).
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface  we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal nonadmissible fat graphs. Another major result of this thesis is that there are infinitely many minimal nonadmissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal nonplanar graphs).
In this very informal seminar, I will discuss various aspects of ongoing work to build an automated theorem proving system using, among other things, machine learning.
Let G be a 2group, and let Z(G) denote the equivariant cobordism algebra of Gmanifolds with finite stationary point sets.A cobordism class in Z(G) is said to be indecomposable if it cannot be expressed as the sum of products of lower dimensional cobordism classes.Indecomposable classes generate the cobordism algebra Z(G). We discuss a sufficient criteria for indecomposability of cobordism classes. Using the above mentioned criterion, we show that the classes of Milnor manifolds (i.e., degree 1 hypersurfaces of the product of two real projective spaces) give nontrivial, indecomposable elements in Z(G) in degrees up to 2^n  5. This talk is based on joint work with Samik Basu and Goutam Mukherjee.
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$
. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(d\sigma)$
, where $d\sigma$
is the surface measure on the sphere $S^{n1}\subset\mathbb{R}^n$
. Then
It follows that $\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$
for all
$p>2n/(n1)$
. This result can be extended to compactly
supported measure on $(n1)$dimensional manifolds with
appropriate assumptions on the curvature. Similar results are
known for measures supported in lower dimensional manifolds in
$\mathbb{R}^n$
under appropriate curvature conditions. However, the
picture for fractal measures is far from complete. This thesis is
a contribution to the study of asymptotic properties of the
Fourier transform of measures supported in sets of fractal
dimension $0<\alpha<n$
for $p\leq 2n/\alpha$
.
In 2004, Agranovsky and Narayanan proved that if $\mu$ is a
measure supported in a $C^1$
manifold of dimension $d<n$
, then
$\widehat{fd\mu}\notin L^p(\mathbb{R}^n)$
for $1\leq p\leq \frac{2n}{d}$
. We
prove that the Fourier transform of a measure $\mu_E$ supported in
a set $E$ of fractal dimension $\alpha$ does not belong to
$L^p(\mathbb{R}^n)$
for $p\leq 2n/\alpha$
. We also study $L^p$
asymptotics
of the Fourier transform of fractal measures $\mu_E$ under
appropriate conditions on $E$ and give quantitative versions of
the above statement by obtaining lower and upper bounds for the
following:
Our vision is unsurpassed by machines because we use a sophisticated object representation. This representation is unlike the retinal image: on the one hand, two outofphase checkerboards, maximally different in image pixels, appear perceptually similar. On the other hand, two faces, similar in their image pixels, appear perceptually distinct. What is then the nature of perceptual space? Are there principles governing its organization? To address these questions, we have been using visual search to measure similarity relations between objects.
I will summarize a line of research from our laboratory indicative of a surprising linear rule governing distances in perceptual space. In the first study, we found that search time is inversely proportional to the feature difference between the target and distracters. The reciprocal of search time is therefore linear and interestingly, it behaved like a mathematical distance metric. It also has a straightforward interpretation as a saliency signal that drives visual search (Arun, 2012). In a second study, complex searches involving multiple distracters were explained by a linear sum of pairwise dissimilarities measured from simpler searches involving homogeneous distracters (Vighneshvel & Arun, 2013). In a third study, dissimilarities between objects differing in multiple features were found to combine linearly. This was also true for integral features such as the length and width of a rectangle upon including aspect ratio as an additional feature (Pramod & Arun, 2014). Finally, I will describe some recent results extending these findings to more naturalistic objects.
Arun and Olson (2010) conducted a visual search experiment where human subjects were asked to identify, as quickly as possible, an oddball image embedded among multiple distractor images. The reciprocal of the search times for identifying the oddball (in humans) and an ad hoc neuronal dissimilarity index, computed from measured neuronal responses to component images (in macaques), showed a remarkable correlation. In this talk, I will describe a model, an active sequential hypothesis testing model, for visual search. The analysis of this model will suggest a natural alternative neuronal dissimilarity index. The correlation between the reciprocal of the search times and the new dissimilarity index continues to be equally high, and has the advantage of being firmly grounded in decision theory. I will end the talk by discussing the many gaps and challenges in our modeling and statistical analysis of visual search. The talk will be based on ongoing work with Nidhin Koshy Vaidhiyan (ECE) and S. P. Arun (CNS).
In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for signaltointerferenceplusnoiseratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be nonzero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.
Unlike integer factorization, a reducible holomorphic eta quotient may not factorize uniquely as a product of irreducible holomorphic eta quotients. But whenever such an eta quotient is reducible, the occurrence of a certain type of factor could be observed: We conjecture that if a holomorphic eta quotient f of level M is reducible, then f has a factor of level M. In particular, it implies that rescalings and AtkinLehner involutions of irreducible holomorphic eta quotients are irreducible. We prove a number of results towards this conjecture: For example, we show that a reducible holomorphic eta quotient of level M always factorizes nontrivially at some level N which is a multiple of M such that rad(N) = rad(M) and moreover, N is bounded from above by an explicit function of M. This implies a new and much faster algorithm to check the irreducibility of holomorphic eta quotients. In particular, we show that our conjecture holds if M is a prime power. We also show that the level of any factor of a holomorphic eta quotient f of level M and weight k is bounded w.r.t. M and k. Further, we show that there are only finitely many irreducible holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients. Finally, we give an example of an infinite family of irreducible holomorphic eta quotients of prime power levels.
FareyFord Packings are a special case of more general circle packings called Apollonian Circle Packings (ACP). These packings have some very interesting properties, for exmaple, if any four mutually tangent circles have integer curvatures, then so do all others in the packing. This has led to many important problems like prime number theorem in this setting. Kontorovich and Oh explore it from dynamics point of view whereas Bourgain, Fuchs and Sarnak look at them more number theoretically. In this talk, our focus will be on the specialized packings FareyFord Packings. We consider some basic statistics associated to these circles and answer some questions about their distributions and asymptotic behavior. One can ask similar questions in the general setting for ACP, and if time permits, we will discuss it in this talk. Some of this is joint work with Athreya, Chaubey and Zaharescu.
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3manifolds
$\mathbb{R P}^3$
, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$
,
$S^{\hspace{.2mm}2} \times S^1$
, $\TPSS$
and $S^{\hspace{.2mm}3}/Q_8$
,
where $Q_8$
is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq1,q)$ with $4(q+k1)$ vertices for $q \geq 3$
, $k \geq 2$
and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$manifolds from a given presentation $\langle S \mid R \rangle$
with
$\#S=\#R=2$
. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$
with $4(m+n+k3)+ 2\delta_n^2 + 2
\delta_k^2$
vertices for $m\geq 3$
and $m \geq n \geq k \geq 2$
, where
$\delta_i^j$
is the Kronecker delta.
If $n=2$ or $k\geq 3$
and $m \geq 4$
then these crystallizations
are vertexminimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$
, $S^2 \times S^2$
, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4manifolds which are
homeomorphic
but not plhomeomorphic.
TBA
TBA
Weyl discovered a decomposition of the invariant probability measure on a connected compact Lie group, according to elements of a maximal torus and their conjugacy classes. In the case of the unitary group, it allowed Diaconis and Shahshahani to access the distribution of the eigenvalues of a random unitary matrix. We explore these ideas in the lecture.
Consider the problem of n cards labelled 1 through n, lying face up on a table. Suppose two integers a and b are chosen independently and uniformly between 1 and n. The cards labelled a and b are switched. If many such transpositions are made the row of cards will tend to appear in random arrangement. Then question is how many steps are required until the deck is well mixed up (i.e. the permutation is close to random)? Diaconis and Shahshahani used tools of representation theory of symmetric groups to prove that at least 1/2(n log n) steps are required before the deck will be well mixed up for large n. I will explain ideas of their proof and few related problems.
A Kronecker coefficient counts the number of times a representation of a symmetric group occurs in the tensor product of two others. Finding a fast algorithm to determine when a Kronecker coefficient is positive is an open problem. There has been an increased interest in this problem over the last few years as it comes up in the geometric approach to the complexity conjecture P = NP due to Mulmuley and Sohoni. I will explain how a higher dimensional analogue of the Robinson–Schensted–Knuth correspondence relates Kronecker coefficients to the problem of counting the number of integer arrays with specified slice sums.
In this twopart talk, we reconsider Burnside algebras, a classical tool in the theory of finite groups, from a computational perspective. Using modern computer algebra systems, many of the results about these rings that were proved in the second half of the last century can be transformed into effective algorithms.
In this twopart talk, we reconsider Burnside algebras, a classical tool in the theory of finite groups, from a computational perspective. Using modern computer algebra systems, many of the results about these rings that were proved in the second half of the last century can be transformed into effective algorithms.
In this talk, I will give a proof of Alexandrovs theorem on the Gauss curvature prescription of Euclidean convex body. The proof is mainly based on mass transport. In particular, it doesnt rely on pdes method nor convex polyhedra theory. To proceed, I will discuss generalizations of wellknown results for the quadratic cost to the case of a cost function which assumes infinite value.
Four axioms (A1)–(A4) link estimators and distance functions on a set of admissible refinements together and imply optimality of a standard finite element routine on an abstract level with a loop: solve, estimate, mark, and refine. The presentation provides proofs and examples of the recent review due to C. Carstensen, M. Feischl, M. Page, and D. Praetorius: The axioms of adaptivity, Comput. Math. Appl. 67 (2014) 1195 –1253 and so discusses the current literature on the mathematics of adaptive finite element methods. The presentation concludes with an overview over several applications of the set of axioms. If time permits, some recent developments are discussed on ongoing joint work with Hella Rabus on separate marking.
In the first half of the talk, I will define the dimer model on planar graphs and prove Kasteleyn’s groundbreaking result expressing the partition function (i.e. the generating function) of the model as a Pfaffian. I will then survey various results arising as a consequence, culminating in the beautiful limit shape theorems of Kenyon, Okounkov and coworkers.
In the second half, I will define a variant of the monomerdimer model on planar graphs and prove that the partition function of this model can be expressed as a determinant. I will use this result to calculate various quantities of interest to statistical physicists and end with some open questions.
TBA
In this talk we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the Rboundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the L^p boundedness of operators M which are known to be bounded on L^2: We obtain sufficient conditions on the kernel of the operator M so that it satisfies weighted L^p estimates. As an application we prove L^p boundedness of Hermite pseudomultipliers.
I will report on work in progress with Radhika Ganapathy. One wishes to study irreducible smooth' complex representations of symplectic and (split) orthogonal groups over local function fields, i.e., fields of the form F_q((t)). The theory of
close local fields’ proposes to do this by studying the representation theory of these groups over (varying) finite extensions of Q_p. We will discuss an approach to using this philosophy to study the local Langlands correspondence for these groups.
TBA
TBA
TBA
The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. VershikKerov and LoganShepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of BaikDeiftJohansson who related this length to the TracyWidom distribution.
I will start with an overview of classical Iwasawa theory and give a formulation of main conjectures in noncommutative Iwasawa theory. If time permits I will say something about proof of main conjectures in noncommutative Iwasawa theory.
In 1998, Don Zagier studied the ‘modified Bernoulli numbers’ $B_n^{*}$ whose 6periodicity for odd $n$ naturally arose from his new proof of the EichlerSelberg trace formula. These numbers satisfy amusing variants of the properties of the ordinary Bernoulli numbers. Recently, Victor H. Moll, Christophe Vignat and I studied an obvious generalization of the modified Bernoulli numbers, which we call ‘Zagier polynomials’. These polynomials are also rich in structure, and we have shown that a theory parallel to that of the ordinary Bernoulli polynomials exists. Zagier showed that his asymptotic formula for $B_{2n}^{*}$ can be replaced by an exact formula.
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed by the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.
We recall some facts about Rational Homotopy Theory, from both Sullivan and Quillen points of view. We show how to find a Sullivan model of a homogeneous space G/H . Let M be a closed oriented smooth manifold of dimension d and LM= map(S^1, M) denote the space of free loops on M . Using intersection products, Chass and Sullivan defined a product on \\mathbb{H}*(LM)=H{+d} (LM) that turns to be a graded commutative algebra and defined a bracket on \\mathbb{H}_(LM) making of it a Gerstenhaber algebra. From the work of Jones, Cohen, F\\‘elix, Thomas and others there is an isomorphism of Gerstenhaber algebras between the Hochschild cohomology HH^(C^(M), C^(M)) and \\mathbb{H}_(LM) . Using a Sullivan model (\\land V, d) of M , we show that that the Gerstenhaber bracket can be computed in terms of derivations on (\\land V, d) . Precisely, we show that HH^(\\land V, \\land V) is isomorphic to H_(\\land(V)\\otimes \\land Z , D) , where Z is the dual of V . We will illustrate with computations for homogeneous spaces.
Finite difference formulas approximate the derivatives of a function given its values at a discrete set of grid points. Much of the theory for choosing grid points is concerned with discretization errors and rounding errors are ignored completely. However, when the grids become fine the rounding errors dominate. Finding finite difference formulas which optimize the total error leads to a combinatorial optimization problem with a large search space. In this talk, we will describe a random walk based strategy for tackling the combinatorial optimization problem.
simpcomp is an official extension to the computer algebra system GAP, that is, simpcomp is part of every full GAP installation. The software allows the user to compute numerous properties of abstract simplicial complexes such as the f , g and hvectors, the fundamental group, the automorphism group, (co)homology, the intersection form, and many more. It provides functions to generate simplicial complexes from facet lists, orbit representatives, or permutation groups. In addition, functions related to slicings and polyhedral Morse theory as well as a combinatorial version of algebraic blowups and the possibility to resolve isolated singularities of 4manifolds are implemented.
In this talk, I will give an overview over the existing features of the software and also discuss some ongoing projects such as (i) support for Forman’s discrete Morse theory for homology and fundamental group computations, and (ii) an extension of the features to analyse tightness of triangulated manifolds.
In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles.
We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit “real world” polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for “practical” inputs by running just a linear number of linear programs.
Abstract: See https://drive.google.com/file/d/0Bx6ccM3uL81GNHZqQUtic3g5TndEMmF3Y0JKSDl3LTBTeFU4/view?usp=sharing.
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the GromovHausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result  more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
GromovHausdorffLeviProkhorov distance. A distance measure space is a triple
$(X, d, μ)$
, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the DeligneMumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised GromovHausdorffLeviProkhorov distance. The DeligneMumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudoholomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudoholomorphic curves is an important tool in symplectic topology, its applicability is limited due to the nonexistence of a general method to construct pseudoholomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the GaussBonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the DeligneMumford compactification for the space of Riemann surface laminations.
I shall begin with the mapping properties of classical Riesz potentials acting on $L^p$spaces. After reviewing the literature, I shall present our new ‘‘almost’’ Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential) in the socalled supercritical exponent. Finally, I shall show how one could apply these estimates to deduce Sobolev embedding theorems. This is joint work with Daniel Spector and is available on arXiv:1404.1563.
I will prove the following interesting result about finite dimensional complex semisimple algebra. Let A be a finitedimensional complex semisimple algebra without an M_{2}(C) summand and let S be an involutive unital Calgebra antiautomorphism of A. Then there exists an element a in A such that a and S(a) generate A as an algebra. In the proof I will use some basic results in linear algebra
The notion of Ncomplexes goes back to a 1996 preprint of Kapranov, in which he considered chains of composable morphisms satisfying d^N = 0 (as opposed to the usual $d^2 = 0$ which gives the usual chain complexes). Later on, much of the usual homological algebra for chain complexes (homotopy of morphisms, spectral sequences, etc) was generalized to Ncomplexes, mostly by DuboisViolette. Recently in 2014 there has been a burst of interest in this topic with work of Iyama,Kato,Miyachi defining the corresponding Nderived category. We shall begin the talk with simple definitions of $N$ complexes and their homology groups. Then gradually we will move to the paper of Iyama explaining derived category of $N$complexes.
The goal of these talks is to discuss the history, the circumstances, and (the main aspects of) the proof of a recent result of ours, which says: a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism.
Results of this character are motivated by a remarkable theorem of H. Alexander about proper holomorphic selfmaps of the Euclidean ball. The first part of the talk will focus on the meanings of the terms in the above theorem. We will give an elementary proof that the automorphism group of a bounded symmetric domain acts transitively on it. But the existence of an involutive symmetry at each point is restrictive in many other ways. For instance, it yields a complete classification of the bounded symmetric domains. This follows from the work of Elie Cartan.
