In the first part of the talk, we will discuss the main statement of local class field theory that describes the abelian extensions of a non-archimedean local field $F$ in terms of the arithmetic of the field $F$. Then we will discuss the statement of the local Langlands conjectures, a vast generalization of local class field theory, that gives a (conjectural) parametrization of the irreducible complex representations of $G(F)$, where $G$ is a connected, reductive group over $F$, in terms of certain Galois representations. We will then discuss a philosophy of Deligne and Kazhdan that loosely says that to obtain such a parametrization for representations of $G(F’)$, with $F’$ of characteristic $p$, it suffices to obtain such a parametrization for representations of $G(F)$ for all local fields $F$ of characteristic $0$. In the second half of the talk, we will mention some instances where the Deligne-Kazhdan philosophy has been applied successfully to obtain a Langlands parametrization of irreducible representations of $G(F’)$ in characteristic $p$ and focus on some recent work on variants/generalizations of the work of Kazhdan.
Of fundamental importance in number theory is the question of non-vanishing of central L-values of L-functions. One approach, explained in the talk, is to make use of the Relative trace formula (which will be introduced from scratch); a basic example of interest involves twists of L-functions of classical modular forms. If time permits, we will explain the recent work with Michel and Yang on $U(2)$-twists of $U(3)$ L-functions.
We study the spectral gap phenomena for weighted $L^p$-Hardy inequalities on $C^{1,\gamma}$-domain with a compact boundary, where $\gamma\in (0,1]$. We show that the weighted Hardy constant is attained by some appropriate minimizer if and only if the spectral gap (the difference between the weighted Hardy constant and the weighted Hardy constant at infinity ) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. In this talk, we will try to understand how the ideas in criticality theory help us to extend the spectral gap phenomena from $C^2$-domains to $C^{1,\gamma}$-domains. This talk is based on the joint work with Yehuda Pinchover, Baptiste Devyver.
Any compact Riemann surface is shown to have a canonical projective structure (which is different from the canonical one given by the uniformization theorem). Some properties of this projective structure are established. (Joint work with Elisabetta Colombo, Paola Frediani and Gian Pietro Pirola.)
Recent advances in the nonconforming FEM approximation of elliptic PDE eigenvalue problems include the guaranteed lower eigenvalue bounds (GLB) and its adaptive finite element computation. The first part of the talk explains the derivation of GLB for the simplest second-order (and fourth-order) eigenvalue problems with relevant applications, e.g., for the localization of the critical load in the buckling analysis of the Kirchhoff plates. The second part mentions an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace (and bi-Laplace) operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. Numerical experiments in the third part of the presentation shows benchmarks in which the naive adaptive mesh-refining and the post processed GLB do not lead to efficient GLB. The fourth part outlines a new extra-stabilised scheme based on extended Crouzeix-Raviart (resp. Morley) finite elements that directly computes approximations as GLB and that allows optimal convergence rates at the same time.
The presentation is on joint work with Dr. Sophie Puttkammer.
Let $N$
be a prime number $>3$
. Mazur has defined, from the theory of modular forms, a unit $u$
in $\mathbb{Z}/N$
. This unit turned out to be, up to a $6$
-th root of unity,$\prod_{k=1}^{(N-1)/2}k^k$
.
In this talk we will describe how the unit is connected to various objects in number theory. For instance:
–The unit $u$
can be understood as a derivative of the zeta function at $-1$
, (despite living in a finite field).
– Lecouturier has shown that this unit is the discriminant of the Hasse polynomial: $\sum_{i=0}^{(N-1)/2}a_i X^i$
modulo $N$
, where $a_i$
is the square of the $i$
-th binomial coefficient in degree $N$
.
– Calegari and Emerton have related $u$
to the class group of the quadratic field $\mathbb{Q}(\sqrt{-N})$
.
For every prime number $p$
dividing $N-1$
, It is important to determine when $u$
is a $p$
-th power in $(\mathbb{Z}/N)^*$
.
If time allows, I will describe the connections to modular forms and Galois representations, and the general theory that Lecouturier has developed from this unit. For instance,when $u$
is not a $p$
-th power, a certain Hecke algebra acting on modular forms is of rank $1$
over the ring of $p$
-adic integers $\mathbb{Z}_p$
(the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.
Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak h$. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak h$. These include fixed point subalgebras of $\mathfrak h$ under subgroups of diagram automorphisms of $\mathfrak g$. This is joint work with K.N.Raghavan, R.Venkatesh and S.Viswanath.
Application of the Hahn-Banach Theorem to the space of bounded sequences with a specific sub linear functional $p$ defined on it gives rise to linear functionals which are dominated by $p$ and are extensions of limits of convergent sequences. These are called Banach Limits and were studied by Banach (1932), and their uniqueness is called almost convergence and was characterised by Lonentz (1948).
In the present lecture we will discuss about the absolute analogue of almost convergence which generalizes lp spaces.
The two concepts of variational inequality and complementarily problems are essentially the same concepts which are studied by two different groups of mathematicians: applied mathematics on one hand and operations researchers on the other hand. The proof existence of variational inequality problem uses Hahn-Banach Theorem or Fixed Point Theorem.
In this lecture we will discuss about the existence of solutions of the complementarily problem, under the most general conditions on the operator and the cone.
In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume. Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a harmonic function, where $n\geq 2$. One way to quantify the growth of $u$ in the ball $B(0,1) \subset \mathbb{R}^n$ is via the doubling index $N$, defined by \begin{equation} \sup_{B(0,1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|. \end{equation} I will present a result, obtained jointly with A. Logunov and A. Sartori, where we prove an almost sharp result, namely: \begin{equation} \mathcal{H}^{n-1}({u=0} \cap B(0,2)) \gtrsim_{n,\varepsilon} N^{1-\varepsilon}, \end{equation} where $\mathcal{H}^{n-1}$ denotes the $(n-1)$ dimensional Hausdorff measure.
The Kubota-Leopoldt $p$-adic $L$-function is an important concept in number theory. It takes special values in terms of generalized Bernoulli numbers, and helps solve Kummer congruences. It is also used in Iwasawa theory. Formalization of $p$-adic $L$-functions has been done for the first time in a theorem prover called Lean 3. In this talk, we shall briefly introduce the concept of formalization of mathematics, the theory behind $p$-adic $L$-functions, and its formalization.