Earlier results in the literature that are special cases of our result focused on different classes of domains in the Cartan classification, and have mutually irreconcilable proofs. One aspect of our work is a unified proof: a set of techniques that works across the Cartan classification. One of those techniques is the use of an algebraic gadget: the machinery of Jordan triple systems.
We will present definitions and examples related to this machinery in the first part of the talk. In the second part of the talk, we shall state a result on finitedimensional Jordan triple systems that is essential to our work. After this, we shall discuss our proof in greater depth.
See http://www.math.iisc.ernet.in/~arvind/BakloutiAbst.Bangalore14.pdf
In this talk we discuss about the Fourier transforms of functions satisfying Lipschitz conditions of certain order. We cover Fourier transforms on Euclidean spaces, non compact symmetric spaces and also certain hyper groups.
In dynamics, one studies properties of a map from a space to itself, up to change of coordinates in the space. Hence it is important to understand invariants of the map under change of coordinates. An important such invariant is Poincare’s rotation number, associated to invertible maps from the circle to itself. Ghys and others have abstracted the rotation number to give many other important invariants of dynamical systems by viewing it in terms of so called quasihomomorphisms. Quasihomomorphisms are like homomorphisms, except that a bounded error is allowed in the definition. In this expository lecture I will introduce quasihomomorphisms and show some interesting properties, constructions and application, including an alternative construction of the real numbers (due to Ross Street). I shall then show how these can be used to construct dynamical invariants, in particular the rotation number. Only basic algebra and analysis are needed as background for this lecture.
Given n nonzero real numbers v_1,…,v_n, what is the maximum possible number of subsets {i_1,…,i_k} that can have the same subset sum v_{i_1}+…+v_{i_n}? Littlewood and Offord raised the question, showed that the answer is at most 2^nlog(n)/\sqrt{n} and conjectured that the log(n) can be removed. Erdos proved the conjecture and raised more questions that have continued to attract attention, primarily relating the arithmetic structure of v_1,…,v_n and the maximum number of subsets with a given subsetsum.
In the first lecture, we shall review and prove several of these results (due to Littlewood–Offord, Erdos, Moser, Stanley, Halasz, Tao–Vu, Rudelson–Vershynin…) and show an application to random matrices.
In the second part, we start with a question in random polynomials and see how it leads to a variant of the Littlewood–Offord problem that has not been considered before.
Most of the material presented should be accessible to undergraduate students. Much of the lecture is an outcome of my joint study in summer with Sourav Sarkar (ISI, Kolkata).
By using the language of diffusion semigroups, it is possible to define and study classical operators in harmonic analysis. We introduce and develop this idea, and show two applications. First, we investigate fractional integrals and Riesz transforms in compact Riemannian spaces of rank one. Secondly, we carry out the study of operators associated with a discrete Laplacian, namely the fractional Laplacian, maximal heat and Poisson semigroups, square functions, Riesz transforms and conjugate harmonic functions.
This talk will concern asymptotic estimates at infinity for positive harmonic functions in a large class of nonsmooth unbounded domains. These are domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns. We will also mention a few related results in probability, e.g harmonic measure (distribution of exit position of Brownian motion) estimates.
HarishChandra is a mathematician whose name in the history of mathematics is permanently etched in stone. I will begin by giving a brief account of his life, and in the second half of the lecture, explain some of the technical terms used in the first half. This lecture is aimed mainly at Ph.D. students.
In this talk we shall briefly discuss the theory of onesided maximal functions. Onesided maximal function is a variant of the classical HardyLittlewood maximal function. The theory of onesided maximal functions is an active research area. Specially, in the past two decades there has been a lot of research activities in this area.
In this talk we consider a linear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’’s method combined with the technique of twoscale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.
In noncommutative geometry (NCG) one typically treats unital Calgebras as noncommutative compact spaces via GelfandNaimark duality. In various applications of NCG to problems in geometry or topology it is customary to first reformulate these problems in terms of certain (co)homology theories for noncommutative spaces. The celebrated BaumConnes conjecture, that reduces the Novikov conjecture to an assertion in bivariant Ktheory, is a prime example of this principle. However, the category of Calgebras is wellknown to be de ficient from the viewpoint of homotopy theory or index theory. In this talk I am going to first survey certain (co)homology theories for noncommutative spaces, then present my proposed solution to the aforementioned problem, and finally (time permitting) discuss some applications. I will try to keep it nontechnical and accessible to a wide range of audience.
The nonbacktracking matrix of a graph is a nonsymmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also be proposed recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the ErdosRenyi graph G(n,c/n) and other simple inhomogeneous random graphs. This is a joint work with Marc Lelarge and Laurent Maussouli.
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
As a sequel to the talk on July 16th, I shall talk about a recent result (joint work with Shinpei Baba) concerning fibers of the holonomy map from P(S) to the PSL(2,C) character variety. The proof involves some threedimensional hyperbolic geometry, and traintracks.
We discuss the question of rank of symmetric and nonsymmetric matrices when the entries are i.i.d. nondegenerate random variables. In particular we show that for an $n \times n$ symmetric matrix the probability that it is singular is of the order $O(n^{ (1/4) + \epsilon})$. This is joint work with Paulo Manrique and Victor PerezAbreu.
As in other ancient civilisations, mathematics in India began in counting and drawing: numbers and plane geometry. The earliest textually recorded geometry is that of the Sulbasutra (around 800 BC onwards) which are manuals for the construction of Vedic altars, its mathematical high point being the `theorem of the diagonal’, Pythagoras’ theorem to you and me. I will state its earliest formulation and touch briefly on some of the ideas around it as found in the texts.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface  we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal nonadmissible fat graphs. Another major result of this thesis is that there are infinitely many minimal nonadmissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal nonplanar graphs).
Complex projective structures on a surface S arise in the complexanalytic description of Teichmller space. The operation of grafting parametrizes the space P(S) of such structures, and yields paths in Teichmller space called grafting rays. I shall introduce these, and describe a result concerning their asymptotic behavior. This talk shall be mostly expository.
Informally, a preferential attachment scheme is a dynamic (reinforcement) process where a network is grown by attaching new vertices to old ones selected with probability proportional to their weight, a function of their degrees. In these two talks, we give an introduction to these schemes and discuss a couple of approaches to the study of the large scale degree structure of these graphs. One fruitful approach is to view the scheme in terms of branching processes. Another is to understand it in terms of Markov decompositions and fluid limits.
We will use transcendental shiftlike automorphisms of C^k ,k>2 to construct two examples of nondegenerate entire mappings with prescribed ranges. The first example generalizes a result of DixonEsterle in C^2, i.e., we construct an entire mapping of C^k, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. The second example shows the existence of a FatouBieberbach domain in C^k, k>2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and RosayRudin.
Unlike the partial differential equations, the solutions of variational inequalities exhibit singularities even when the data is smooth due to the existence of free boundaries. Therefore the numerical procedure of these problems based on uniform refinement becomes inefficient due to the loss of the order of convergence. A popular remedy to enhance the efficiency of the numerical method is to use adaptive finite element methods based on computable a posteriori error bounds. Discontinuous Galerkin methods play a very important role in the local mesh adaptive refinement techniques.
The main focus in this thesis has been on the derivation of reliable and efficient error bounds for the discontinuous Galerkin methods applied to elliptic variational inequalities. The variational inequalities can be split into two kinds, namely, inequalities of the first kind and the second kind. We study an elliptic obstacle problem and a Signorini contact problem in the category of the first kind, while the frictional plate contact problem in the category of the fourth order variational inequalities of second kind. The mathematical analysis of error estimation in this class of problems crucially depends on a suitable nonlinear smoothing function that enriches the smoothness of the numerical solution. Another remarkable advantage of discontinuous Galerkin methods has been realized in the applications to higher order problems. Numerical experiments support the theoretical results and exhibit optimal convergence.
1961 Frankel in a novel approach used the curvature of Complex Projective space to show complex submanifolds of complimentary dimension intersect. Based on this approach Scheon and Wolfson reproved the BarthLefshetz type theorems using the Morse Theory on the space of paths, in the case the ambient space is a Hermitian Symmetric Space. In this talk we describe how to extend their work to a much larger class of Homogenous spaces.
In this presentation we briefly recall some results on harmonic maps. Subsequently, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, certain partial results regarding the uniqueness of such a soap film are discussed.
An automorphism T of a locally compact group is said to be distal if the closure of the Torbit of any nontrivial element stays away from the identity. We discuss some properties of distal actions on groups. We will also relate distal groups with behaviour of powers of probability measures on it.
An automorphism T of a locally compact group is said to be distal if the closure of the Torbit of any nontrivial element stays away from the identity. We discuss some properties of distal actions on groups. We will also relate distal groups with behaviour of powers of probability measures on it.
This thesis investigates two different types of problems in multivariable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.
I. We introduce the defect sequence for a contractive tuple of
Hilbert space operators and investigate its properties. We show that there
are upper bounds for the defect dimensions. The upper bounds are
different in the noncommutative and in the commutative case. The tuples
for which these upper bounds are obtained are called maximal contractive
tuples. We show that the creation operator tuple on the full Fock space
and the coordinate multipliers on the
DruryArveson space are maximal. We also show that if M is an
invariant subspace under the creation operator tuple on the full Fock
space, then the restriction is always maximal. But the situation is
starkly different for coinvariant subspaces. A characterization for a
contractive tuple to be maximal is obtained. We define the notion of
maximality for a submodule of the DruryArveson module on the
ddimensional unit ball. For $d=1$
, it is shown that every submodule of the
Hardy module over the unit disc is maximal. But for $d>2$
, we prove that any
homogeneous submodule or a submodule generated by polynomials is not
maximal. We obtain a characterization of maximal submodules of the
DruryArveson module. We also study pure tuples and see how the defect
dimensions play a role in their irreducibility.
II. We investigate the following question : Let $(T_1, ....., T_n)$
be a
commuting $n$tuple of bounded linear operators on a Hilbert space $H$. Does
there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$
? We
got some affirmative answers for the doubly commuting
invariant subspaces of the Bergman space and the Dirichlet space over the
unit polydisc. We show that for any doubly commuting invariant subspace
of the Bergman space or the Dirichlet space over polydisc, the tuple
consisting of restrictions of coordinate multiplication operators
always possesses a generating wandering subspace.
The dominant problem in applied mathematics is the application of linear operators and solving linear equations. Dense linear systems arise in numerous applications: Boundary integral equations for elliptic partial differential equations, covariance matrices in statistics, Bayesian inversion, Kalman filtering, inverse problems, radial basis function interpolation, density functional theory, multifrontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements scaling as $O(N^2)$ and extensive computational time to perform matrix algebra scaling as $O(N^2)$ or $O(N^3)$ make computations impractical, where $N$ is the underlying number of degrees of freedom. This talk will discuss new fast algorithms that scale as $O(N)$ for a class of problems, given a prescribed tolerance. Applications in the context of Gaussian processes, Integral equations for electromagnetic scattering, Symmetric factorization for Brownian dynamics, Bayesian inversion, Kalman filtering, multifrontal solvers will also be presented.
In this expository talk, I will focus on CMC surfaces which are not minimal surfaces. I will talk about the link between CMC surfaces and integrable systems. I will then talk about how CMC surfaces fall out of a constrained optimization problem. I will give examples of rotational and helicoidal CMC surfaces and the isometry between them. If time permits I will talk about the classification of CMC surfaces, namely its Weirstrass representation.
We construct a model of capital inflow in a two and multicountry framework. A capitalscarce country, typically a developing country with a high return on capital borrows from a capitalrich country, typically a developed country to finance domestic investment. In the process both the countries gain, raising the world welfare. We formulate the problem in terms of utility maximization with respect to both develop and developing countries’ perspective over infinite time horizon and numerically solve for optimal interest rate, borrowing/lending amount, exchange rate using dynamic programming principle.
In this expository talk, I will first review prequantization of symplectic manifolds. I will then talk of polarizations and geometric quantization. I will then focus on geometric quantization of Kahler manifolds and the Rawnsley coherent states. If time permits I will talk of coadjoint orbits and coherent states.
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the GromovHausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result  more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
GromovHausdorffLeviProkhorov distance. A distance measure space is a triple
$(X, d, μ)$
, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the DeligneMumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised GromovHausdorffLeviProkhorov distance. The DeligneMumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudoholomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudoholomorphic curves is an important tool in symplectic topology, its applicability is limited due to the nonexistence of a general method to construct pseudoholomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the GaussBonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the DeligneMumford compactification for the space of Riemann surface laminations.
One looks at a certain sum G involving the pth roots of unity (where p is a prime number), called the quadratic gaussian sum. It is easy to see that G^2=p, which means that G itself is either the positive or the negative square root of p. Which one ? It took Gauss many years to find the answer and to prove the result. Since then some other proofs of this result have been given, and it has become the central example of what is called the root number of an Lfunction. So the result is very important.
We study homomorphisms $\rho_{V}$($\rho_{V}(f)=\left (
\begin{smallmatrix}
f(w)I_n& \sum_{i=1}^{m} \partial_if(w)V_{i} \\
0 & f(w)I_n
\end{smallmatrix}\right ), f \in \mathcal O(\Omega_\mathbf A)$
) defined on
$\mathcal
O(\Omega_\mathbf A)$
, where $\Omega_\mathbf A$
is a bounded
domain of the form:
for some choice of a linearly independent set of $n\times n$
matrices $\{A_1, \ldots, A_m\}.$
From the work of V. Paulsen and E. Ricard, it follows that if
$n\geq 3$
and $\mathbb B$
is any ball in $\mathbb C^m$
, then there exists
a contractive linear map which is not complete
contractivity. It is known that contractive homomorphisms of the
disc and the bidisc algebra are completely contractive, thanks
to the dilation theorem of B. Sz.Nagy and Ando. However, an
example of a contractive homomorphism of the (Euclidean) ball
algebra which is not completely contractive was given by G. Misra. The
characterization of those balls in $\mathbb C^2$
for which
contractive linear maps which are always comletely contractive
remained open. We answer this question.
The class of homomorphism of the form $\rho_V$
arise from
localization of operators in the CowenDouglas class of $\Omega.$ The
(complete) contractivity of a homomorphism in this class
naturally produces inequalities for the curvature of the
corresponding CowenDouglas bundle. This connection and some of
its very interesting consequences are discussed.
The famous RSA public key cryptosystem is possibly the most studied topic in cryptology research. For efficiency and security purposes, some variants of the basic RSA has been proposed. In this talk, we will first discuss two such RSA variants: Prime Power RSA and Common Prime RSA.
The dynamics of excursions of Brownian motion into a set with more than one boundary point , which no longer have the structure of a Poisson process, requires an extension of Ito’s excursion theory , due to B.Maisonneuve. In this talk we provide a `bare hands’ calculation of the relevant objects  local time, excursion measure  in the simple case of Brownian excursions into an interval (a,b), without using the Maisonneuve theory. We apply these computations to calculate the asymptotic distribution of excursions into an interval, straddling a fixed time t, as t goes to infinity.
In this presentation we briefly recall some results on harmonic maps. Subsequently, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, certain partial results regarding the uniqueness of such a soap film are discussed.
By a triangulation of a topological space $X$, we mean a simplicial complex $K$ whose geometric carrier is homeomorphic to $X$. The topological properties of the space can be expressed in terms of the combinatorics of its triangulation. Simplicial complexes have gained in prominence after the advent of powerful computers as they are especially suitable for computer processing. In this regard, it is desirable for a triangulation to be as efficient as possible. In this thesis we study different notions of efficiency of triangulations, namely, minimal triangulations, tight triangulations and tight neighborly triangulations.
a) Minimal Triangulations: A triangulation of a space is called minimal if
it contains minimum number of vertices among all triangulations of the
space. In general, it is hard to construct a minimal triangulation, or to
decide if a given triangulation is minimal. In this work, we present
examples of minimal triangulations of connected sums of sphere bundles
over the circle.
b) Tight Triangulations: A simplicial complex (triangulation) is called
tight w.r.t field $\mathbb{F}$
if for any induced subcomplex, the induced
homology maps from the subcomplex to the whole complex are all injective.