Let $p$
be an odd prime, $f$
be a $p$
-ordinary newform of weight $k$
and $h$
be a normalized cuspidal $p$
-ordinary Hecke eigenform of weight $\ell < k$
. Let $p$
be an Eisenstein prime for $h$
i.e. the residual Galois representation of $h$
at $p$
is reducible. In this talk, we show that the $p$
-adic $L$
-function and the characteristic ideal of the $p^{\infty}$
-Selmer group of the Rankin-Selberg convolution of $f$
, $h$
generate the same ideal modulo $p$
in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$
holds modulo $p$
. This is a joint work with Somnath Jha and Sudhanshu Shekhar.
Given a closed, oriented surface with genus greater that 2, we study quasi-Fuchsian hyperbolic 3-manifolds homeomorphic to this surface times the interval. Different properties of these manifolds have been carefully studied in previous important works on 3 manifold geometry and topology and some interesting questions about them still remain to be answered. In this talk, we will focus on a new geometric invariant associated to them which we call the measured foliations at infinity. These are horizontal measured foliations of a holomorphic quadratic differential ( the Schwarzian derivative ) associated canonically with each of the two connected component of the boundary at infinity of a quasi-Fuchsian manifold. We ask whether given any pair of measured foliations (F,G) on a surface, is there a quasi-Fuchsian manifold with F and G as it measured foliations at infinity. The answer is affirmative under certain assumptions; first, (F,G) satisfy the property of being an “arational filling pair” and second, the quasi-Fuchsian manifold should be very close to being “Fuchsian” . The goal of this talk would be introducing the concepts and outlining the proof idea.
The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.
By the resolution of the Poincare conjecture in 3D, we know that the only closed three-manifold with the trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation $\pi_1(M)\rightarrow SU(2)$ is the trivial one. Does this imply that $\pi_1(M)$ is trivial? The class of manifolds $M$ for which this question is interesting (and open) are integer homology spheres. We prove a result in this direction: the half-Dehn surgery on any non-trivial fibered knot $K$ in $S^3$ admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the strategy. This is based on work in progress, some jointly with Zhenkun Li and Fan Ye.
Associated to every reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, obtained by killing all “sufficiently long” braid words, as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal.
Motivated by the classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture, and continuing beyond the previous work of Khare, we attempt to obtain a classification of the finite-dimensional nil-Hecke algebras for all reflection groups $W$. These include the usual nil-Coxeter algebras for $W$ of finite type, their “fully commutative” analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). We further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\overline{12}$-avoiding signed permutations), and classify the Frobenius nil-Hecke algebras in the aforementioned cases. (Joint with Apoorva Khare.)
This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.
The focus of the first part is when the domain is the complex plane. In this case, given a finite cyclic configuration of points $P \subset \partial\mathbb{H}^3=\mathbb{CP}^1$, we construct a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ that is asymptotic to a twisted ideal polygon with ideal vertices contained in $P$. Moreover, we prove that given any ideal twisted polygon in $\mathbb{H}^3$ with rotational symmetry, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.
In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $PSL_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely marked and bordered surfaces (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (PSL_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$.
In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.
Motivated by optimization considerations and the (matrix theory) inequalities of Ky Fan and von Neumann,
we introduce a Fan-Theobald-von Neumann system as a triple $(V,W,\lambda)$, where $V$ and $W$ are real
inner product spaces and $\lambda:V\rightarrow W$ is a (nonlinear) map satisfying the following condition:
For all $c,u\in V$,
$$\max \{\langle c,x\rangle: x\in [u] \}=\langle \lambda(c),\lambda(u)\rangle,$$
where $[u]:= \{x:\lambda(x)=\lambda(u)\}$.
This simple formulation happens to be equivalent to the Fenchel conjugate formula of the form $(\phi\circ \lambda)^*=\phi^*\circ \lambda$ and a subdifferential formula in some settings and becomes useful in addressing linear/distance optimization problems over “spectral sets” which are of the form $\lambda^{-1}(Q)$, where $Q$ is a subset of $W$. Three standard examples of FTvN systems are: $(\mathbb{R}^n,\mathbb{R}^n,\lambda)$ with $\lambda(x):=x^\downarrow$ (the decreasing rearrangement of the vector $x\in \mathbb{R}^n$); $({\cal H}^n,\mathbb{R}^n,\lambda)$, where ${\cal H}^n$ is the space of $n$ by $n$ complex Hermitian matrices with $\lambda$ denoting the eigenvalue map; and $(M_n,\mathbb{R}^n, \lambda)$, where $M_n$ is the space of $n$ by $n$ complex matrices with $\lambda$ denoting the singular value map. Other examples come from Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). In the general framework of Fan-Theobald-von Neumann systems, we introduce and elaborate on the concepts of commutativity, automorphisms, majorization, etc. We will also talk about “transfer principles” where properties (such as convexity) of $Q$ are carried over to $\lambda^{-1}(Q)$, leading to a generalization of a celebrated convexity theorem of Chandler Davis.
n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.
I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino.
This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.
Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.
We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.
The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.
For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).
This thesis consists of two parts. In the first part, we introduce coupled K¨ahler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Amp`ere vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the K¨ahlerness of the symplectic form which arises in the moment map interpretation of Calabi-Yang-Mills equations.
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss the structures of the pairs of commuting $C_0$-semigroups of isometries in generality as well as under certain additional assumptions like double commutativity or dual double commutativity.
The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a factorization of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.
Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over $\mathbb{Q}$. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group.
Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree $d$?” To obtain more refined asymptotics, one may fix the Galois structure of the number fields in question. There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods to prove many new instances of these conjectures, including some joint results with Altug, Bhargava, Ho, Shankar, and Wilson.
I will discuss how the inverse problem of recovering conductances in an electrical network from its response matrix can be solved using an automorphism of the positive Grassmannian called the twist.
The origins of disordered actomyosin network contraction such as in the cellular cortex remain an active topic of research. We derive an agent-based mathematical model for the evolution of two-dimensional networks. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use simulations of disordered networks and find that both protein friction and actin filament bending are sufficient for contraction.
Asymptotic analysis of a special case of this model implies that bending induces a geometric asymmetry that enables motors to move faster close to filament plus-ends, inhibiting expansion.
We also explore a minimal model for pattern formation through biased turnover of actin filaments. The resulting discrete-time interacting particle system can be interpreted as voter model with continuous opinion space. We fully characterise the asymptotic shape of solutions which are characterised by transient clusters.