We normally take the field to be $\mathbb{Z}_2$
. Tight triangulations have
several desirable properties. In particular any simplexwise linear
embedding of a tight triangulation (of a PL manifold) is “as convex” as
possible. Conjecturally, tight triangulations of manifolds are minimal,
and it is known to be the case for most tight triangulations of manifolds.
Examples of tight triangulations are extremely rare, and in this thesis we
present a construction of an infinite family of tight triangulations,
which is only the second of its kind known in literature.
c) Tight Neighborly Triangulations: For dimensions three or more, Novik and Swartz obtained a lower bound on the number of vertices in a triangulation of a manifold, in terms of its first Betti number. Triangulations that meet this bound are called tight neighborly. The examples of tight triangulations constructed in the thesis are also tight neighborly. In addition, it is proved that there is no tight neighborly triangulation of a manifold with first Betti number equal to two.
In this report, after recalling the definition of the M\\obius group, we define homogeneous operators, that is, operators $T$ with the property $\\varphi(T)$ is unitarily equivalent to $T$ for all $\\varphi$ in the M\\obius group and prove some properties of homogeneous operators. Following this, (i) we describe isometric operators which are homogeneous. (ii) we describe the homogeneous operators in the CowenDouglas class of rank 1. Finally, Multiplier representations which occur in the study of homogeneous operators are discussed. Following the proof of Kobayashi, the multiplier representations are shown to be irreducible.
Motivated by applications, we introduce a class of optimization problems with constraints. Difficulties in solving these problems are highlighted. Mathematical developments to overcome these difficulties are discussed.
I will discuss the negative curvature properties of certain intrinsic metrics in complex analysis. The talk will be accessible to graduate students.
The aim of this thesis is to give explicit descriptions
of the set of proper holomorphic mappings between two complex
manifolds with reasonable restrictions on the domain and target
spaces. Without any restrictions, this problem is intractable
even when posed for domains in $C^n$
. We present results for
special classes of manifolds. We study, broadly, two types of
structure results:
I. Descriptive: Our first result is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result of Remmert and Stein adapted to the above setting. However, the presence of factors that have no boundary, or boundaries that consist of a discrete set of points, necessitates the use of alternative techniques. Specifically: we make use of a finiteness theorem of Imayoshi.
II. Rigidity:
A famous theorem of H. Alexander proves the nonexistence of
noninjective proper holomorphic selfmaps of the unit ball
in $C^n,\ n > 1$
. Several extensions of this result for various
classes of domains have been established since the appearance
of Alexander’s result. Our first rigidity result establishes
the nonexistence of noninjective proper holomorphic selfmaps
of bounded, balanced pseudoconvex domains of finite type (in
the sense of D’Angelo) in $C^n,\ n > 1$
. This generalizes a result
in $C^2$
due to Coupet, Pan and Sukhov to higher dimensions. In
higher dimensions, several aspects of their argument do not work.
Instead, we exploit the circular symmetry and a recent result in
complex dynamics by Opshtein.
Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, allencompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply some of our techniques to more general classes of domains. We illustrate this through a rigidity result for certain convex balanced domains whose automorphism groups are only assumed to be noncompact. For the bounded symmetric domains, our key tool is that of Jordan triple systems.
A recent theorem of S. Alesker, S. ArtsteinAvidan and V. Milman characterises the Fourier transform on R^n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. Analogously, we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. This is a joint work with Prof. S. Thangavelu.
A recent theorem of S. Alesker, S. ArtsteinAvidan and V. Milman characterises the Fourier transform on R^n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. Analogously, we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. This is a joint work with Prof. S. Thangavelu.
Generalised polygons are incidence structures that generalise projective planes (generalised triangles) and are closely related to finite groups. The FeitHigman theorem states that any generalised ngon is either an ordinary polygon or n = 2, 3, 4, 6, 8 or 12.
A great progress on the twin prime problem was made last year by Yitang Zhang and it made him famous in mathematical circles almost overnight. He proved the existence of a positive real number M such that there are infinitely many pairs of primes that differ by at most M. The twin prime conjecture predicts that M can be taken to be two. I shall give an overview of the works leading up to Zhang’s spectacular achievement.
Quadrature domains are those on which holomorphic functions satisfy a generalized mean value equality. The purpose of this talk will be to reflect on the Schwarz reflection principle and to understand how it leads to examples of quadrature domains.
Discontinuous Galerkin methods have received a lot of attention in the past two decades since these are high order accurate and stable methods which can easily handle complex geometries, irregular meshes with hanging nodes and different degree polynomial approximation in different elements. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main tools to steer the adaptive mesh refinement. In this talk, we present a posteriori error analysis of discontinuous Galerkin methods for variational inequalities of the first kind and the second kind. Particularly, we study the obstacle problem and the Signorini problem in the category of variational inequalities of the first kind and the plate frictional contact problem for the variational inequality of the second kind. Numerical examples will be presented to illustrate the theoretical results.
Non linear difference equations of order more than one is a relatively new and dynamic area of research in applied mathematics. In particular, the theory of Rational difference equations emerged in the last two decades and is an ongoing challenging field of study. In this talk we will present a brief introduction to nonlinear difference equations with some applications to various fields. Some techniques used in the qualitative analysis of the global character of solutions will be outlined. We will also discuss the results of a recent paper in which the speaker has addressed and generalized, an open problem posed by E. Camouzis and G. Ladas.
Elliptic problems with discontinuous nonlinearity has its own difficulties due to the nondifferentiability of the associated functional. Hence, a generalized gradient approach developed by Chang has been used to solve such problems if the associated functional is known to be Lipchitz continuous. In this talk, we will consider critical elliptic problem in a bounded domain in $\\mathbb{R}^2$ with the simultaneous presence of a Heaviside type discontinuity and a powerlaw type singularity and investigate the existence of multiple positive solutions. Here discontinuity coupled with singularity does not fit into any of the known framework and we will discuss our approach employed to obtain positive solutions.
While discontinuous Galerkin (DG) methods had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. Furthermore, if boundary conditions are not chosen isotropically, convergence can only be expected in relatively weak topology. While the latter result is a property of the transport model, asymptotic analysis reveals, that the forcing into a continuous subspace can be avoided. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened. It has been known for long time, that the so called diffusion limit of radiative transfer is the solution to a diffusion equation; it turns out, that by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit. Finally, we will discuss an efficient and robust multigrid method for the resulting discrete problems.
Discontinuous Galerkin methods have received a lot of attention in the past two decades since these are high order accurate and stable methods which can easily handle complex geometries, irregular meshes with hanging nodes and different degree polynomial approximation in different elements. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main tools to steer the adaptive mesh refinement. In this talk, we present a posteriori error analysis of discontinuous Galerkin methods for variational inequalities of the first kind and the second kind. Particularly, we study the obstacle problem and the Signorini problem in the category of variational inequalities of the first kind and the plate frictional contact problem for the variational inequality of the second kind. Numerical examples will be presented to illustrate the theoretical results.
We discuss the Gromov hyperbolicity of the Kobayashi metric on smooth convex domains in Euclidean space. We prove that if the boundary contains a holomorphic disc then the domain is not hyperbolic. On the other hand we give examples of convex (but not strongly pseudoconvex) domains which are hyperbolic. This is a joint work with Harish Seshadri.
We will describe quite carefully two classes of open threemanifolds. For one of them almost nothing is known about its Riemannian geometry and we will state open questions. For the second class a theorem is available as well as a complete classification.
We discuss the Gromov hyperbolicity of the Kobayashi metric on smooth convex domains in Euclidean space. We prove that if the boundary contains a holomorphic disc then the domain is not hyperbolic. On the other hand we give examples of convex (but not strongly pseudoconvex) domains which are hyperbolic. This is a joint work with Harish Seshadri.
On page 336 in his lost notebook, S. Ramanujan proposes an identity that may have been devised to attack a divisor problem. Unfortunately, the identity is vitiated by a divergent series appearing in it. We prove here a corrected version of Ramanujan’s identity. While finding a plausible explanation for what may have led Ramanujan to consider a series that appears in this identity, we are led to a connection with a generalization of the famous summation formula of Vorono. One of the ramifications stemming from this work allows us to obtain a onevariable generalization of two double Bessel series identities of Ramanujan that were proved only recently. This is work in progress and is joint with Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.
Grothendieck and Verdier introduced the notion of triangulated category to extract homological informations. This structure appear in many branches of Mathematics. Balmer introduced the notion of spectrum of tensor triangulated category as classifying support data. This opens a way for geometric study of theses abstract categorical data. Balmer also defined the structure sheaf on this spectrum and as an application reconstructed large class of schemes from the category of perfect complexes associated with them. In this talk we’ll recall Balmer’s construction of spectrum and its application to reconstruction problem. We’ll also discuss computation of Balmer spectrum for some other tensor triangulated categories obtained in a joint work with Vivek Mallick.
In this talk we address the problem of integrating entire functions (of several complex variables) in ‘polar coordinates’!
Wolfgang Kuhnel introduced the notion of tightness and conjectured that all tight triangulations of closed manifolds are strongly minimal. In a recent paper with B. Datta (European J. Comb. 36 (2014), 294–313), the speaker obtained a very general combinatorial criterion for tightness and found results in partial confirmation of this conjecture. We shall discuss these results.
It is known that there are only finitely many perfect powers in non degenerate binary recurrence sequences. However explicitly finding them is an interesting and a difficult problem for binary recurrence sequences. A recent breakthrough result of Bugeaud, Mignotte and Siksek states that Fibonacci sequences (F_n) given by F_0 = 0; F_1 = 1 and F_{n+2} = F_{n+1} + F_n for n >= 0 are perfect powers only for F_0 = 0; F_1 = 1; F_2 = 1; F_6 = 8 and F_12 = 144.
The talk is based on a joint paper with T.Alberts and J.Quastel. We consider directed polymers in a random environment. However the inverse temperature is scaled with the length of a polymer. It turn out that with a proper critical scaling one can get a nontrivial universal behaviour for the partition function and the end point distribution. Moreover the limiting partition function distribution asymptotically converges to the TracyWidom law as the rescaled inverse temperature tends to infinity.
Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this talk. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues in a completely different way. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.
Most algebraist believe they know Linear Algebra. The purpose of this talk is to indicate that this is not necessarily true. We show a substantial amount of little known basic Linear Algebra and its connection to Algebraic Geometry, in particular to the theory of zerodimensional subschemes of affine spaces, and to Computer Algebra, in particular to the task of solving zerodimensional polynomial systems. Here are some questions which we will answer in this talk: What are the big kernel and the small image of an endomorphism? What are its eigenspaces and generalized eigenspaces if it has no eigenvalues? What is the kernel of an ideal? What is a commendable endomorphism? And what is a commendable family of endomorphisms? How is this connected to curvilinear and Gorenstein schemes? And how can you use this to solve polynomial systems?
In cooperative game theory , Shapley value and the nucleolus are two fundamental solution concepts. They associate a unique imputation for the players whose coalitional worths are given a priori.
If the data defining a problem and at least one solution to the problem lie in the same Archimedean ordered field induced by the data, we say that the problem has order field property. When this property holds one may hope to do only finitely many arithmetic operations on the data to arrive at one such a solution. For example if we start with rational data, the value and a pair of optimal strategies for the players in a zero sum two person game have rational entries. This was first noticed by Herman Weyl , and it was a precursor to solving them via the simplex method. For bimatrix games while Nash exhibited an equilibrium in mixed strategies, it was Vorobev and Kuhn who checked that the order field property holds for bimatrix games. Later Lemke and Howson gave the so called linear complementarity algorithm to locate an equilibrium pair in the same data field. For three person games, Nash with Shapley constructed simple counter examples for lack of order field property. In general stochastic games fail to have order field property.
A Boolean function is a mapping from the set of all binary ntuples to the set {0, 1}. Boolean functions are important building blocks in designing secure cryptosystems known as stream ciphers. Boolean functions also form an important class of linear codes, known as the ReedMuller codes. Over the last few decades, a lot of research has been done on Boolean function for its applications in cryptography and coding theory.
Given a L1 function f over an infinite measure space S how does one estimate the integral of f using statistical tools? In this talk we propose using regenerative sequences with heavy tails to do this. We obtain a consistent estimator and show the rate of convergence is slower than $1/\sqrt{n}$. When S is countable the SSRW works.
I will talk about techniques to approximate real functions such as $x^s,$ $x^{1}$ and $e^{x}$ by simpler functions and how these results can be used in the design of fast algorithms. The key lies in the fact that such approximations imply faster ways to approximate primitives such as $A^sv,$ $A^{1}v$ and $\exp({A})v$, and in the computation of matrix eigenvalues and eigenvectors. Indeed, many fast algorithms reduce to the computation of such primitives, which have proved useful for speeding up several fundamental computations, such as random walk simulation, graph partitioning, solving linear systems of equations, and combinatorial approaches for solving semidefinite programs.
This thesis investigates two different types of problems in multivariable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.
I. We introduce the defect sequence for a contractive tuple of
Hilbert space operators and investigate its properties. We show that there
are upper bounds for the defect dimensions. The upper bounds are
different in the noncommutative and in the commutative case. The tuples
for which these upper bounds are obtained are called maximal contractive
tuples. We show that the creation operator tuple on the full Fock space
and the coordinate multipliers on the
DruryArveson space are maximal. We also show that if M is an
invariant subspace under the creation operator tuple on the full Fock
space, then the restriction is always maximal. But the situation is
starkly different for coinvariant subspaces. A characterization for a
contractive tuple to be maximal is obtained. We define the notion of
maximality for a submodule of the DruryArveson module on the
ddimensional unit ball. For $d=1$
, it is shown that every submodule of the
Hardy module over the unit disc is maximal. But for $d>2$
, we prove that any
homogeneous submodule or a submodule generated by polynomials is not
maximal. We obtain a characterization of maximal submodules of the
DruryArveson module. We also study pure tuples and see how the defect
dimensions play a role in their irreducibility.
II. We investigate the following question : Let $(T_1, ....., T_n)$
be a
commuting $n$tuple of bounded linear operators on a Hilbert space $H$. Does
there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$
? We
got some affirmative answers for the doubly commuting
invariant subspaces of the Bergman space and the Dirichlet space over the
unit polydisc. We show that for any doubly commuting invariant subspace
of the Bergman space or the Dirichlet space over polydisc, the tuple
consisting of restrictions of coordinate multiplication operators
always possesses a generating wandering subspace.
Modular forms, particularly cusp forms, are ubiquitous objects in mathematics. A natural way to understand a cusp form is to study its Lp norms, for in principle the distribution of a function can be recovered from the knowledge of its moments. In this talk I will describe a new bound for the L4 norm of a cusp form of prime level q, as q tends to infinity. This work is joint with Jack Buttcane.
We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP). We introduce microlocal analysis and use it to explain why the wellknown streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP operator is a standard pseudodifferential operator, and so it does not add artifacts. We provide reconstructions to illustrate our mathematical results. This is joint work with Juergen Frikel, Helmholtz Zentrum, Muenchen.
A family of sets is said to be unionclosed if the union of any two sets from the family remains in the family. Frankl’s conjecture, aka the unionclosed sets conjecture, is the remarkable statement that for any finite unionclosed family of finite sets, there exists an element that belongs to at least half of the sets in the family. Originally stated in 1979, it is still wide open. This will be an informal discussion on progress towards the conjecture.
The aim of this thesis is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for domains in C^n. We present results for special classes of manifolds. We study, broadly, two types of structure results:
Fundamental groups of smooth projective varieties are called projective groups. We shall discuss (cohomological) conditions on dimension that force such a group to be the fundamental group of a Riemann surface.
The Lectures will focus on topics that go beyond the classical framework of Hilbert spaces of holomorphic functions (Bergman spaces, Hardy spaces) on the unit ball or more general bounded symmetric domains. The main point is to include vectorvalued functions and even cohomology classes for nonconvex domains. The plan of the lectures is as follows:
We study the problem of characterizing functions, which when applied entrywise, preserve Loewner positivity on distinguished submanifolds of the cone of positive semidefinite matrices. Following the work of Schoenberg and Rudin (and several others), it is wellknown that entrywise functions preserving positivity in all dimensions are necessarily absolutely monotonic. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations are known only in the $n=2$ case.
Tits theory of spherical buildings gives a uniform geometric context to study all finite simple groups (except the alternating groups and the sporadic simple groups) and simple algebraic groups. More generally, the theory of buildings is central to the Lie theory associated with infinite root systems. These structures are `built of’ two basic objects: Coxeter complexes and Moufang generalized polygons. The generalized polygons (which includes projective planes) are rank 2 geometries (incidence geometries with 2 kind of objects  points and lines  and an incidence relation among them) whose classification is fundament, difficult and perhaps impossible.
Tits theory of spherical buildings gives a uniform geometric context to study all finite simple groups (except the alternating groups and the sporadic simple groups) and simple algebraic groups. More generally, the theory of buildings is central to the Lie theory associated with infinite root systems. These structures are `built of’ two basic objects: Coxeter complexes and Moufang generalized polygons. The generalized polygons (which includes projective planes) are rank 2 geometries (incidence geometries with 2 kind of objects  points and lines  and an incidence relation among them) whose classification is fundament, difficult and perhaps impossible.
A multiqueue system serves a multiclass population. Classes differ in their valuation of time. Oblivious routing in which routing is not informed by current queue status or past decisions is assumed. First, we explore the structure of the routing fractions that maximise social welfare. We then analyse the case customer are strategic and the queues have an admission price. We then argue that admission prices can be set to achieve optimal routing.
The conjecture is about various subgroup that can occur as the inertia subgroup of an etale Galois cover of an affine line at a point above infinity over an algebraically closed field of positive characteristic. I will explain this conjecture and mention some positive results supporting this conjecture.
This talk will be an elementary introduction to (Hilbert) module approach to operator theory. We explore the relationship of the classical von NeumannWold decomposition theorem and the BeurlingLaxHalmos theorem for isometries. We will also discuss a unified approach to the invariant subspace problem for bounded linear operators on Hilbert spaces. The talk will be accessible to general audience including graduate students.
This talk will be an elementary introduction to (Hilbert) module approach to operator theory. We explore the relationship of the classical von NeumannWold decomposition theorem and the BeurlingLaxHalmos theorem for isometries. We will also discuss a unified approach to the invariant subspace problem for bounded linear operators on Hilbert spaces. The talk will be accessible to a general audience including graduate students.
A conjecture of Gelander states that there is an effective triangulation in a compact deformation retract of a given locally symmetric space, giving linear bounds on the full homotopy type. This talk will explain some of this construction.
A conjecture of Gelander states that there is an effective triangulation in a compact deformation retract of a given locally symmetric space, giving linear bounds on the full homotopy type. This talk will explain some of this construction.
The mean curvature flow is a process under which a submanifold is deformed in its mean curvature vector’s direction. This process received more attention since it is an efficient way to construct submanifold which minimizes the volume : it is the negative gradient flow of volume functional. In this firs talk. I will discuss about basic tools in the study of mean curvature and describe some examples .
We consider nonlocal currents in the context of quantized affine algebras. In two special cases, these currents can be identified with configurations in the sixvertex and Izergin–Korepin nineteenvertex models. Mapping these to their corresponding Temperley–Lieb loop models, we directly identify nonlocal currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for nonlocal currents. Joint with Y. Ikhlef, R. Weston and M. Wheeler.
Around 1960, Grothendieck developed the theory of descent. The aim of this theory is to construct geometric objects on a base space – in particular bundles, sheaves and their sections – in terms a generalized covering space which is visualized to lie upstairs' over the base space. The objects over the base space are obtained by
descending’ similar objects from the covering space. In late 1960searly 1970s, Deligne addressed the problem of how to understand cohomology of the base space via cohomology of a covering, by this time descending cohomology classes (instead of just descending global sections of sheaves, which is the case of 0th cohomology). The theory developed by Deligne, known as `Cohomological Descent’ has found important applications to Hodge theory and to cohomology of algebraic stacks. In these two expository lectures, I will begin with a quick look at Grothendieck’s theory of descent, and then go on to give a brief introduction to Deligne’s theory of Cohomological Descent.
Around 1960, Grothendieck developed the theory of descent. The aim of this theory is to construct geometric objects on a base space – in particular bundles, sheaves and their sections – in terms a generalized covering space which is visualized to lie upstairs' over the base space. The objects over the base space are obtained by
descending’ similar objects from the covering space. In late 1960searly 1970s, Deligne addressed the problem of how to understand cohomology of the base space via cohomology of a covering, by this time descending cohomology classes (instead of just descending global sections of sheaves, which is the case of 0th cohomology). The theory developed by Deligne, known as `Cohomological Descent’ has found important applications to Hodge theory and to cohomology of algebraic stacks. In these two expository lectures, I will begin with a quick look at Grothendieck’s theory of descent, and then go on to give a brief introduction to Deligne’s theory of Cohomological Descent.
The KadisonSinger problem is a question in operator theory which arose in 1959 while trying to make Dirac’s axioms for quantum mechanics mathematically rigorous. Over the course of several decades, this question was reduced to several equivalent conjectures about finite matrices, and shown to have significant implications in applied mathematics, computer science, and various branches of pure mathematics.
We prove that for any finite dimensional vector space $V$ over an algebraically closed field $K$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the order $G$ is a unit in $K$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes G}$, where $\mathcal{O}(1)$ denotes the ample generator of the Picard group of $\mathbb P(V)$. We also prove that for the standard representation $V$ of the Weyl group $W$ of a semisimple algebraic group of type $A_n , B_n , C_n , D_n , F_4$ and $G_2$ over $\mathbb{C}$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes W}$, where $V^m$ denote the direct sum of $m$ copies of $V.$
In this lecture we will study the well posedness problem for the nonlinear Schr\\{o}dinger equation for the magnetic Laplacian on $\\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\\C^n$. We establish the well posednes in certain first order Sobolev spaces associated to the twisted Laplacian, and also in $L^2(\\C^n)$.
Points, lines and circles are among the most primitive and fundamental of mathematical concepts, yet few geometric objects have generated more beautiful and nontrivial mathematics. Deceptively simple in their formulation, many classical problems involving sets of lines or circles remain open to this day. I will begin with a sample that has spearheaded much of modern research, and explore connections with analysis, geometry, combinatorics  maybe also parallel parking.
In recent times, exponential type cost structure has become popular in control theory. In this talk we formulate and discuss a risk sensitive type control problem for a multiclass queuing system under the moderate deviation scaling. It is known that the rate function corresponding to the moderate deviation scaling is of Gaussian type. This property of the rate function is often useful for mathematical analysis. We show that the limiting game corresponding to our control problem is solvable. Also the limiting game has a similarity to the wellstudied Brownian control problems. This problem is also related to a conjecture of Damon Wischik (2001). The main difficulty in working with G/G/1 queuing network is that the underlying state dynamics is not Markov. Markov property has proven useful for these type of problems (see e.g., AtarGoswamiShwartz(2012)). The standard way to solve these problems is to look at the pde associated with the state dynamics and sandwich the limiting value between the upper and lower value of the game. This technique does not work when the state dynamics is not Markov. We will see that the special structure of the rate function and moderate deviation settings will be helpful to overcome such difficulties. Extension to manyserver models will also be discussed.
In this talk I will mainly focus on the existence and uniqueness of weak solutions to the nonlinear continuous coagulation and multiple fragmentation equations. In addition, the convergence analysis of two numerical methods (the fixed pivot technique and the cell average technique) for solving nonlinear coagulation or Smoluchowski equation is introduced. At the end, the convergence rates obtained from both the techniques are compared and mathematical results of the convergence analysis are also demonstrated numerically.
We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain lowdimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics. This works gives some possible notions of `randomness’ of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Leliever, and with Y.Cheung.
Basic Algebraic Topology is built on considering objects (such as differential forms) up to boundary (e.g., exterior derivative). However, there is more structure in chains and boundaries that can be extracted by not immediately passing to quotients. We will discuss elementary examples of this, such as the Hopf invariant and Milnor’s higher linking.
The classical (as well as axiomatic) potential theory has been developed in the context of the study of analytic functions on Riemann surfaces, Newton potentials and Markov processes. This talk aims to develop a discrete version of potential theory on infinite graphs, based on the examples of electrical networks and infinite graphs.
We will continue our discussion of the Homotopy Transfer Theorem for Ainfinity algebras. We will introduce the notion of a minimal model for a dg algebra. This will allow us to relate quasiisomorphisms between strict dg algebras and quasiisomorphisms between Ainfinity algebras.
Homotopy Type Theory (HoTT), developed recently in a large collaboration centered at IAS, Princeton, combines elements from type theory, logic and topology to give alternative foundations of mathematics which are much closer to mathematical practice (useful for Automated Reasoning). HoTT also gives new insights into topology.
Homotopy Type Theory (HoTT), developed recently in a large collaboration centered at IAS, Princeton, combines elements from type theory, logic and topology to give alternative foundations of mathematics which are much closer to mathematical practice (useful for Automated Reasoning). HoTT also gives new insights into topology.
We will introduce the notion of a differential graded algebra that is not strictly associative, but associative up to chain homotopy. Such objects are called Ainfinity algebras or strong homotopy associative algebras. The goal is to state and prove the Homotopy Transfer Theorem, which states that any chain complex that is chain homotopic to an Ainfinity algebra has an Ainfinity structure and the maps in the chain homotopy can be lifted to a map of Ainfinity algebras.
Starting with the Euler characteristic in graph theory/combinatorics, we trace a brief history, first viewing it as an Euler class in topology, then as an obstruction to splitting of vector bundles and finally get to the more recent notion of the Euler class in algebra/geometry and its use as an obstruction to the splitting of projective modules. This recent notion has two approaches, Euler class groups and ChowWitt groups, the second of which uses the GerstenWitt complex as mentioned in the title. Time permitting, we hope to state results about both approaches.
HOTT (homotopy type theory) is logic built on type theory (mostly from Computer Science) and ideas from topology to give foundations of mathematics that are very elegant and much closer to mathematical practice. This makes HOTT very useful for computer proof systems, and also gives a very nice new synthetic treatment of homotopy theory.
HOTT (homotopy type theory) is logic built on type theory (mostly from Computer Science) and ideas from topology to give foundations of mathematics that are very elegant and much closer to mathematical practice. This makes HOTT very useful for computer proof systems, and also gives a very nice new synthetic treatment of homotopy theory.
Integral geometry is a field of mathematics that studies inversions and various properties of transforms, which integrate functions along curves, surfaces and hypersurfaces. Such transforms arise naturally in numerous problems of medical imaging, remote sensing, and nondestructive testing. The most typical examples include the Radon transform and its generalizations. The talk will discuss some problems and recent results related to generalized Radon transforms, and their applications to various problems of tomography.
The informal seminar on mathematical reasoning continues. This meetingwill be selfcontained (i.e., not dependent on previous sessions).
Many commonly used mixed finite elements for the Stokes problem only conserve mass approximately. We show that if we supplement polynomial basis functions with divergence free rational functions in the finite element method we can conserve mass exactly. Similarly, we show how to supplement polynomial basis in the finite element space with rational functions in when approximating linear elasticity problems in order to preserve angular momentum exactly. This is joint work with Michael Neilan (University of Pittsburgh).
Let u be a weak solution to a pharmonic system with vanishing Neumann data on a portion of the boundary of a domain which is convex. We show that subsolution type arguments for some uniformly elliptic PDE’s can be used to deduce that the modulus of the gradient is bounded depending on the Lipschitz character of the domain. In this context, I would like to mention that classical results on the boundedness of the gradient require the domain to be C^{1, Dini}. However, in our case, since the domain is convex, one can make use of the fundamental inequality of Grisvard which can be thought of as an analogue of the use of the barriers for Dirichlet problems in convex domains. Our arguments replaces an argument based on level sets in recent important works of Mazya, CianchiMazya and GengShen involving similar problems. I also intend to indicate some open problems in the regularity theory of degenerate elliptic and parabolic systems. This is a joint work with Prof. John L. Lewis.
We discuss the renormalization and reverse renormalization constructions used in studying the dynamics of germs of holomorphic diffeomorphisms fixing the origin in C with linear part a rotation. Such a germ is said to be linearizable if it is analytically conjugate to its linear part.
Motivated by rigidity problems for negatively curved manifolds, we study Moebius and conformal maps $f : \\partial X \\to \\partial Y$ between boundaries of CAT(1) spaces $X, Y$ equipped with visual metrics.

The systole of a compact Riemannian manifold (M,g) is the length of the shortest non contractible loop of M, it is attained by a periodic geodesic. A systolic inequality is a lower on the volume of any Riemannian metric depending only on the systole. If one consider the systole as a kind of ‘belt size’ of (M,g), a systolic inequality just says ‘the bigger the belt, the bigger the guy’. In this talk, we will discuss systolic inequalities for surfaces. We will prove optimal results for the torus and the projective plane (which go back to Loewner and Pu in the 50’s) and non optimal results for higher genus surfaces (due to Hebda and Gromov in the 80’s). This topic involve a nice mixture of metric geometry and elementary topology. If time permits, we will say a few words about higher dimensions.
In the seqries of 4 lectures, we will cover some aspects of the LittlewoodOfford problem. This problem concerns the anticoncentration phenomenon of sums of independent random variables and has applications to invertibility of random matrices and statistics of real zeros of random polynomials. The subject overlaps probability, combinatorics, additive combinatorics and some algebra.
The geometry of Metric spaces equipped with a probability measure is a very dynamic field. One motivation for the study of such spaces is that they are the natural limits of Riemannian manifolds in many contexts. In this talk, I will introduce basic properties of metric measure spaces and the GromovProhorov distance on them. I will also discuss joint work with Manjunath Krishnapur in which we show that independently sampling points according to the given measure gives an asymptotically biLipschitz correspondence between Metric measure spaces and Random matrices. Finally, I will briefly discuss work with Divakaran in which we study the compactification of the Moduli space of Riemann surfaces in terms of metric measure spaces.
In this thesis we investigate single and multiplayer stochastic dynamic optimization problems in both discrete and continuous time. In the multiplayer setup we investigate zerosum games with both complete and partial information. We study partially observable stochastic games with average cost criterion and the state process being discrete time controlled Markov chain. We establish the existence of the value of the game and also obtain optimal strategies for both players. We also study a continuous time zerosum stochastic game with complete observation. In this case the state is a pure jump Markov process. We investigate the finite horizon total cost criterion. We characterise the value function via appropriate Isaacs equations. This also yields optimal Markov strategies for both players.
We discuss two topics in this talk. First we study compact Ricciflat 4 manifolds without boundary and obtain pointwise restrictions on curvature (not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricciflat Khler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.
Tools from additive combinatorics are finding their way into numerous areas of mathematics and applied mathematics, and in particular have been central to recent developments in both random matrix theory and harmonic analysis. The goal of this short course is to understand some of these tools. (If time permits and the audience is willing to pitch in with some talks, we may also cover selected applications.)
Any real physical problem arising in fluid mechanics, when it is translated to mathematical approach generally govern a differential equation with some assumptions. The solution for such a study of the system is solved using analytical or numerical techniques. An attempt has been made to understand the characteristics or behavior or analysis of the physical system using different methods like Lighthills method, perturbation methods, rational approximation or Pad approximation, shooting technique and also highlights on the advantages and limitations of each method are discussed.
Let ${H}$ be a separable Hilbert space over the complex field. The class $S := \lbrace N_{M} : N$ is normal on ${H}$ and ${M}$ is an invariant subspace for $N \rbrace$ of operators was introduced by Halmos and consists of subnormal operators. Each subnormal operator possesses a unique minimal normal extension $\hat{N}$ as shown by Halmos. Halmos proved that $\sigma(\hat{N}) \subseteq \sigma(S)$ and then Bram proved that $\sigma(S)$ is obtained by filling certain number of holes in the spectrum $\sigma(\hat{N})$ of the minimal normal extension $\hat{N}$ of a subnormal operator in ${S}$.
I will describe work of Jonathan Williams giving a description of smooth 4manifolds in terms of certain collections of closed curves on surfaces and certain moves on them. This builds on constructions using Lefschetz pencils and their variants, but is a completely elementary description.