In this talk, we first introduce the basic structure of tumors and consequently present some fundamental modelling aspects of tumor growth based on ODE / PDE models. We then introduce the biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. We introduce what is called in-vivo and in-vitro tumors considering an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are saturated by a physiological extracellular fluid. The mass and momentum equations for both the phases are coupled due to the interaction term. Well-posedness results will be discussed in brief. The criterion for necrosis will be shown in terms of the nutrient transport.
Prime numbers have been studied by Humankind for centuries and have applications in Internet Cryptography. We will outline this connection and also talk about how prime numbers give rise to different number systems.
A “buckyball” or “fullerene” is a trivalent graph embedded in the sphere, all of whose rings have length 5 or 6. The term originates from the most famous buckyball, “Buckminsterfullerene,” a molecule composed of 60 carbon atoms. In this talk, I will explain why there are exactly 1203397779055806181762759 buckyballs with 10000 carbon atoms.
Nonlocal conservation laws are gaining interest due to their wide range of applications in modeling real world phenomena such as crowd dynamics and traffic flow. In this talk, the well-posedness of the initial value problems for certain class of nonlocal conservation laws, scalar as well as system, will be discussed and monotone finite volume approximations for such PDEs will be proposed. Strong compactness of the proposed numerical schemes will be presented and their convergence to the entropy solution will be proven. Some numerical results illustrating the established theory will also be presented.
If we had two extra thumbs, how would we check if “2024” is divisible by eleven? Or by “11”? We will see a simple test in any base $B$, i.e. usable by species having any number of fingers (whether shaped like hot-dogs or not); and for any divisor $d$. That is, the test works for everything ($d$), everywhere ($B$), all at once.
We will then move to recurring decimals. Note that 1/3 = 0.3333… and 1/3x3 = 0.1111… have the same number of digits - one - in their recurring parts. (Is 3 the only prime with this property in base 10?) More generally, we will see how many digits $1/d$ has in its recurring “decimal” expansion, for us or for any species as above.
Finally, for a species with a given number of fingers (= digits!), are there infinitely many primes $p$ for which the recurring part of $1/p$ has $p-1$ digits? (E.g. for us, 1/7 has the decimal recurring string (142857).) And what does this have to do with Gauss, Fermat, and one of the Bernoullis? Or with Artin and a decimal number starting with 0.3739558136… ? I will end by mentioning why this infinitude of primes holds for at least one species among humans (10), emus (6), ichthyostega (14), and computers (2) - but, we don’t know which one!
We classify similarity classes of tetrahedra whose dihedral angles are all rational multiples of $\pi$ (when measured in radians), answering a question of Conway-Jones from 1976. In the process, we also classify collections of vectors in $\mathbb{R}^3$ whose pairwise angles are rational. The proof uses a mixture of theoretical arguments, exact computations in computer algebra, and floating-point numerical computations. (Joint with Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein.)
We will define one of the most famous functions in all of mathematics, the Riemann zeta function, whose properties are the subject of one of the Millenium Problems. We will also look at some of its analogues for other objects.
We introduce an important family of polynomials, the cyclotomic polynomials, whose roots are the roots of unity of a fixed order. We explore the structure of these polynomials and the number fields that they generate, including a brief look at Gauss sums.
This talk will be a lucid introduction to the formal mathematics behind Euclidean Constructions, which we all learn in our middle school curriculum. The rules, regulations and restrictions of this type of construction will be discussed in detail. An alternative will also be suggested. We shall also find out how a completely geometric question can be answered using purely algebraic techniques giving rise to an elegant theory introduced in the nineteenth century by a famous French mathematician named Évariste Galois.
We shall discuss Reeb’s Theorem and basic differential topology of Morse functions. This was used by Milnor to prove the existence of exotic spheres in 7 dimensions. We shall propose a generalization of Reeb’s Theorem and discuss a proof of it. This is joint work with Sachchidanand Prasad.
The problem of algorithmically computing the volumes of convex bodies is a well studied problem in combinatorics and theoretical computer science. The best known results are perhaps those concerning the use of Markov Chain Monte Carlo techniques for approximately computing the volumes of general convex bodies. There are also results of a different kind: Deterministic (approximate) computation of the volumes of (certain)polytopes. In this direction, Alexander Barvinok and John Hartigan gave an algorithm based upon the Maximum Entropy heuristic from Statistical Physics that provides good approximations for certain classes of polytopes, that includes the transportation polytopes.
The Maximum Entropy heuristic, originally introduced by Jaynes in 1957 says the following: Suppose one is faced with an unknown probability distribution over a product space. Further suppose we know the expectations of a certain number of random variables with respect to this measure. Then the Maximum Entropy heuristic says that it ‘is natural’ to work with that probability distribution that has max entropy subject to the given linear constraints. Barvinok and Hartigan’s work uses this idea and combines it with some fundamental results about the computability of entropies of these max entropy distributions.
In this talk, I will show how to adapt this approach to Spectrahedra, which are a naturally occurring class of convex sets, defined as slices of the cone of Positive Semidefinite matrices. The case of spectrahedra shows up several surprises. As a byproduct of this work it will follow that central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. I will then give several examples to illustrate the utility of this method.
This is joint work with Jonathan Leake (U Waterloo) and Mahmut Levent Dogan (T U Berlin).
This talk comprises two parts. In the first part, we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few “labeled points”, our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian often perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data. We introduce a class of graph-based higher order fractional Sobolev spaces (H^s) and establish their consistency in the large data limit, along with applications to the SSL problem. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of a weighted Laplace-Beltrami operator in the continuum.
Obtaining optimal convergence rates for such spectra has so-far been an open question in stochastic homogenization. In the rest of the talk, we answer this question by obtaining optimal, state-of-the-art results for the case of a Poisson point cloud on a bounded domain in Euclidean space with Dirichlet or Neumann boundary conditions.
The first half is joint work with Dejan Slepcev (CMU), and the second half is joint work with Scott Armstrong (Courant).
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems. The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and L1 data in a circular oscillating domain. As the source term we are considering is in L1, we will examine the renormalized solutions. The second problem we will investigate is an optimal control problem governed by a second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the oscillations occur in $m$ directions, with $1< m < n$. We will explore the asymptotic behavior of this problem by homogenizing the corresponding optimality systems.
The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.
The (tame) class field theory for a smooth variety $X$
is the
study of describing the abelianized (tame) {'e}tale fundamental group of
$X$
in terms of some groups which are defined using algebraic cycles of $X$
.
In this talk, we study the tame class field theory for smooth varieties
over local fields. We will begin with defining few notions and recalling
various results from the past to overview the historical background of the
subject. We will then study abelianized tame fundamental group denoted as
$\pi^{ab,t}_{1}(X)$
, with the help of reciprocity map $\rho^{t}_{X} :
C^{t}(X) \rightarrow \pi^{ab,t}_{1}(X)$
and will describe the kernel and
topological cokernel of this map. This talk is based on a joint work with
Prof. Amalendu Krishna and Dr. Rahul Gupta.
Minimal Lagrangian tori in CP^{2} are the expected local model for particular point singularities of Calabi-Yau 3-folds and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods we construct new examples of minimal Lagrangian surfaces of genus 1/2(k-1)(k-2) for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit.
This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following.
(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.
(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate.
(3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.
Nanomedicine is an offshoot of nanotechnology that involves many disciplines, including the manipulation and manufacturing of materials, imaging, diagnosis, monitoring, and treatment. An efficient iterative reconstruction algorithm,together with Total Variation (TV), and a good mathematical model, can be used to enhance the spatial resolution and predictive capabilities. In this webinar, I will start with our current results using integrated approach for predicting efficient biomarkers for Acute respiratory distress syndrome (ARDS) and then move to PDE based (Total variation flow) approach for Image denoising which can have promising applications in denoising medical images from different modalities. In principle, I will be discussing the below-mentioned topics and their important concepts in dealing with the main markers of cardiovascular diseases, specifically Pulmonary Hypertension.
1. 4D FlowMRI Data Assimilation: Integrated approach reveals new biomarkers for Experimental ARDS conditions. The purpose of this study is to characterize flow patterns and several other hemodynamic parameters (WSS, OSI, Helicity) using computational fluid dynamics model by combining imaging data from 4D-Flow MRI with hemodynamic pressure and flow waveforms from control and hypertensive subjects (related to acute respiratory distress syndrome). This work mainly concerns how to facilitate bench-bedside approach using integrated approach by combining CFD and AI.
2. An adaptive $C^0$ interior penalty discontinuous galerkin approximation of second order total variation problems.
Singular nonlinear fourth order boundary value problems have significant applications in image processing and material science.
We consider an adaptive $C^0$ Interior Penalty Discontinuous Galerkin (C0IPDG) method for the numerical solution of singular
nonlinear fourth order boundary value problems arising from the minimization of functionals involving the second order total
variation. The mesh adaptivity will be based on an aposteriori error estimator that can be derived by duality arguments. The
fourth order elliptic equation reads as follows:
\begin{align}
u + \lambda \nabla \cdot \nabla \cdot \frac{D^2 u}{|D^2 w|} = & \ 0 \quad \mbox{in} \ Q := \Omega, \\
u = & \ 0 \quad \mbox{on} \ \Gamma,\\
n_{\Gamma} \cdot\frac{D^2 u} {n_{\Gamma}} = & \ 0 \quad \mbox{on} \ {\Gamma}.
\end{align}
ChatGPT and other advances in Artificial Intelligence have become popular sensations. In parallel with this has been an enormous advance in the digitization of mathematics through Interactive Theorem Provers and their libraries. Artificial Intelligence has started entering mathematics through these and other routes.
This session will have some presentations/demos about present use of Computer Proofs, Artificial Intelligence together and separately in Mathematics and related fields (including software), both in research and in teaching. After that everyone is welcome to discuss their work, ideas, wish-lists etc related to these themes.
I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.
For decades, mathematicians have been using computers to calculate. More recently there has been some interest in trying to get them to reason. What is the difference? An example of a calculation: compute the first one million prime numbers. An example of reasoning: prove that there are infinitely many prime numbers. Tools like ChatGPT can prove things like this, because they have seen many proofs of it on the internet. But can computers help researchers to come up with new mathematics? Hoping that a computer will automatically prove the Riemann Hypothesis is still science fiction. But new tools and methods are becoming available. I will give an overview of the state of the art.
(This is a Plenary talk in the EECS Research Students’ Symposium)
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
Let $K$
be an imaginary quadratic field of class number $1$
such that both $p$
and $q$
split in $K$
. We show that under appropriate hypotheses, the $p$
-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $\mathbb{Z}_q$
-extension of $K$
. This is joint work with Antonio Lei.
The Virasoro algebra, which can be realized as a central extension of (complex) polynomial vector fields on the unit circle, plays a key role in the representation theory of affine Lie algebras, as it acts on almost every highest weight module for the affine Lie algebra. This remarkable phenomenon eventually led to constructing the affine-Virasoro algebra, which is a semi-direct product of the affine Lie algebra and the Virasoro algebra with a common extension. The representation theory of the affine-Virasoro algebra has been studied extensively and is an extremely well-developed classical object.
In this talk, we shall consider a natural higher-dimensional analogue of the affine-Virasoro algebra, popularly known as the full toroidal Lie algebra in the literature and henceforth classify the irreducible Harish-Chandra modules over this Lie algebra. As a by-product, we also obtain the classification of all possible irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, thereby proving Eswara Rao’s conjecture (conjectured in 2004). These directly generalize the well-known result of O. Mathieu for the classical Virasoro algebra and also the recent work of Billig–Futorny for the higher rank Witt algebra.
Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition, for instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure.
Consider a finite group $G$ and a prime number $p$ dividing the order of $G$. A $p$-regular element of $G$ is an element whose order is coprime to $p$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. The quasi $p$-Steinberg character is a generalization of the well-known $p$-Steinberg character. A group, which does not have a non-linear quasi $p$-Steinberg character, can not be a finite group of Lie type of characteristic $p$. Therefore, it is natural to ask for the classification of all non-linear quasi $p$-Steinberg characters of any finite group $G$. In this joint work with Digjoy Paul and Pooja Singla, we classify quasi $p$-Steinberg characters of all finite complex reflection groups.
We report on new ideas of Ki-Seng Tan and myself towards the construction of a $p$
-adic $L$
-function associated to an automorphic overconvergent $F$
-isocrystal over a curve over a finite field. This function should be of interest in the Iwasawa theory for such coefficients.
Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Mike Wolf.