(Bio)chemical reaction networks are used in systems biology to model gene regulatory, protein, metabolic, and other cellular networks. Since existence of multiple steady states (MSS) provides the underpinnings for switching in chemical reaction networks, it is a fundamental problem to determine which network structures permit MSS. There exist several criteria which, when satisfied, establish that a network does not permit MSS regardless of the parameter values. On the other hand, results that establish that a network does permit MSS are rare. I will describe our recent work which provides a novel approach towards solving this problem. In the second part of the talk, I will describe stochastic switching which occurs in a network of neurons which is responsible for the distinct brain states of sleep and wake and for the transitions between the two states.
(Bio)chemical reaction networks are used in systems biology to model gene regulatory, protein, metabolic, and other cellular networks. Since existence of multiple steady states (MSS) provides the underpinnings for switching in chemical reaction networks, it is a fundamental problem to determine which network structures permit MSS. There exist several criteria which, when satisfied, establish that a network does not permit MSS regardless of the parameter values. On the other hand, results that establish that a network does permit MSS are rare. I will describe our recent work which provides a novel approach towards solving this problem. In the second part of the talk, I will describe stochastic switching which occurs in a network of neurons which is responsible for the distinct brain states of sleep and wake and for the transitions between the two states.
We consider a real analytic map f from R^4 to R^2 with a singularity at 0. One method to investigate the singularity is to work on its link L. If 0 is an isolated singularity then it is well known that L is a fibered link in the 3sphere S^3. This describes immediately a contact structure on S^3. In this talk we suggest that even if 0 is not an isolated singularity, we can associate to the singularity a welldefined stable Hamiltonian structure on S^3, provided that f describes a Seifert fibration on S^3, L being a multilink in this fibration. This condition is satisfied, for example, when f is complex analytic or f is given as g\\bar{h} with g and h being complex analytic. If the link is already fibered, the stable Hamiltonian structure is nothing but the contact structure mentioned above. Our construction is in fact far more general: given a Seifert multilink (not necessarily associated to a map from R^4 to R^2) in a Seifert fibered 3manifold, we build a welldefined stable Hamiltonian structure on the 3manifold.
In this talk, we shall discuss the complex of HNN  extensions associated to the free group F_n of finite rank n. We shall sketch a proof the following result The group of simplicial automorphisms of this complex is isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n of rank n.
This talk will be on two themes that illustrate the rigidity and regularity of holomorphic mappings. The first part will deal with results concerning the smoothness of continuous CR (Cauchy – Riemann) mappings; in particular, that of Lipschitz continuous CR mappings from hextendible/semiregular hypersurfaces into certain Levi corank one hypersurfaces, in C^n. The second part will deal with the classification of Kobayashi hyperbolic, finite type rigid polynomial domains with abelian automorphism group in C^3.
One of the main goals of classical metric Diophantine approximation is to quantify the denseness of the rational numbers in the real numbers, or more generally, of Q^d in R^d. An equally natural problem is to quantify the denseness of the rational points on the sphere, and more generally, rational points in other compact and noncompact algebraic subvarieties in R^d. We will describe a solution to this problem for a large class of homogeneous varieties.
We will trace the evolution of the mean ergodic theorem, from its original formulation by vonNeumann to some very recent formulations valid in the context of algebraic groups and their lattice subgroups. We will then present a variety of recent applications of mean ergodic theorems, particularly to counting lattice points.
The affine stratification number of a variety is a measure of how far a variety is from being affine and how close it is to being projective. I shall talk about a certain filtration on the moduli space of curves and the affine stratification number of the open sets occurring in the filtration. This will lead to some cohomology computations where we shall make use of the natural operad structure on the homology of the moduli spaces as well as the some mixed hodge theory.
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. We shall discuss the relationship between Jones Polynomial and representation of Knots through Tangles.
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. We shall discuss the relationship between Jones Polynomial and representation of Knots through Tangles.
The curvature of a contraction T in the CowenDouglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the CowenDouglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the CowenDouglas class.
The talk will consist of two distinct parts. We will firt study the hyperbolicity of some domains in an almost complex manifold (M,J). In the second part we will study the question of the embeddability of compact almost complex manifolds in complex projective spaces.
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. Though can be computed easily in terms of what are called skein relations, it originally arose from a studying a particular kind of a ‘trace’ of braid representations into an algebra derived as the quotient of a group ring of braid group. We shall discuss this construction of Jones Polynomial in the first talk. In the second part, we shall discuss the construction of Jones Polynomial from the tangle representation of Knots.
Grothendieck published an extraordinary paper entitled Resume de la theorie metrique des produits tensoriels topologiques in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality. Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We reformulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class.
Ricci flow is a PDE that deform the metric of a Riemannian manifold in the direction of its Ricci curvature. For compact smooth manifolds, there is a well established existence and uniqueness theory. However for some applications it can be useful to consider Ricci flows of nonsmooth spaces, or metric spaces whose distance doesn’t come from a Riemannian metric. We will show that existence and uniqueness holds for the Ricci flow of compact singular surfaces whose curvature is bounded from below in the sense of Alexandrov.
The WhitneyGrauert theorem states that regular curves in R^2 (i.e. immersions of S^1) are classified up to regular homotopy by the winding number of the derivative. I will present Eliashberg and Geiges’s simple proof of this theorem, in which regular plane curves are realized as projections of curves in R^3 satisfying a certain geometric condition (they’ll be Legendrian curves in the standard contact structure). This is one of the simplest examples of the general pattern of lifting a purely topological problem to an equivalent but simpler problem in contact/symplectic geometry. No knowledge of contact geometry is assumed; the only prerequisite is differential forms on R^3.
I will discuss some constructions, results, questions and applications relating to the space of hyperbolic structures on a surface and its natural compactifications. I will assume that the audience understands what `hyperbolic structures on surfaces’ means.
We characterize the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimensionfree bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
We characterize the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimensionfree bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
We study risksensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterise the value function via HJB equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.
This is an informal lecture to celebrate the unprecedented event that I understand some piece of the work of some economists.
This is an informal lecture to celebrate the unprecedented event that I understand some piece of the work of some economists.
String topology is the study of the free loop space of a manifold LM. The loop product, defined on the homology of LM, is described intuitively as a combination of the intersection product on M and loop concatenation in the based loop space of M. However, since the intersection product is welldefined only on transversally intersecting chains, this description is incomplete. Brown’s theory of twisting cochains provides a chain model of a bundle in terms of the chains on the base and chains on the fiber. We extend this theory so that it can be applied to provide a model of the free loop space. We give a precise definition of the loop product defined at the chain level.
Google has recently released open source software called `coursebuilder’ (at https://code.google.com/p/coursebuilder/) to make interactive online courses consisting of videos interleaved with quizzes. In this brief presentation, I will show as an example a coursebuilder course I made and describe the process of making such courses. My goal is to convince that making online courses is both easy and worthwhile.
Necessity to understand the role of additional food as a tool in biological control programs is being increasingly felt, particularly due to its Ecofriendly nature. In this present talk, we develop/analyse a variation of standard predatorprey model with Holling type II function response which presents predatorprey dynamics in presence of some additional food to predators. The aim is to study the consequences of providing additional food on the system dynamics. A thorough mathematical analysis reveals that handling times for the available foods play a vital role in determining the eventual state of the system. It is interesting to observe that by varying the quality (characterised by the handling times) and quantity of additional food we can not only control and limit the prey, but also limit and eradicate the predators. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. We further study the controllability aspects of the predatorprey system by considering quality of the additional food as the control variable. Control strategies are offered to steer the system from a given initial state to a required terminal state in a minimum time by formulating Mayer problem of optimal control. It is observed that an optimal strategy is a combination of bangbang controls and could involve multiple switches. Properties of optimal paths are derived using necessary conditions for Mayer problem. In the light of the results evolved in this work it is possible to eradicate the prey from the system in the minimum time by providing the predator with high quality additional food, which is relevant in the pest management. In the perspective of biological conservation this study highlights the possibilities to drive the state to an admissible interior equilibrium (irrespective of its stability nature) of the system in a minimum time.
Credit risk refers to the potential losses that can arise due to the changes in the credit quality of financial instruments. There are two approaches to pricing credit derivatives, namely the structural and the reduced form or intensity based models. In the structural approach explicit assumptions are made about the dynamics of a firm’s assets, its capital structure, debt and share holders. A firm defaults when its asset value reaches a certain lower threshold, defined endogenously within the model. In the intensity based approach the firm’s asset values and its capital structure are not modelled at all. Instead the dynamics of default are exogenously given through a default rate or intensity.
We study several kinds of matching problems between two point processes. First we consider the set of integers $\\mathbb{Z}$. We assign a color red or blue with probability 1/2 to each integer. We match each red integer to a blue integer using some algorithm and analyze the matched edge length of the integer zero. Next we go to $\\mathbb{R}^{d}$. We consider matching between two different point processes and analyze a typical matched edge length $X$. There we see that the results vary significantly in different dimensions. In dimensions one and two (d=1,2), even $E[X^{d/2}]$ does not exist. On the other hand in dimensions more than two (d>2), $E[\\exp(cX^{d})]$ exist, where $c$ is a constant depends on $d$ only.
Rational points on elliptic curves have found applications in cryptography and in the solution to some problems dating back to antiquity. However, we still do not know how to find an elliptic curve with as many points as possible. In this talk, we will see how the theory of modular forms (of Ramanujan) along with a recently developed theory enables one to understand this problem. Along the way, we will see how congruences between coefficients of modular forms shed light on this problem.
We will discuss some questions concerning the existence of families of Siegel modular forms with some prescribed nonzero Fourier coefficients. If time permits, we also plan to discuss the question of characterizing cusp forms by the growth of their Fourier coefficients. Both are recent joint works with Siegfried Boecherer.
We will discuss some questions concerning the existence of families of Siegel modular forms with some prescribed nonzero Fourier coefficients. If time permits, we also plan to discuss the question of characterizing cusp forms by the growth of their Fourier coefficients. Both are recent joint works with Siegfried Boecherer.
Firstly, regarding Carry Value Transformation (CVT) some mathematical observations will be discussed. In using mathematical tools in Genomics we adopted twoway path. One is model based, another is issue based. Both these approaches will be discussed on using Human Olfactory receptors.
In this talk we will discuss some of the impact of Subramanian Chandrasekhar’s work on modern applied mathematics. One of the highlight of the talk will be a discussion of Chandrasekhar’s radiative transfer theory where he developed a number of breathtaking mathematical structures such as nonlinear integral equations for the Chandrasekhar H functions and X, Y functions as well as infinite dimensional Riccati (integropartial differential differential ) equations for the scattering matrix long before the infinite dimensional systems theory.
Some basic algebraic structures on the set of 1dimensional IVTs are introduced. Then discrete dynamical systems are defined through IVT maps and a report on convergence of the dynamical systems has been made. Finally, some problems which are unsolved yet in the domain are discussed.
Polynomial knots were introduced to represent knots in 3 space by simple polynomial equations. In this talk we will discuss how the degrees of these equations can be used in deriving information of some important knot invariants.
We wish to study those domains in $\mathbb{C}^n$, for $n\geq 2$, the socalled domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\mathbb{C}$ by discussing some examples of special types of domains in $\mathbb{C}^n$, $n\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullen’s construction of a domain (not necessarily in $\mathbb{C}^n$) spread over $\mathbb{C}^n$, the socalled envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\mathbb{C}^n$.The aforementioned characterization (holomorphic convexity) is very difficult to check. This calls for other (equivalent) criteria for a domain in $\mathbb{C}^n$, $n\geq 2$, to be a domain of holomorphy. We shall survey these criteria. We shall sketch those proofs of equivalence that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain $\Omega\subset \mathbb{C}^n$, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than $\Omega$ to which all functions holomorphic on $\Omega$ continue analytically. One tool that is used most often in such constructions is called “Kontinuitaetssatz”. It has been invoked, without any clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a folk theorem. We shall provide a precise statement of this folk Kontinuitaetssatz and give a proof of it.
I will discuss some instances of the interplay between dynamics on homogeneous spaces of algebraic groups and Diophantine approximation, with an emphasis on recent developments. The latter includes joint work with Gorodnik and Nevo, and with Einsiedler and Lytle.
The curvature of a contraction T in the CowenDouglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the CowenDouglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the CowenDouglas class.
The $4$genus of a knot is an important measure of complexity, related to the unknotting number. A fundamental result used to study the $4$genus and related invariants of homology classes is the \emph{Thom Conjecture}, proved by KronheimerMrowka, and its symplectic extension due to OzsvathSzabo, which say that \textit{closed} symplectic surfaces minimize genus.
We shall discuss a new result that relates grafting, which are certain deformations of complex projective structures on a surface, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any fixed Fuchsian holonomy, such geometric structures are dense in moduli space.
We shall discuss a new result that relates grafting, which are certain deformations of complex projective structures on a surface, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any fixed Fuchsian holonomy, such geometric structures are dense in moduli space.
Let Y be a complex, projective manifold and X a smooth hyperplane section in Y. Given a submanifold Z in X, under what conditions is it cut out by a submanifold Z’ in Y. An analogous quesion can be asked for vector bundles on X: namely when is a bundle on X, the restriction of a bundle on Y.
In this thesis we study some Questions on vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension $\\geq 2$, we study the extension problem of Vector bundles. We try to find some conditions under which a vector bundle over an ample divisor of nonsingular projective variety, extends as a vector bundle to an open set containing that ample divisor.
We derive a form of the KPZ equation, which governs the fluctuations of a class of interface heights, in terms of a martingale problem, as the scaling limit of fluctuation fields with respect to some particle systems such as zero range processes. This is joint work in progress with P. Goncalves and M. Jara.
Tempered stable distributions were introduced in Rosinski 2007 as models that look like infinite variance stable distributions in some central region, but they have lighter (i.e. tempered) tails. We introduce a larger class of models that allow for more variety in the tails. While some cases no longer correspond to stable distributions they serve to make the class more flexible, and in certain subclasses they have been shown to provide a good fit to data. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs. We will also characterize the weak limits of sequences of tempered stable distributions. If time permits, we will motivate why distributions that are stablelike in some central region but with lighter tails may show up in applications.
von Neumann algebras are non commutative analogue of measure spaces. The study of maximal abelian subalgebras (masas) in finite von Neumann algebras is classical to the subject from its birth and is closely tied up with Ergodic theory. Dixmier introduced two types of masas, namely, regular (Cartan) and singular. The philosophies of these two kinds of masas until recently were regarded as being different from each other. After an introduction on the subject, we justify that the existing theories can be unified. Using techniques from Free Probability and playing with suitable amenable groups we exhibit: For each subset S of $\\mathbb{N}$ (could be empty), there exist uncountably many pairwise non conjugate (by automorphism) singular masas in the free group factors for each of which $S\\cup {\\infty\\}$ arises as its Pukanzsky invariant (multiplicity function). If time permits, some other issues related to mixing, coarse bimodules, and Banach’s problem on simple Lebesgue spectrum will be addressed.
von Neumann algebras are non commutative analogue of measure spaces. The study of maximal abelian subalgebras (masas) in finite von Neumann algebras is classical to the subject from its birth and is closely tied up with Ergodic theory. Dixmier introduced two types of masas, namely, regular (Cartan) and singular. The philosophies of these two kinds of masas until recently were regarded as being different from each other. After an introduction on the subject, we justify that the existing theories can be unified. Using techniques from Free Probability and playing with suitable amenable groups we exhibit: For each subset S of $\\mathbb{N}$ (could be empty), there exist uncountably many pairwise non conjugate (by automorphism) singular masas in the free group factors for each of which $S\\cup {\\infty\\}$ arises as its Pukanzsky invariant (multiplicity function). If time permits, some other issues related to mixing, coarse bimodules, and Banach’s problem on simple Lebesgue spectrum will be addressed.
This talk will be on two themes that illustrate the rigidity and regularity of holomorphic mappings. The first part will deal with results concerning the smoothness of continuous CR (Cauchy – Riemann) mappings; in particular, that of Lipschitz continuous CR mappings from hextendible/semiregular hypersurfaces into certain Levi corank one hypersurfaces, in C^n. The second part will deal with the classification of Kobayashi hyperbolic, finite type rigid polynomial domains in C^3.
A simplicial cell complex is roughly speaking a CW complex whose cells are all simplices. This notion is equivalent to that of simplicial poset in combinatorics. A simplicial complex is a simplicial cell complex, but two simplices in a simplicial cell complex may be glued together along more than one simplex on their boundaries. In this talk, I will discuss the characterization of face numbers of simplicial cell decompositions of some manifolds.