Let $F$
be a totally real field. Let $\pi$
be a cuspidal cohomological automorphic representation for $\mathrm{GL}_2/F$
. Let $L(s, \mathrm{Ad}^0, \pi)$
denote the adjoint $L$
-function associated to $\pi$
. The special values of this $L$
-function and its relation to congruence primes have been studied by Hida, Ghate and Dimitrov. Let $p$
be an integer prime. In this talk, I will discuss the construction of a $p$
-adic adjoint $L$
-function in neighbourhoods of very decent points of the Hilbert eigenvariety. As a consequence, we relate the ramification locus of this eigenvariety to the zero set of the $p$
-adic $L$
-functions. This was first established by Kim when $F=\mathbb{Q}$
. We follow Bellaiche’s description of Kim’s method, generalizing it to arbitrary totally real number fields. This is joint work with John Bergdall and Matteo Longo.
From the longest increasing subsequence in a random permutation to the shortest distance in a randomly weighted two dimensional Euclidean lattice, a large class of planar random growth models are believed to exhibit shared large scale features of the so-called Kardar-Parisi-Zhang (KPZ) universality class. Over the last 25 years, intense mathematical activity has led to a lot of progress in the understanding of these models, and connections to several other topics such as algebraic combinatorics, random matrices and partial differential equations have been unearthed. I shall try to give an elementary introduction to this area, describe some of what is known as well as many questions that remain open.
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about the multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
The Siegel-Veech transform is a basic tool in homogeneous as well as Teichmuller dynamics. I will introduce the transform and explain how it can be used in counting problems.
It is well known that solvability of the complex Monge- Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s celebrated resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others (including some at IISc). A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. Results of this nature are ubiquitous in Kaehler geometry - existence of weak Kaehler-Einstein metrics on normal varieties and Hermitian-Einstein metrics on reflexive sheaves to name a couple. Much closer to the present theme, is the work of Boucksom-Eyssidieux-Guedj-Zeriahi on solving the complex Monge-Ampere equation in big classes. In the talk, I will first speak about some joint and ongoing work with Ramesh Mete and Jian Song, that offers a reasonably complete resolution in complex dimension two, at least for the J-equation and the dHYM equations. Next, I will discuss some conjectures on what one can expect in higher dimensions.
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
This thesis consists of two parts. In the first part, we introduce coupled Kähler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Ampère vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the Kählerness of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations.
This will be an introductory talk on some matters relating to Fatou-Bieberbach domains and uniformizing stable manifolds.
In the late 1950s, an important problem in number theory was to extend the notion of $L$
-functions attached to cuspforms on the upper-half
plane to automorphic forms on reductive groups. Langlands’s work on non-abelian Harmonic analysis, namely the problem of the spectral decomposition of automorphic forms, led him to a general notion of $L$
-functions
attached to cuspforms. We give an introduction to the spectral decomposition of automorphic forms and discuss some contemporary problems.
Over an unramified extension $F/\mathbb{Q}_p$
, by the works of Fontaine, Wach, Colmez and Berger, it is well-known that a crystalline representation of the absolute Galois group of $F$
is of finite height. Moreover, in this case, to a crystalline representation one can functorially attach a lattice inside the associated etale $(\varphi, \Gamma)$
-module called the Wach module. Berger showed that the aforementioned functor induces an equivalence between the category of crystalline representations and Wach modules. Furthermore, this categorical equivalence admits an integral refinement. In this talk, our goal is to generalize the notion of Wach modules to relative $p$
-adic Hodge theory. For a “small” unramified base (in the sense of Faltings) and its etale fundamental group, we will generalize the result of Berger to an equivalence between crystalline representations and relative Wach modules as well as establish its integral refinement.
Cauchy’s determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the rank-one matrix $(u_i v_j)$. This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials $p(t)$ with a negative coefficient that entrywise preserve positivity. A key novelty here has been the application of Schur polynomials, which essentially arise from the expansion of $\det(p[{\bf u}{\bf v}^T])$, to positivity.
In the first half of the talk, I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion of $\det(f[{\bf u}{\bf v}^T])$ for all power series $f$. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.) In the second half, joint with Siddhartha Sahi, I will explain how to extend the above determinantal identities to (a) any subgroup $G$ of signed permutations; (b) any character of $G$, or even complex class function; (c) any commutative ground ring $R$; and (d) any power series over $R$.
Andreatta, Iovita, and Pilloni have proven the existence of an adic eigencurve, which includes characteristic $p$
points at the boundary. In joint work with Ruochuan Liu, using the theory of Crystalline periods, we show that the Galois representations associated to these points satisfy an appropriate trianguline property.
In 1976 Bernstein, Gelfand, and Gelfand introduced Category $\mathcal{O}$ for a semi-simple Lie algebra $\mathfrak{g}$. This is roughly the smallest sub-category of $\mathfrak{g}$-mod containing the Verma modules and such that the simple modules have projective covers. After work of Beilinson–Bernstein and Beilinson–Ginzburg–Soergel it became clear that the the good homological properties of this category were due to the fact that it can be identified with a category of perverse sheaves on the flag variety $G/B$.
In this talk I will show how this story fits into the physics of 3d mirror symmetry. This leads to conjectural 2-categorifications of category $\mathcal{O}$ that can be computed explicitly for $\mathfrak{g} = \mathfrak{sl}_2$.
The geometry, and the (exposed) faces, of $X$ a “Root polytope” or “Weyl polytope” over a complex simple Lie algebra $\mathfrak{g}$, have been studied for many decades for various applications, including by Satake, Borel–Tits, Casselman, and Vinberg among others. This talk focuses on two recent combinatorial analogues to these classical faces, in the discrete setting of weight-sets $X$.
Chari et al [Adv. Math. 2009, J. Pure Appl. Algebra 2012] introduced and studied two combinatorial subsets of $X$ a root system or the weight-set wt $V$ of an integrable simple highest weight $\mathfrak{g}$-module $V$, for studying Kirillov–Reshetikhin modules over the specialization at $q=1$ of quantum affine algebras $U_q(\hat{\mathfrak{g}})$ and for constructing Koszul algebras. Later, Khare [J. Algebra 2016] studied these subsets under the names “weak-$\mathbb{A}$-faces” (for subgroups $\mathbb{A}\subseteq (\mathbb{R},+)$) and “$212$-closed subsets”. For two subsets $Y\subseteq X$ in a vector space, $Y$ is said to be $212$-closed in $X$, if $y_1+y_2=x_2+x_2$ for $y_i\in Y$ and $x_i\in X$ implies $x_1,x_2\in Y$.
In finite type, Chari et al classified these discrete faces for $X$ root systems and wt $V$ for all integrable $V$, and Khare for all (non-integrable) simple $V$. In the talk, we extend and completely solve this problem for all highest weight modules $V$ over any Kac–Moody Lie algebra $\mathfrak{g}$. We classify, and show the equality of, the weak faces and $212$-closed subsets in the three prominent settings of $X$: (a) wt $V$ $\forall V$, (b) the hull of wt $V$ $\forall V$, (c) wt $\mathfrak{g}$ (consisting of roots and 0). Moreover, in the case of (a) (resp. of (b)), such subsets are precisely the weights falling on the exposed faces (resp. the exposed faces) of the hulls of wt $V$.
While statistical decision theory led me to game theory, certain war duel models, and the close connection between the Perron–Frobenius theorem and game theory led me to the works of M.G. Krein on special classes of cones, and spectral properties of positive operators. The influence of Professors V.S. Varadarajan, K.R Parthasarathy and S.R.S Varadhan in early 60’s at ISI is too profound to many of us as young graduate students in 1962-66 period. The talk will highlight besides the theorems, the teacher-student interactions of those days.
The study of diluted spin glasses may help solve some problems in computer science and physics. In this talk, I shall introduce the diluted Shcherbina–Tirozzi (ST) model with a quadratic Hamiltonian, for which we computed the free energy at all temperatures and external field strengths. In particular, we showed that the free energy can be expressed in terms of the weak limits of the quenched spin variances and identified these weak limits as the unique fixed points of a recursive distributional operator. The talk is based on a joint work with Wei-Kuo Chen and Arnab Sen.
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal
control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems.
The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and $L^1$
data in a circular oscillating domain. As the source term we are considering is in $L^1$, we will examine the
renormalized solutions. The second problem we will investigate is an optimal control problem governed by a
second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the
oscillations occur in $m$ directions, with $1<m<n$
. We will explore the asymptotic behavior of this problem by
homogenizing the corresponding optimality systems.
In the first half of the talk I will recall two classical theorems - Dirichlet’s class number formula and Stickelberger’s theorem. Stark and Brumer gave conjectural generalisations of these statements. We will see formulations of some of these conjectures. In the second half of the talk we will restrict to a special case of the Brumer-Stark conjecture. Here p-adic techniques can be used to resolve the conjecture. We will see a sketch of this proof. This is joint work with Samit Dasgupta.
Let $\mathfrak g$ be a Borcherds–Kac–Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak n_+ \oplus \mathfrak h \oplus \mathfrak n_{-}$, and each root space $\mathfrak g_{\alpha}$ is either contained in $\mathfrak n_+$ or $\mathfrak n_{-}$. In particular, each $\mathfrak g_{\alpha}$ involves only the relations (2) and (3). In this talk, we will discuss the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially using heaps of pieces and construct two different bases for these root spaces of $\mathfrak g$.
The Thom conjecture, proven by Kronheimer and Mrowka in 1994, states that complex curves in $\mathbb{C}{\rm P}^2$ are genus minimizers in their homology class. We will show that an analogous statement does not hold for complex hypersurfaces in $\mathbb{C}{\rm P}^3$. This is joint work with Ruberman and Strle.
The intersection theory of the Grassmannian, known as Schubert calculus, is an important development in geometry, representation theory and combinatorics. The Quot scheme is a natural generalization of the Grassmannian. In particular, it provides a compactification of the space of morphisms from a smooth projective curve C to the Grassmannian. The intersection theory of the Quot scheme can be used to recover Vafa-Intriligator formulas, which calculate explicit expressions for the count of maps to the Grassmannian subject to incidence conditions with Schubert subvarieties.
The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with symplectic (or symmetric) bilinear form. I will present explicit formulas for certain intersection numbers of the symplectic and the orthogonal analogue of Quot schemes. Furthermore, I will compare these intersection numbers with the Gromov–Ruan–Witten invariants of the corresponding Grassmannians.
Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the $\ell$
-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the $\ell$
-primary part of principally polarized Abelian $3$
-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.
We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over $\mathbb{Q}$
defined over at least $10$
variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).
We shall discuss Legendre Pairs, an interesting combinatorial object related to the Hadamard conjecture. We shall demonstrate the exceptional versatility of Legendre Pairs, as they admit several different formulations via concepts from disparate areas of Mathematics and Computer Science. We shall mention old and new results and conjectures within the past 20+ years, as well as potential future avenues for investigation.
The video of this talk is available on the IISc Math Department channel.
$\mathrm{Per}_n $ is an affine algebraic curve, defined over $\mathbb Q$, parametrizing (up to change of coordinates) degree-2 self-morphisms of $\mathbb P^1$ with an $n$-periodic ramification point. The $n$-th Gleason polynomial $G_n$ is a polynomial in one variable with $\mathbb Z$-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of $\mathbb C$ with an $n$-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is $\mathrm{Per}_n$ connected? (2) Is $G_n$ irreducible over $\mathbb Q$?
We show that if $G_n$ is irreducible over $\mathbb Q$, then $\mathrm{Per}_n$ is irreducible over $\mathbb C$, and is therefore connected. In order to do this, we find a $\mathbb Q$-rational smooth point of a projective completion of $\mathrm{Per}_n$. This $\mathbb Q$-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits an interpretation in terms of tropical geometry.
(This talk will be pitched at a broad audience.)
Given a bipartite graph $G$ (subject to a constraint), the “cross-ratio degree” of G is a non-negative integer invariant of $G$, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings. Finally, time permitting, I may discuss the tropical side of the story.
The theory of Lie superalgebras have many applications in various areas of Mathematics and Physics. Kac gives a comprehensive description of mathematical theory of Lie superalgebras, and establishes the classification of all finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. In the last few years the theory of Lie superalgebras has evolved remarkably, obtaining many results in representation theory and classification. Most of the results are extension of well known facts of Lie algebras. But the classification of all finite dimensional nilpotent Lie superalgebras is still an open problem like that of finite dimensional nilpotent Lie algebras. Till today nilpotent Lie superalgebras $L$ of $\dim L \leq 5$ over real and complex fields are known.