The notion of a GKM graph was introduced by GuilleminZara [1], motivated by a result of GoreskyKottwitzMacPherson [2]. A GKM graph is a regular graph with directions assigned to edges satisfying certain compatibility condition. The 1skeleton of a simple polytope provides an example of a GKM graph. One can associate a GKM graph $\\mathcal{G}_M$ to a closed manifold $M$ with an action of a compact torus satisfying certain conditions (those manifolds are often called GKM manifolds). Many important manifolds such as toric manifolds and flag manifolds are GKM manifolds. The GKM graph $\\mathcal{G}_M$ contains a lot of geometrical information on $M$, e.g. the (equivariant) cohomology of $M$ can be recovered by $\\mathcal{G}_M$. I will present an overview of some facts on GKM graphs.
The Grushin operator is defined as $G:=\Deltax^2\partial_t^2$ on $\mathbb{R}^{n+1}$. We study the boundedness of the multipliers $m(G)$ of $G$ on $L^p(\mathbb{R}^{n+1})$. We prove the analogue of the HormanderMihlin theorem for $m(G)$. We also study the boundedness of the solution of the wave equation corresponding to $G$ on $L^p(\mathbb{R}^{n+1})$. The main tool in studying the above is the operatorvalued Fourier multiplier theorem by Lutz Weis.
We study several kinds of matching problems between two point processes. First we consider the set of integers $\\mathbb{Z}$. We assign a color red or blue with probability 1/2 to each integer. We match each red integer to a blue integer using some algorithm and analyze the matched edge length of the integer zero. Next we go to $\\mathbb{R}^{d}$. We consider matching between two different point processes and analyze a typical matched edge length $X$. There we see that the results vary significantly in different dimensions. In dimensions one and two (d=1,2), even $E[X^{d/2}]$ does not exist. On the other hand in dimensions more than two (d>2), $E[\\exp(cX^{d})]$ exist, where $c$ is a constant depends on $d$ only.
We prove the Structure Theorem of the entropy solution. Furthermore we obtain the shock regions each of which represents a single shock at infinity. Using the structure Theorem we construct initial data $u_0\\in C_c^\\f$ for which the solution exhibits infinitely many shocks as $t \\rightarrow \\f$. Also we have generalized the asymptotic behavior (the work of Dafermos, Liu, Kim) of the solution and obtain the rate of decay of the solution with respect to the $N$wave.
Given a domain D in the complex plane and a compact subset K, Runge’s theorem provides conditions on K which guarantee that a given function that is holomorphic in some neighbourhood of K can be approximated on K by a holomorphic function on D. We look at an analogous theorem on noncompact Riemann surfaces, i.e., Runge’s approximation theorem, stated and proved by Malgrange. We revisit Malgrange’s proof of the theorem, invoking a very basic result in distribution theory: Weyl’s lemma. We look at two main applications of Runge’s theorem. Firstly, every open Riemann surface is Stein and secondly the triviality of holomorphic vector bundles on noncompact Riemann surfaces. Next, we look at the GunningNarasimhan theorem which states that every open (connected) Riemann surface can be immersed into $\\mathbb{C}$. We discuss the proof of this theorem as well, which depends on Runge’s theorem too. Finally we contrast the compact case to the noncompact case, by showing that every compact Riemann surface can be embedded into a large enough complex projective space.
Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. We wish to gain a deeper understanding of complex dynamics using the tools and techniques of potential theory, and we will restrict ourselves to the iteration of holomorphic polynomials.
Let $M$ be a closed smooth manifold. Consider the space of Riemannian metrics $\\mathcal{M}$ on $M$. A real valued function on $\\mathcal{M}$ is called a Riemannian functional if it remains invariant under the action of the group of diffeomorphisms of $M$ on $\\mathcal{M}$. We will discuss some geometric properties of the critical points of certain natural Riemannian functionals in this lecture.
We consider functions $f$ on $\mathbb{R}^n$ for which both $f$ and their Fourier transforms $\hat{f}$ are bounded by the Gaussian $e^{\frac{a}{2}x^2}$ for some $0<a<1$. Using the Bargmann transform, we show that their FourierHermite coefficients have exponential decay. This is an extension of the one dimensional result of M. K. Vemuri, in which sharp estimates were proved. In higher dimensions, we obtain the analogous result for functions $f$ which are $O(n)$finite. Here by an $O(n)$finite function we mean a function whose restriction to the unit sphere $S^{n1}$ has only finitely many terms in its spherical harmonic expansion. Some partial results are proved for general functions. As a corollary to these results, we obtain Hardy’s uncertainty principle. An analogous problem is studied in the case of Beurling’s uncertainty principle.
I will discuss some of the results on projective modules and complete intersections spanning a few decades. So, necessarily, the details have to be sketchy. But, I hope to give the flavour and some of the still persistent questions in the field.
Let R be an integral domain such that every nonzero quotient R/mR is finite. Consider the unitaries and isometries on \\ell^{2}(R) induced by the addition and the multiplication operation of the ring R. The Calgebra generated by these unitaries and isometries is called the ring Calgebra and was studied by Cuntz and Li.
This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on $\\mathbb{R}^n$, as described in the following two paragraphs respectively.
This talk is on a topic in number theory, but should be accessible to a general mathematical audience.
In a foundational paper Operators Possessing an Open Set of Eigenvalues written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space H possessing an open set W (in complex plane) of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and viceversa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that dimension of the eigenspace at each point w in W is 1, then the map f on W, sending w to ker(Tw), admits a nonzero holomorphic section, say S, and therefore defines a line bundle LT on W.It is well known that the curvature KL of a line bundle LT is a complete invariant for the line bundle LT . On the other hand, define
We wish to study those domains in $\\mathbb{C}^n$, for $n\\geq 2$, the socalled domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\\mathbb{C}$ by discussing some examples of special types of domains in $\\mathbb{C}^n$, $n\\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullen’s construction of a domain (not necessarily in $\\mathbb{C}^n$) spread over $\\mathbb{C}^n$, the socalled envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\\mathbb{C}^n$.
Recently, the financial world witnessed a series of major defaults by several institutions and banks. Therefore, it is not at all surprising that credit risk analysis has turned out to be one of the most important aspects of study in the finance community. As credit derivatives are long term instruments, they are affected by the changes in the market conditions. Thus, it is appropriate to take into consideration the cyclical effects of the market. This thesis addresses some of the important issues in credit risk analysis for a regimeswitching market.
Two key steps devised by Harish Chandra for his construction of the global characters of discrete series for a noncompact real semisimple Lie group involve
Two key steps devised by Harish Chandra for his construction of the global characters of discrete series for a noncompact real semisimple Lie group involve
Please see the attachment to this email.
Please see the attachment to this email.
Title: Topological Rigidity
The operators $f(t) \\rightarrow f(ta)$ and $f(t) \\rightarrow e^{2\\pi bt} f(t)$ on $L^2(\\mathbb R)$ generate unitary representations of the discrete Heisenberg group $H$ with central character $e^{2\\pi abz}$. What are the irreducible representations of $H$ with this central character, and how can one synthesize the representation just described from them ? When $ab$ is rational, the answers are quite straightforward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.
The operators $f(t) \\rightarrow f(ta)$ and $f(t) \\rightarrow e^{2\\pi bt} f(t)$ on $L^2(\\mathbb R)$ generate unitary representations of the discrete Heisenberg group $H$ with central character $e^{2\\pi abz}$. What are the irreducible representations of $H$ with this central character, and how can one synthesize the representation just described from them ? When $ab$ is rational, the answers are quite straightforward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.
A fibred category consists of a functor $p:\mathbf N\longrightarrow \mathbf M$ between categories $\mathbf N$ and $\mathbf M$ such that objects of $\mathbf N$ may be pulled back along any arrow of $\mathbf M$. Given a fibred category $p:\mathbf N\longrightarrow \mathbf M$ and a model structure on the base category $\mathbf M$, we show that there exists a lifting of the model structure on $\mathbf M$ to a model structure on $\mathbf N$. We will refer to such a system as a fibred model category and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base category $\mathbf M$ may be lifted to right homotopic maps in the fibred category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf N$ and $\mathbf M$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf M$ lifts to a Quillen pair on $\mathbf N$.
Hurwitz equivalence is a simple algebraic relation on the set of ntuples in a group G. This and its generalizations are related to important problems in topology. I discuss some approaches to understanding Hurwitz equivalence.
Let D_1,…D_n be a system of commuting, formally selfadjoint, left invariant operators on a Lie group G. Under suitable hypotheses, we show that D_1,…D_n are essentially selfadjoint on L^2(G) and admit a joint spectral resolution, and we characterize their joint L^2 spectrum as the set of eigenvalues corresponding to a class of generalized joint eigenfunctions. Moreover, in the case G is a homogeneous group and D_1,…D_n are homogeneous, we obtain L^pboundedness results for operators of the form m(D_1,…D_n), analogous to the MihlinHormander and Marcinkiewicz multiplier theorems.
Conformal field theories (CFTs) are related (in Mathematics) to algebraic geometry, infinitedimensional Lie algebras and probability, and (in Physics) to critical phenomena and string theory. From a mathematical point of view, much of the formalisation has been from the point of view of algebra – in fact using formal power series. I will give a denition of the simplest chiral or holomorphic CFT using elementary function theory. If time permits, I will also explain operator product expansions.
The $Cos^\\lambda$ transform on real Grassmann manifolds was first studied in convex geometry. The definition of this integral transform has been later extended to Grassmann manifolds over $\\mathbb K$, where $\\mathbb K$ denotes the reals, the complex numbers or the quaternions.
We introduce the concept of optimal test functions that guarantee stability of resulting numerical schemes. PetrovGalerkin methods seek approximate solutions of boundary value problems in a trial space by weakly imposing all equations via a (possibly different) test space. A basic design principle is that while trial spaces must have good approximation properties, the test space must be chosen for stability. The optimal test functions are those that realize discrete stability constants equal to those in the wellposedness estimates for the undiscretized boundary value problem. When such functions are used within an ultraweak variational formulation, we obtain Discontinuous PetrovGalerkin (DPG) methods that exhibit remarkable stability properties. We present the first complete theory for the DPG for Laplace’s equation as well as numerical results for other more complex applications.
In this talk, we will begin with the definition of a unimodular row and its relation to Serre’s problem on projective modules. We will then see under what conditions group structures exist on orbit spaces of unimodular rows under elementary action.
Let $F$ be a closed orientable surface of genus $g \\geq 2$ and $C$ be a simple closed curve in $F$. Let $t_C$ denote a lefthanded Dehn twist about $C$. When $C$ is a nonseparating curve, D. Margalit and S. Schleimer showed the existence of such roots by finding elegant examples of roots of $t_C$ whose degree is $2g + 1$ on a surface of genus $g + 1$. This motivated an earlier collaborative work with D. McCullough in which we derived conditions for the existence of a root of degree $n$. We also showed that MargalitSchleimer roots achieve the maximum value of $n$ among all the roots for a given genus. Suppose that $C$ is a separating curve in $F$. First, we derive algebraic conditions for the existence of roots in Mod$(F)$ of the Dehn twist $t_C$ about $C$. Finally, we show that if $n$ is the degree of a root, then $n \\leq 4g^2 + 2g$, and for $g \\geq 10$, $n \\leq \\frac{16}{5}g^2+ 12g + \\frac{45}{4}$.
In this talk, we shall define $\\Gamma$contractions, which were introduced by Jim Agler and Nicholas Young. We shall construct an explicit $\\Gamma$isometric dilation of a $\\Gamma$contraction and produce a genuine functional model. A crucial operator equation has to be solved for constructing such a dilation. We shall show how the existence of such a solution characterizes a $\\Gamma$contraction. This solution, which is unique, also provides a complete unitary invariant.
Large dimensional data presents many challenges for statistical modeling via Bayesian nonparametrics, both with respect to theroetical issues and computational aspects. We discuss some models that can accomodate large dimensional data and have attractive theoretical properties, specially focussing on kernel partition processes, which are a generalization of the well known Dirichlet Processes. We discuss issues of consistency. We then move onto some typical computational problems in Bayesian nonparametrics, focussing initially on Gaussian processes (GPs). GPs are widely used in nonparametric regression, classification and spatiotemporal modeling, motivated in part by a rich literature on theoretical properties. However, a well known drawback of GPs that limits their use is the expensive computation, typically O($n^3$) in performing the necessary matrix inversions with $n$ denoting the number of data points. In large data sets, data storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with $n$. To address these problems, a rich variety of methods have been proposed, with recent options including predictive processes in spatial data analysis and subset of regressors in machine learning. The underlying idea in these approaches is to use a subset of the data, leading to questions of sensitivity to the subset and limitations in estimating fine scale structure in regions that are not well covered by the subset. Partially motivated by the literature on compressive sensing, we propose an alternative random projection of all the data points onto a lowerdimensional subspace, which also allows for easy parallelizability for further speeding computation. We connect this with a wide class of matrix approximation techniques. We demonstrate the superiority of this approach from a theoretical perspective and through the use of simulated and real data examples. We finally consider extensions of these approaches for dimension reduction in other non parametric models.
In this talk, we discuss two problems: the sampling problem for a given class of functions on $\mathbb{R}$ (direct problem) and the reconstruction of a function from finite samples (inverse problem). In the sampling problem, we are given a class of functions $V\subset L^2(\mathbb{R})$ and one seeks to find sets of discrete samples such that any $f$ in $V$ can be completely recovered from its values at the sample points. Here we address the sampling problem for the class of functions $V = V(\varphi)$, the (integer) shift invariant space defined by a generator $\varphi$ with some general assumption. In the second problem, we discuss the problem of reconstruction of a realvalued function $f$ on $X\subset \mathbb{R}^d$ from the given data $\{(x_i,y_i)\}_{i=1}^n\subset X\times\mathbb{R}$, where it is assumed that $y_i=f(x_i)+\xi_i$ and $(\xi_1,…,\xi_n)$ is a noise vector. In particular, we are interested in reconstructing the function at points outside the closed convex hull of $\{x_1,…,x_n\}$, which is the socalled extrapolation problem. We consider this problem in the framework of statistical learning theory and regularization networks. In this framework, we address the major issues: how to choose an appropriate hypothesis space and regularized predictor for given data through a metalearning approach. We employ the proposed method for blood glucose prediction in diabetes patients. Further, using real clinical data, we demonstrate that the proposed method outperforms the stateofart (time series and neuralnetworkbased models).
The empirical spectral distribution (ESD) of the sample variance covariance matrix of i.i.d. observations under suitable moment conditions converges almost surely as the dimension tends to infinity. The limiting spectral distribution (LSD) is universal and is known in closed form with support [0,4].
I will explain how the purely combinatorial RobinsonSchenstedKnuth correspondence can be used to give a simple proof of the classification of irreducible representations of symmetric groups in the semisimple case. It turns out that all the standard results in the representation theory of symmetric groups can be recovered using this approach.
We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 1/2 , to a general first order HJB equation. Our results include some regularity estimates for the viscosity solutions of such perturbations, making the solutions classically welldefined. Most importantly, we use these regularity estimates to study the vanishing viscosity approximation to first order HJB equations by 1/2Laplacian and derive an explicit rate of convergence for the vanishing viscosity limit.
I shall give more details of the relation of intersections to hyperbolic geometry and give a sketch of the proof of the main theorem. I shall also outline to the relation of intersection numbers to the socalled Hurwitz equivalence, and hence to smooth 4manifolds.
The Goldman bracket associates a Lie Algebra to closed curves on a surface. I shall describe the bracket and its basic properties. I shall also sketch some joint work with Moira Chas, where we show that the Goldman bracket together with the operation of taking powers determines geometric intersection and selfintersection numbers.
Let f be a smooth function on the real line. The divided difference matrices of order n, whose, (i,j)th entries are the divided differences of f at (\\lambda_i,\\lambda_j) – where \\lambda_1,…,\\lambda_n are prescribed real numbers – are called Loewner matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this talk we describe some of this work.
Heuristics indicate that point processes exhibiting clustering of points have larger critical radii for the percolation of their continuum percolation models than spatially homogeneous point processes. I will explain why the dcx ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical radius is increasing in dcx order. We will prove the conjecture for some nonstandard critical radii; however it is false for the standard critical radii. I will discuss the implications of these results. A powerful implication is that point processes dcxsmaller than a homogeneous Poisson point process admit uniformly nondegenerate lower and upper bounds on their critical radii. In fact, all the above results hold under weaker assumptions of ordering of moment measures and void probabilities of the point processes. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with traceclass integral kernels. Perturbed lattices are the most general examples of dcx sub and superPoisson point processes. More generally, we show that point processes dcxsmaller than a homogeneous Poisson point process exhibit phase transitions in certain percolation models based on the levelsets of additive shotnoise fields of these point process. Examples of such models are kpercolation and SINRpercolation. This is a joint work with Bartek Blaszczyszyn.
Model Theory, as a subject, has grown tremendously over the last few decades. Starting out as a branch of mathematical logic, it has now wide applications in most branches of mathematics, with algebra, algebraic geometry, number theory, combinatorics and even analysis, to name a few. In this talk, I will give a brief introduction to model theory, talk about the compactness theorem (one of the main tools in model theory) and how it is used, and give one famous application of model theory to algebra, namely, the AxKochen Theorem, the answer to Artin’s Conjecture. Time permitting, I will talk about a few more recent results in this direction.
Heuristics indicate that point processes exhibiting clustering of points have larger critical radii for the percolation of their continuum percolation models than spatially homogeneous point processes. I will explain why the dcx ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical radius is increasing in dcx order. We will prove the conjecture for some nonstandard critical radii; however it is false for the standard critical radii. I will discuss the implications of these results. A powerful implication is that point processes dcxsmaller than a homogeneous Poisson point process admit uniformly nondegenerate lower and upper bounds on their critical radii. In fact, all the above results hold under weaker assumptions of ordering of moment measures and void probabilities of the point processes. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with traceclass integral kernels. Perturbed lattices are the most general examples of dcx sub and superPoisson point processes. More generally, we show that point processes dcxsmaller than a homogeneous Poisson point process exhibit phase transitions in certain percolation models based on the levelsets of additive shotnoise fields of these point process. Examples of such models are kpercolation and SINRpercolation. This is a joint work with Bartek Blaszczyszyn.
Let G be the absolute Galois group of a field that contains a primitive pth root of unity. This is a profinite group which is a central object of study in arithmetic algebraic geometry. In joint work with Ido Efrat and Jan Minac, we have shown that a remarkably small quotient of this big group determines the entire Galois cohomology of G. As application of this result, we give new examples of profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work. ALL ARE CORDIALLY INVITED
The classical Fej\\‘erRiesz Theorem states that a nonnegative trigonometric polynomial can be factored as the absolute square of an analytic polynomial. Indeed, the factorization can be done with an outer polynomial. Various generalizations of this result have been considered. For example, Rosenblum showed that the result remained true for operator valued trigonometric polynomials. If one instead considers operator valued polynomials in several variables, one obtains factorization results for strictly positive polynomials, though outer factorizations become much more problematic. In another direction, Scott McCullough proved a factorization result for socalled hereditary trigonometric polynomials in freely noncommuting variables (strict positivity not needed). In this talk we consider an analogue of (hereditary) trigonometric polynomials over discrete groups, and give a result which includes a strict form of McCullough’s theorem as well as the multivariable version of Rosenblum’s theorem.
The classical Fej\\‘erRiesz Theorem states that a nonnegative trigonometric polynomial can be factored as the absolute square of an analytic polynomial. Indeed, the factorization can be done with an outer polynomial. Various generalizations of this result have been considered. For example, Rosenblum showed that the result remained true for operator valued trigonometric polynomials. If one instead considers operator valued polynomials in several variables, one obtains factorization results for strictly positive polynomials, though outer factorizations become much more problematic. In another direction, Scott McCullough proved a factorization result for socalled hereditary trigonometric polynomials in freely noncommuting variables (strict positivity not needed). In this talk we consider an analogue of (hereditary) trigonometric polynomials over discrete groups, and give a result which includes a strict form of McCullough’s theorem as well as the multivariable version of Rosenblum’s theorem.
Beginning with the attempts of Heisenberg and Pauli in the 1920’s, the subject grewat an astonishing speed, culminating in the remarkable predictive successof Quantum Electrodynamics.Different attemptsto bring the subjectto a sound mathematical footing (comparable to that of Quantum Mechanics)– whetheranalytic, operatoralgebraic or geometric – have tastedonly partial success. These talks will try to give a bird’seye view of the mathematical areas (ideas)spawned by these attempts, keeping in view the recent book of Folland on this subject.
ALL ARE INVITED
In this talk, we shall study about the leaking rate of the SinaiRuelleBowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps (open dynamics). The dependence of this rate on the size and position of the hole shall be explained. For an easier and better understanding, the simple quadratic map restricted on the unit circle will be analysed thoroughly. Time permitting, we will also compute the Hausdorff dimension of the survivor set.
Let A be a selfadjoint operator. We are interested in the discrete spectrum of B = A+M where B is nonselfadjoint. If the resolvent difference is in the Schatten class S_p, then we have an estimate on the distribution of the eigenvalues of B. By means of this estimate we can give qualitative estimates for the number of eigenvalues of B or their moments. That can be applied to Schodinger operators with complex potentials.
In finitedimensional control theory, feedback control using controllability Gramian goes back to results by Kleinman and Lukes in 1968 and 1970. Some years before, R. Bass characterized controls of minimal norms also using controllability Gramians. The extension of these results to infinitedimensional systems has a long history. Surprisingly, only reversible infinitedimensional systems have been considered in those results. We shall present existing results in the literature and we shall characterize stabilizing controls of minimal norms for parabolic systems. This is a joint work with S. Kesavan. Application to the stabilization of the NavierStokes equations will be given.
Certain classes of nonlocal pseudodifferential operators can be associated with Markov processes and this result has an infinite dimensional counterpart too. The best known example is the Levy process with its generator an integrodifferential operator. In this talk we will give an introduction to stochastic NavierStokes equation with jump (Levy) noise and point out opportunities for harmonic as well as stochastic analysis to gain understanding in solvability theory and applications such as control and filtering theory.
Stochastic partial differential equations (SPDE) are partial differential equations (PDE) with a `noise term’. One can think of these as a semimartingale in a function space or a space of distributions with a drift (a bounded variation process involving a second order elliptic partial differential operator ) and a noise term which is a martingale. When the martingale term is suitably structured, the solutions of these SPDE’s are closely related to certain finite dimensional diffusion processes and may be viewed as generalized solutions of the classical stochastic differential equations of Ito, StroockVaradhan and others. In this talk [based on Rajeev and Thangavelu (2008) and Rajeev (2010)], we describe how the expected values of the solutions give rise to solutions of PDE’s associated with the diffusion.
I will describe some joint work with Todorcevic on the Tukey theory of ultrafilters on the natural numbers. The notion of Tukey equivalence tries to capture the idea that two directed posets look cofinally the same, or have the same cofinal type. As such, it provides a device for a rough classification of directed sets based upon their cofinal type, as opposed to an exact classification based on their isomorphism type. This notion has recently received a lot of attention in various contexts in set theory. As background, I will illustrate the idea of rough classification with several examples, and explain how rough classification based on Tukey equivalence fits in with other work in set theory. The talk will be based on the paper Cofinal types of ultrafilters. A preprint of the paper is available on my website: http://www.math.toronto.edu/raghavan .
Serre’s mass formula counts the number of totally ramified degree$n$ extensions $E$ of a local field $F$, each extension being assigned some weight depending upon how ramified it is. We will present an elementary proof of this formula when the degree $n$ is prime.The background material will be covered, so that the talk should be accessible to a broad mathematical audience.
Serre’s mass formula counts the number of totally ramified degree$n$ extensions $E$ of a local field $F$, each extension being assigned some weight depending upon how ramified it is. We will present an elementary proof of this formula when the degree $n$ is prime.The background material will be covered, so that the talk should be accessible to a broad mathematical audience.
An analogue of Brylinski’s knot beta function is defined for a submanifold of ddimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.
An analogue of Brylinski’s knot beta function is defined for a submanifold of ddimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wavenumbers in excess of a threshold $\kg$ exhibit unexpected features. The study is carried out for both the onedimensional Burgers equation and the twodimensional incompressible Euler equation. At large $\kg$, for smooth initial conditions, the first symptom of truncation, a localized shortwavelength oscillation which we call a “tyger”, is caused by a resonant interaction between fluid particle motion and truncation waves generated by smallscale features (shocks, layers with strong vorticity gradients, etc). These tygers appear when complexspace singularities come within one Galerkin wavelength $\lambdag = 2\pi/\kg$ from the real domain and typically arise far away from preexisting smallscale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first  in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to $\kg ^{2/3}$ and $\kg ^{1/3}$ respectively  but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T.D. Lee in 1952. The sudden dissipative anomalythe presence of a finite dissipation in the limit of vanishing viscosity after a finite time $\ts$, which is well known for the Burgers equation and sometimes conjectured for the 3D Euler equation, has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit $\kg\to\infty$. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscidlimit solution. There are indications that it may be possible to purge the tygers and thereby to recover the correct inviscidlimit behaviour.
One of the fundamental problems in Extremal Combinatorics concerns maximal set systems with forbidden subconfigurations. One such open problem concerns the conjecture due to Anstee and Sali on the order of maximal configurations with certain forbidden subconfigurations. I shall talk about some well known results, talk about the AnsteeSali conjecture, and finally talk about some of my recent work concerning Steiner designs occuring as maximal forbidden configurations for certain natural choices of subconfigrations. This generalizes a result of Anstee and Barekat.
I will report on the existence and properties of Hilbert spaces based on different families of complex, holomorphic or not, functions, like Hermite polynomials. The resulting coherent state quantizations of the complex plane will be presented. Some interesting issues will be examined, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules.
Higher cognitive functions require the coordination of large assemblies of
spatially distributed neurons in ever changing constellations. It is
proposed that this coordination is achieved through temporal coordination
of oscillatory activity in specific frequency bands. Since there is no
supraordinate command centre in the brain, the respective patterns of
synchronous activity selforganize, which has important implications on
concepts of intentionality and top down causation. Evidence will be
provided that synchronisation supports response selection by attention,
feature binding, subsystem integration, shortterm memory, flexible
routing of signals across cortical networks and access to the workspace
of consciousness. The precision of synchronisation is in the millisecond
range, suggesting the possibility that information is encoded not only in
the covariation of discharge rates but also in the precise timing of
individual action potentials. This could account for the high speed with
which cortical circuits can encode and process information. Recent studies
in schizophrenic patients indicate that this disorder is associated with
abnormal synchronisation of oscillatory activity in the high frequency
range (beta and gamma). This suggests that some of the cognitive deficits
characteristic for this disease result from deficient binding and
subsystem integration.
An important question in complex analysis is to solve the inhomogenous CauchyRiemann equations (also called the dbar equation) in a domain in C^n. The question of boundary smoothness in the dbar problem is classically dealt with by solving the associated dbar Neumann problem and showing that the solution operator, the $\\overline{\\partial}$Neumann operator is compact. For many domains of interest, in particular the product domains, this approach fails. We discuss in this talk some new results on the regularity of the dbar problem in product domains. This work is joint with MeiChi Shaw.
A geometry is a collection of lines and points satisfying the usual incidence axioms. By a theorem of Gromov, given an almost complex structure (which is `tame’) on the complex projective plane CP^2, we obtain a geometry by declaring appropriate holomorphic curves to be the lines. Ghys asked whether Desargues’s theorem (a Euclidean geometry result related to symmetry) characterises the standard complex structure. We show that this is indeed the case.
The abstract of this talk has been posted at:
The space of all complex structures on a surface, and the DeligneMumford compactification of this space, play an important role in many areas of mathematics. We give a combinatorial description of a space that is homotopy equivalent to the DeligneMumford compactification, in the case of surfaces with at least one puncture.
Why are soap bubbles spherical? Why do double soap bubbles have the shape that they do (three spherical caps meeting along a circle at 120 degree angles)? The single bubble problem was solved in the 19th century, and the double bubble problem was solved a few years ago. The analogous problem for triple soap bubbles remains a mystery. We will give an introduction to these problems and their solutions (when known).
It is very difficult in general to determine when a given compact in C^n, n>1, is polynomially convex. In this talk, we shall discuss polynomial convexity of some classes of sets. First, we shall consider two totallyreal surfaces in C^2 that contain the origin and have distinct tangent planes there. We shall discuss how the local polynomial convexity of the union of the tangent planes at (0,0) influences local polynomial convexity of the union of the surfaces at (0,0). Secondly, we will present a condition for local polynomial convexity of unions of more than two totallyreal planes in C^2 containing the origin. Next, we shall talk about pluri subharmonicity. Using this notion we shall give a new proof of an approximation theorem of Axler and Shields and also generalize it. Polynomial convexity plays a very central role in our proof. Finally we shall discuss a characterization for (large) compact patches of smooth totallyreal graphs in C^{2n} to be polynomially convex.
Let D be a smoothly bounded pseudoconvex domain in C^n, n > 1. Using G(z, p), the Green function for D with pole at p in D, associated with the standard sumofsquares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kahler metric (the socalled Lambda–metric) using the Robin function arising from G(z, p). The purpose of this thesis is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the Lambda–metric is comparable to the Kobayashi (and hence to the Bergman and Caratheodory metrics) when D is strongly pseudoconvex. The unit ball in C^n is also characterized among all smoothly bounded strongly convex domains on which the Lambda–metric has constant negative holomorphic sectional curvature. This may be regarded as a version of LuQi Keng’s theorem for the Bergman metric.
Kunze Stein inequality can be thought of as a semisimple version of Young’s inequality. A remarkable observation of M.G.Cowling and S.Meda shows that these inequalities can be naturally extended to Lorentz spaces. The final result in this direction was proved by A. Ionescu. In this talk we will try to explain the central ideas behind Kunze Stein type convolution inequalities for Lorentz spaces.
It is well known that there exist domains in C^n, n > 1, such that all functions holomorphic therein extend holomorphically past the boundary. In this talk, we shall examine this surprising phenomenon by discussing refinements of the fundamental example of Hartogs. We shall look at a generalization of Hartogs’ construction discovered by Chirka. Finally, we shall provide a partial answer to a related question raised by Chirka.
We derive the conservation form of equations of evolution of a front propagating in three dimensions. We obtain a system of six conservation laws, known as 3D kinematical conservation laws (KCL) in a ray coordinate system. The conservative variables of 3D KCL are also constrained by a stationary vector constraint, known as geometric solenoidal constraint, which consists of three divergencefree type conditions. The 3D KCL is an underdetermined system, and therefore, additional closure relations are required to get a complete set of equations. We consider two closure relations for 3D KCL: (1) energy transport equation of a weakly nonlinear ray theory (WNLRT) to study the propagation of a nonlinear wavefront, (2) transport equations of a shock ray theory (SRT) to study the propagation of a curved weak shock front. In both the cases we obtain a weakly hyperbolic system of balance laws. For the numerical simulation we use a highresolution semidiscrete central scheme. The second order accuracy of the scheme is based on MUSCL type reconstructions and TVD RungeKutta time stepping procedures. A constrained transport technique is used to enforce the geometric solenoidal constraint and in all the test problems considered, the constraint is satisfied up to very high accuracy. We present the results of extensive numerical experiments, which confirm the efficiency and robustness of the method and also its ability to capture many physically realistic features of the fronts.
This thesis considers two themes, both of which emanate from and involve the Kobayashi and the Carath\\‘{e}odory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\\mathbf C^2$ and on convex finite type domains in $\\mathbf C^n$ using the scaling method. Applications include an alternate proof of the WongRosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in $ \\mathbf C^2$ and convex finite type domains in $ \\mathbf C^n$ in terms of Euclidean parameters. Second a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for $C^1$isometries of the Kobayashi and Carath\\‘{e}odory metrics on a smoothly bounded strongly pseudoconvex domain.