Batten introduced and studied Schur multiplier and cover of Lie algebras and later on studied by several authors. We have extended these notation to Lie superalgebra case. Given a free presentation $ 0 \longrightarrow R \longrightarrow F \longrightarrow L \longrightarrow 0 $ of Lie superalgebra $L$ we define the multiplier of $L$ as $\mathcal{M}(L) = \frac{[F,F]\cap R}{[F, R]}$. In this talk we prove that for nilpotent Lie superalgebra $L = L_{\bar{0}} \oplus L_{\bar{1}}$ of dimension $(m\mid n)$ and $\dim L^2= (r\mid s)$ with $r+s \geq 1$, \begin{equation} \dim \mathcal{M}(L)\leq \frac{1}{2}\left[(m + n + r + s - 2)(m + n - r -s -1) \right] + n + 1. \end{equation} Moreover, if $r+s = 1$, then the equality holds if and only if $ L \cong H(1, 0) \oplus A(m-3 \mid n)$ where $A(m-3 \mid n)$ is an abelian Lie superalgebra of dimension $(m-3 \mid n)$, and $H(1, 0)$ is special Heisenberg Lie superalgebra of dimension $(3 \mid 0)$. Then we define the function $s(L)$ as \begin{equation} s(L)= \frac{1}{2}(m+n-2)(m+n-1)+n+1-\dim \mathcal{M}(L). \end{equation} Clearly $s(L) \geq 0$ and structure of $L$ with $s(L)=0$ is known. We obtain classification all finite dimensional nilpotent Lie superalgebras with $s(L) \leq 2$.
We hope, this leads to a complete classification of the finite dimensional nilpotent Lie superalgebras of dimension $6,7$.
In this talk, I will discuss the relation of square-free monomial ideals to combinatorics. In particular, I will explain some combinatorial invariants of hypergraphs that can be used to describe the Castelnuovo–Mumford regularity and componentwise linearity of different kinds of powers of squarefree monomial ideals.
Expanders are a family of finite graphs that are sparse but highly connected. The first explicit examples of expanders were quotients of a Cayley graph of a discrete group with Property (T) by finite index subgroups. This was due to Margulis. In recent years, higher dimensional generalizations of expander graphs (family of simplicial complexes of a fixed dimension) have received much attention. I will talk about a generalization of Margulis’ group theoretic construction that replaces expanders by one of its higher analogs.
I will report on recent work with Lichtenbaum and Suzuki on a new proof of the relation between the arithmetic of an elliptic curve over function fields and surfaces over finite fields.
We discuss the random dynamics and asymptotic analysis of 2D Navier–Stokes equations. We consider two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a linear multiplicative white noise of Ito type on the whole space. We prove that non-autonomous 2D SNSE generates a bi-spatial continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $L^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(L^2(\mathbb{R}^2), \mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $L^2$-norm but also in $\mathbb{H}^1$-norm. Next, we discuss the existence of an invariant measure for the random cocycle associated with autonomous SNSE which is a consequence of the existence of random attractors. We prove the uniqueness of invariant measures by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions.
We construct a pointwise Boutet de Monvel-Sjostrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudo-convex domains in dimension two. Next we present an application where we prove that a weakly pseudoconvex two dimensional domain of finite type with a Kähler-Einstein Bergman metric is biholomorphic to the unit ball. This extends earlier work of Fu-Wong and Nemirovski-Shafikov. Based on joint works with C.Y. Hsiao and M. Xiao.
The system of hyperbolic conservation laws is the first order partial differential equations of the form \begin{equation} \frac{\partial \mathbf{u}}{\partial t}+\sum_{\alpha=1}^d \frac{\partial \mathbf{f}_{\alpha}(\mathbf{u})}{\partial x_{\alpha}} =0,~~~~~~ (\mathbf{x},t)\in \Omega \times (0,T], \qquad \qquad \qquad (1) \end{equation} subject to initial data \begin{equation} \mathbf{u}(\mathbf{x},0)=\mathbf{u}_0(\mathbf{x}), \end{equation} where $\mathbf{u}=(u_1,u_2,\ldots, u_m)\in \mathbb{R}^m$ are the conserved variables and $\mathbf{f}_{\alpha}:\mathbb{R}^m \rightarrow \mathbb{R}^m$, $\alpha=1,2,\ldots,d$ are the Cartesian components of flux. It is well-known that the classical solution of (1) may cease to exist in finite time, even when the initial data is sufficiently smooth. The appearance of shocks, contact discontinuities and rarefaction waves in the solution profile make difficult to devise higher-order accurate numerical schemes because numerical schemes may develop spurious oscillations or sometimes blow up of the solution may occur.
In this talk, we will discuss recently developed Weighted Essentially Non-oscillatory (WENO) and hybrid schemes for hyperbolic conservation laws. These schemes compute the solution accurately while maintaining the high resolution near the discontinuities in a non-oscillatory manner.
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
Let $F$
be a totally real field and $p$
be an odd prime unramified in $F$
. We will give an overview of the problem of determining the explicit mod $p$
structure of a modular $p$
-adic Galois representation and determining the associated local Serre weights. The Galois representations are attached to Hilbert modular forms over $F$
, more precisely to eigenforms on a Shimura curve over $F$
. The weight part of the Serre’s modularity conjecture for Hilbert modular forms relates the local Serre weights at a place $v|p$
to the structure of the mod $p$
Galois representation at the inertia group over $v$
. Thus, local Serre weights give good information on the structure of the modular mod $p$
Galois representation. The eigenforms considered are of small slope at a fixed place $\mathbf{p}|p$
, and with certain constraints on the weight over $\mathbf{p}$
. This is based on a joint work with Shalini Bhattacharya.
Abstract: A fundamental problem in complex geometry is to construct canonical metrics, such as Hermite-Einstein (HE) metrics on vector bundles and constant scalar curvature Kähler (cscK) metrics on Kahler manifolds. On a given vector bundle/manifold, such a metric may or may not exist, in general. The existence question for such metrics has been found to have deep connections to algebraic geometry. In the case of vector bundles, the Hitchin-Kobayashi correspondence proved by Uhlenbeck–Yau and Donaldson show that the existence of a HE metric is captured by the notion of slope stability for the vector bundle. In the case of manifolds, the still open Yau-Tian-Donaldson conjecture relates the existence of cscK metrics to K-stability of the underlying polarised variety.