The most immediate way to use classical homotopy theory to study topological spaces endowed with extra structures, such as complex spaces, is to forget these structures and study the underlying topological space. Usually this procedure is inadequate since what we are forgetting is not homotopy invariant nor topological invariant. There are more sophisticated approaches to use homotopy theory to try to detect different complex structures on the same topological space which prove to be effective, for instance if the topological space is a complex variety and the complex structures endow the tangent bundle of different Chern classes/numbers. A much more thorough approach can be achieved by using model category theory to provide the category of complex spaces with holomorphic maps of an homotopy theory which realizes to the ordinary topological one, it is biholomorphic but not topological invariant. These techniques have been previously implemented by Morel and Voevodsky to create the so called A^1 homotopy theory of algebraic varieties few years ago.
Free groups and the group of their outer automorphisms have been extensively studied in analogy with (fundamental groups of) surfaces and the mapping class groups of surfaces. We study the analogue of intersection numbers of simple curves, namely the ScottSwarup algebraic intersection number of splittings of a free group and we also study embedded spheres in $3$ manifold of the form $ M =\\sharp_n S^2 \\times S^1 $. The fundamental group of $M$ is a free group of rank $n$. This $3$manifold will be our model for free groups. We construct geosphere laminations in free group which are analogues of geodesic laminations on a surface.
One of the most important subclasses of the class of normalized analytic univalent functions on the open unit disc D is the class of convex functions. In this talk we will focus on meromorphic analogues of the results known for this class. I.e. we consider functions that map D conformally onto a set whose complement is a bounded convex set. We shall begin with a brief history of Livingston’s conjecture which concerns the exact set of variability of the Taylor coefficients for concave functions. Thereafter, we shall discuss some new results concerning the closed convex hull of concave functions and extreme points of it.
We discuss applications of computers to prove mathematical theorems. In particular we discuss possible future applications in Topology/Geometry. This will also be the introductory lecture to the new course `Computer Assisted Topology/Geometry’
We discuss applications of computers to prove mathematical theorems. In particular we discuss possible future applications in Topology/Geometry. This will also be the introductory lecture to the new course `Computer Assisted Topology/Geometry’
Given a surface, one can consider the set of free homotopy classes of oriented closed curves (this is the set of equivalence classes of maps from the circle into the surface, where two such maps are equivalent if the corresponding curves can be deformed one into the other.) Given a free homotopy class one can ask what is the minimum number of times (counted with multiplicity) in which a curve in that class intersects itself. This is the minimal selfintersection number of the free homotopy class. Analogously, given two classes, one can ask what is the minimum number of times representatives of these classes intersect. This is the minimal intersection number of these two classes.
The appropriate context for algebraicgeometric realizations of holomorphic representations of a complex semisimple group is that of a compact homogeneous projective flag manifold Z = G/Q. One topic of more current interest is the study of the possibility of realizing (infinitedimensional) unitary representations of a real form G0 of G on function and/or cohomologyspaces of open G_0orbits D in Z (flag domains) and their cycle spaces. After an introduction for nonspecialists, we will indicate a proof by Schubert incidence geometry of the Kobayashi hyperbolicity of the relevant cycle spaces. This will then be applied to give an exact description of the group Aut_O(D) of holomorphic automorphisms.
The classical realization theorem gives various characterizations of functions in the unit ball of $H^\\infty(\\mathbb D)$, the bounded analytic functions on the unit disk $\\mathbb D$, which happens to also correspond to the multipliers of Hardy space $H^2(\\mathbb D)$. This realization theorem yields an elegant way of solving the NevanlinnaPick interpolation problem. In the mid 80s, Jim Agler discovered an analogue of the realization theorem over the polydisk. While for $d=2$, it once again gives a characterization of the elements of the unit ball of $H^\\infty(\\mathbb D^d)$ (and so allows one to solve interpolation problems in $H^\\infty(\\mathbb D^2)$), for $d>2$, the class of functions which are realized is a proper subset of the unit ball of $H^\\infty(\\mathbb D^d)$ — the socalled SchurAgler class.
Clifford analysis is a higher dimensional analogue of single variable complex analysis. Although functions take values in a finite dimensional Clifford algebra, the representation formula for Clifford regular functions is simpler and more powerful than for holomorphic functions of several complex variables. The talk shows how Clifford analysis techniques can be employed in operator theory for a functional calculus of $n$tuples of operators.
Methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices include the moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. We develop this method further and provide a general framework to deal with symmetric patterned matrices with entries coming from an independent sequence. This approach can be extended to cover matrices of the form Ap =XX’/n where X is a pxn matrix with p going infinity and n = n(p) going to infinity and p/n going to y between 0 and infinity. The method can also be used to cover some situations where the input sequence is a suitable linear process. Several new classes of limit distributions arise and many interesting questions remain to be answered.
For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functions on M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the biasymptotic property. Finally we prove that the total Busemann function is continuous in C^\\infty topology. As a consequence of it we show that the uniform divergence of geodesics holds in these spaces.
Consider a system of N interacting particles evolving over time with Markovian dynamics. The interaction occurs only due to a shared resource. I shall briefly sketch the classical tightnessexistenceuniqueness approach used to prove weak convergence of the system to a limiting system as N tends to infinity. I shall illustrate another approach in more detail with the example of a nonlinear Markov chain. In the case of exchangeable particles, weak convergence of the system implies propogation of chaos i.e, in the limiting system particles evolve independently due to a deterministically shared resource.
After a brief introduction to some classical group invariants, we proceed to consider their generalizations with weights. We discuss some recent results on these weighted generalizations.
Heuristics indicate that more the clustering in a point process, worse the percolation will be. We shall in this talk see a first step towards a formal proof of this heuristic. Here is a more precise abstract of the results we shall see in the talk.
The polarization conjecture for R^d is that
Let R be the reflection group of the complex leech lattice plus a hyperbolic cell. Let D be the incidence graph of the projective plane over the finite field with 3 elements. Let A(D) be the Artin group of D: generators of A(D) correspond to vertices of D. Two generators braid if there is an edge between them, otherwise they commute.
Let H :R –> R. with multiple minima on R.For T> 0 consider the probability measure on R with pdf proportional to exp(–H/T). In this talk we discuss the problem of weak limits of this measure as T goes to infinity. It depends on the behaviour of H near its minima locations. Both Gaussian and stable limit laws arise as weak limits.
Consider a multidimensional diffusion model where the drift and the diffusion coefficients for individual coordinates are functions of the relative sizes of their current value compared to the others. Two such models were introduced by Fernholz and Karatzas as models for equity markets to reflect some wellknown empirically observed facts. In the first model, called ‘Rankbased’, the timedynamics is determined by the ordering in which the coordinate values can be arranged at any point of time. In the other, named the ‘Volatilitystabilized’, the parameters are functions of the ratio of the current value to the total sum over all coordinates. We show some remarkable properties of these models, in particular, phase transitions and infinite divisibility. Relationships with existing models of queueing, dynamic spin glasses, and statistical genetics will be discussed. Part of the material is based on separate joint work with Sourav Chatterjee and Jim Pitman.
Gaussian Minkowski functionals (GMFs) for reasonably smooth subsets of Euclidean spaces, are defined as coefficients appearing in the the Gaussiantubeformula in finite dimensional Euclidean spaces. The fact that the measure in consideration here is Gaussian, itself makes the whole analysis an infinite dimensional one. Therefore, one might want to generalize the definition of Gaussian Minkowski functionals to the subsets of Wiener space which arise from reasonably smooth (in Malliavin sense) Wiener functionals. As in the finite dimensional case, we shall identify the GMFs in the infinite dimensional case, as the coefficients appearing in the tube formula in Wiener space. Finally, we shall try to apply this infinite dimensional generalization to get results about the geometry of excursion sets of a reasonably large class of random fields defined on a “smooth” manifold.
ALL ARE INVITED
Coffee/Tea: 3:45 pm
We describe a relationship between cusp excursions of horocycles on the modular surfaces SL(2, R)/SL(2, Z) and diophantine approximation. Some of the work we discuss will be joint with G. Margulis, and some joint with Y. Cheung
Let $V$ be finite dimensional vector space over the field of complex numbers. Let $G$ be a finite subgroup of $GL(V)$, group of all $\mathbb{C}$ linear automorphisms of $V$. Then, the apmple generator of the Picard group of the projective space $\mathbb P(V)$ descends to the quotient variety $\mathbb{P}(V)/G$. Let $L$ denote the descent. We prove that the polarised variety $\mathbb P(V)/G, L$ is projectively normal when $G$ is solvable or $G$ is generated by pseudo reflections.
About thirtyfive years ago, several problems in operator theory concerning almost normal operators led L. G. Brown, P. A. fillmore and me to introduce methods and point of view from algebraic topology to solve them. By the time we were done concrete realization of Khomology was introduced as well as new insight obtained for the index theorem of AtiyahSinger.
We will discuss the notion of spectrum and arithmetic of spaces, and expound on the expectation that they should mutually determine each other for the class of locally symmetric spaces associated to congruent arithmetic lattices.
If X is a positive random variable with a finite mean then the probability distribution with density proprtional to X is called its size biased version. For Markov chains admitting a positive eigen function one can construct a size biased version of this chain which is also Markov.. In this talk we derive conditions for the two chains to be dominated by each other over the full trajectory space.. We then apply this to derive a LLOGL result for supercritical branching processes with arbitrary type space.
We are interested in deriving schemes having some ‘wellbalanced’ and ‘asymptotic preserving’ properties for the approximation of a nonlinear hyperbolic system with source term. In the case of Euler system with friction, the scheme is derived from simple Riemann solvers or equivalently using a relaxation scheme for the enlarged Euler system with ‘potential’. All interested are Welcome
Tits and Weiss in their book Moufang Polygons have conjectured that the groups of rational points of certain forms of E_6 are generated by some inner transformations. These groups occur as groups of similitudes of certain cubic forms in 27 variables. We will explain this conjecture and report on some results and reductions towards a positive answer.
We will discuss the notion of spectrum and arithmetic of spaces, and expound on the expectation that they should mutually determine each other for the class of locally symmetric spaces associated to congruent arithmetic lattices.
If X is a positive random variable with a finite mean then the probability distribution with density proprtional to X is called its size biased version. For Markov chains admitting a positive eigen function one can construct a size biased version of this chain which is also Markov.. In this talk we derive conditions for the two chains to be dominated by each other over the full trajectory space.. We then apply this to derive a LLOGL result for supercritical branching processes with arbitrary type space.
We will discuss the notions of polynomial and rational convexity. In particular, the question of existence of analytic structure on the additional part of the hull will be considered. Applications of polynomial and rational convexity to other problems of complex analysis will also be given.
We will investigate the large deviation rates for sums of the form $\sum_i f(x_i) g(x_{2i})$ where $\{x_i\}$ is a nice Markov process. In other words calculate
$\lim_{n\to\infty}{1\over n}\log E[\exp \sum_{i=1}^n f(x_i) g(x_{2i})]$
where $\{x_i\}$ is Markov Chain with transition probability $\pi(x, y)$.
http://www.univie.ac.at/nuhagphp/scheduler/index.php
Very large datasets occur in the area of machine learning (ML). The tasks are having a computer “learn” to read handwriting, to understand speech, to recognize faces, to filter spam email etc. Mathematically, these problems lead to huge optimization problems, of an, however not unfavorable type, namely convex quadratic programs (QP).
Information about available software to solve a large variety of optimization problems is provided at plato.asu.edu/guide.html while some of this software is evaluated at plato.asu.edu/bench.html. Starting with these sources an overview will be given on codes that are particularly useful for applications in mathematical finance.
I will introduce an exact stochastic representation for certain nonlinear transport equations (e.g. 3DNavierStokes, Burgers) based on noisy Lagrangian paths, and use this to construct a (stochastic) particle system for the NavierStokes equations. On any fixed time interval, this particle system converges to the NavierStokes equations as the number of particles goes to infinity.
The talk will be a report on an ongoing research project (joint with Preena Samuel and K.V.Subrahmanyam). Representation theoretic consequences will be worked out of the combinatorial characterizations of left, right, and twosided KazhdanLusztig cells of the symmetric group. Applications to invariant theory and both ordinary and modular representation theory of the symmetric group will be given.
http://math.iisc.ernet.in/~imi/downloads/gpisier.pdf
We study families of infinitedimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras. Infinitesimal Hecke algebras are deformations of semidirect product Lie algebras, and we study two families over $\mathfrak{sl}(2)$ and $\mathfrak{gl}(2)$. Both of them have a triangular decomposition and a nontrivial center, which allows us to define and study the BGG Category $\mathcal{O}$ over them  including a (central character) block decomposition, and an analog of Duflo’s Theorem about primitive ideals. We then discuss certain related setups.
Reflected diffusions arise in many contexts. We identify a rather general condition under which these diffusions belong to the class of socalled Dirichlet processes, which are a generalization of continuous semimartingales that admit many nice properties, including an Ito formula. We also provide an example arising from applications, in which the reflected diffusion fails to be a semimartingale, but belongs to the class of Dirichlet processes. This is partly based on joint work with Weining Kang.
Embedded contact homology (ECH) is an invariant of threemanifolds due to Hutchings, Sullivan, and Taubes. It uses a contact structure on a threemanifold to produce an invariant of the underlying topological manifold. The invariant is the homology of a chain complex generated by certain closed orbits of the Reeb vector field (of interest in classical dynamics), whose differential counts certain holomorphic curves in the symplectization of the contact threemanifold. Few nontrivial examples of ECH have been computed. In this talk, I will give some background and context on ECH and then describe the computation of the ECH of circle bundles over Riemann surfaces, in which the relevant holomorphic curves are actually meromorphic sections of complex line bundles.
The central question will be focused around the design and implementation of neuronal circuitry involved in the task of decoding sensory information. Sensory information is encoded in the form of sequence of action potential spikes by the peripheral nervous system. This information is then passed onto the central nervous system (CNS) to generate an appropriate response. The action potential spikes are identical in shape and therefore it is assumed that all the information about the environment is embedded in the timing of occurrences of these spikes. The question then is, what neural architecture exists in the CNS to decode this information?
I will outline an approach to noncommutative geometry, due largely to M. Kontsevich, where certain Ainfinity categories play the role of spaces. This noncommutative geometry program draws much of its inspiration from homological mirror symmetry. The talk will be purely expository: it will not contain any new results.
The aim of this series of lectures is to seek analogies of some known uncertainty principles, proved in the case of the real line to certain solvable Lie groups. We will speak about uncertainty principles of Hardy, Cowling Price, Morgan and Beurling and present some recent results on their non commutative analogues. Furthermore, we will also talk about sharpness of the decay condition in Hardys principle and give some new result in that context.
We investigate the (in)consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1\\over 3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function $f$ on $[0,\\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0 \\in (0,\\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function or its least concave majorant, does not have any weak limit in probability. Bootstrapping from a smoothed version of the least concave majorant, however, leads to strongly consistent estimators and the $m$ out of $n$ bootstrap method is also consistent. Our results cast serious doubt on some previous claims about bootstrap consistency (in the class of cube root problems) in the published literature.
In this talk, I will discuss threshold estimation for a regression function in some different settings. The threshold can either be a change–point, i.e. a point of jump discontinuity in an otherwise smooth curve, or the first time that a regression function crosses a certain level. Both problems have numerous applications in a variety of spheres, like biology (pharmacology, doseresponse experiments) and engineering. Our goal is to estimate thresholds of this type given a fixed budget of points to sample from, but with the flexibility that batch sampling can be done in several stages, so that adaptive strategies are possible. Our strategy is to use multistage zoomin procedures to estimate the threshold: an initial fraction of the sample is invested top come up with a first guess, an adequate neighborhood of the first guess is chosen, more points are sampled from this neighborhood and the initial estimate id updated. The procedure continues thus, ending in a finite number of stages. Such zoomin procedures result in accelerated convergence rates over any one–stage method. Approximations to relative efficiencies are computed and optimal allocation strategies, as well as recipes for construction of confidence sets discussed.
Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear.We also discuss the case of arbitrary input sequence.
This is an expository talk whose aim will be to give an introduction to SLE (SchrammLoewner evolution), discovered by Oded Schramm in 2000 to describe many critical statistical mechanical systems.
I will give a brief overview of Wiles’ proof of Fermat’s Last Theorem, and explain the connection between modular forms and elliptic curves via Galois representations e.g. the TaniyamaShimura conjecture. In the second half, I’ll explain some recent results on padic modular forms and deformations of Galois representations. If time permits, I’ll outline a future project on ranks of elliptic curves.