Together with Ruadhaí Dervan, I started a research programme where we study canonical metrics, called Optimal Symplectic Connections, and a notion of stability, on fibrations. We proposed a Hitchin-Kobayashi/Yau-Tian-Donaldson type conjecture in this setting as well. In the case when the fibration is the projectivisation of a vector bundle, we recover the Hermite-Einstein and slope stability notions, respectively, and as such the theory can be seen as a generalisation of the classical bundle theory to more general fibrations. There has recently been great progress on this topic both on the differential and algebraic side, through works of Hallam, McCarthy, Ortu, Hattori, Spotti and Engberg, in addition to the joint works with Dervan. The aim of this talk is to give an introduction to and overview of the status of this programme.
Given any graph, we can uniquely associate a square matrix which stores informations about its vertices and how they are interconnected. The goal of spectral graph theory is to see how the eigenvalues and eigenvectors of such a matrix representation of a graph are related to the graph structure. We consider here (multi)digraphs and define a new matrix representation for a multidigraph and named it as the complex adjacency matrix.
The relationship between the adjacency matrix and the complex adjacency matrix of a multidigraph are established. Furthermore, some of the advantages of the complex adjacency matrix over the adjacency matrix of a multidigraph are observed. Besides, some of the interesting spectral properties (with respect to the complex adjacency spectra) of a multidigraph are established. It is shown that not only the eigenvalues, but also the eigenvectors corresponding to the complex adjacency matrix of a multidigraph carry a lot of information about the structure of the multidigraph.
Abstract: I will discuss the growth of the number of infinite dihedral subgroups of lattices G in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through an even order orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group that’s conjugate to its own inverse. We obtain the asymptotic growth of the number of reciprocal geodesics (or infinite dihedral subgroups) in any orbifold, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.
In this talk, we will discuss the Calderón type inverse problem of determining the coefficients of the nonlocal operators. In the mathematical literature, the method of Electrical Impedance Tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium is known as Calderón’s problem. We will introduce the nonlocal analog of it and further study the connection with the local analog as well.
Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network and are the central objects in graph limit (graphon) theory. In this talk we will derive the higher-order fluctuations (asymptotic distributions) of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can have both Gaussian or non-Gaussian components, depending on a notion of regularity of subgraphs, where the non-Gaussian component is an infinite weighted sum of centered chi-squared random variables with the weights determined by the spectral properties of the graphon. We will also discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs.
(Joint work with Anirban Chatterjee and Svante Janson.)
We introduce a smoothed version of the equivariant $S$
-truncated
$p$
-adic Artin $L$
-function for one-dimensional admissible $p$
-adic Lie
extensions of number fields. Integrality of this smoothed $p$
-adic
$L$
-function, conjectured by Greenberg, has been verified for pro-$p$
extensions (assuming the Equivariant Iwasawa Main Conjecture) as well as
$p$
-abelian extensions (unconditionally). Integrality in the general case
is also expected to hold, and is the subject of ongoing research.
A poset denoted $\mathsf{GTS}_n$ on the set of unlabeled trees with $n$ vertices was defined by Csikvàri. He showed that several tree parameters are monotonic as one goes up this $\mathsf{GTS}_n$ poset. Let $T$ be a tree on $n$ vertices and let $\mathcal{L}_q^T$ be the $q$-analogue of its Laplacian. For all $q\in \mathbb{R}$, I will discuss monotonicity of the largest and the smallest eigenvalues of $\mathcal{L}_q^T$ along the $\mathsf{GTS}_n$ poset.
For a partition $\lambda \vdash n$, let the normalized immanant of $\mathcal{L}_q^T$ indexed by $\lambda$ be denoted as $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$. Monotonicity of $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$ will be discussed when we go up along $\mathsf{GTS}_n$ or when we change the size of the first row in the hook partition $(\lambda=k,1^{n-k})$ and the two row partition $\lambda=(n-k,k)$. We will also discuss monotocity of each coefficients in the $q$-Laplacian immanantal polynomials $\overline{\mathrm{Imm}}_{\lambda}(xI-\mathcal{L}_q^T)$ when we go up along $\mathsf{GTS}_n$. At the end of this talk, I will discuss our ongoing research projects and future plans.
This is a joint work with Prof. A. K. Lal (IITK) and Prof. S. Sivaramakrishnan (IITB).
Marc Rieffel had introduced the notion of quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions of two sphere in this distance, that one finds in many scattered places in the theoretical physics literature. The compact quantum metric spaces and convergence in the quantum Gromov-Hausdorff distance has been explored by a lot of mathematicians in the last two decades. We will define compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman space and prove that it converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance. This is a joint work with Prof. Tirthankar Bhattacharyya.
The video of this talk is available on the IISc Math Department channel.
The Poincaré holonomy variety (or $sl(2, C)$-oper) is the set of holonomy representations of all complex projective structures on a Riemann surface. It is a complex analytic subvariety of the $PSL(2, C)$ character variety of the underlying topological surface. In this talk, we consider the intersection of such subvarieties for different Riemann surface structures, and we prove the discreteness of such an intersection. As a corollary, we reprove Bers’ simultaneous uniformization theorem, without any quasiconformal deformation theory.
It is a natural question to count matrices $A$
with integer entries in an expanding box of side length $x$
with $\det(A) = r$
, a fixed integer; or with the characteristic polynomial of $A = f$
, a fixed integer polynomial; and there are several results in the literature on these problems. Most of the existing results, which use either Ergodic methods or Harmonic Analysis, give asymptotics for the number of such matrices as $x$
goes to infinity and in the only result we have been able to find that gives a bound on the error term, the bound is not very satisfactory. The aim of this talk will be to present an ongoing joint work with Rachita Guria in which, for the easiest case of $2 \times 2$
matrices, we have been able to obtain reasonable bounds for the error terms for the above problems by employing elementary Fourier Analysis and results from the theory of Automorphic Forms.
For a real number $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. The study of the sequence $\|\alpha^n\|$ for $\alpha > 1$ naturally arises in various contexts in number theory. For example, it is not known that the sequence $\|e^n\|$ tends to zero as $n$ tends to infinity. Also, the growth of the sequence $\|(3/2)^n\|$ is linked to the famous Waring’s problem. This was the motivation for Mahler in 1957 to prove that for any non-integral rational number $\alpha > 1$ and any real number $c$ with $0 < c < 1$, the inequality $\|\alpha^n\| < c^n$ has only finitely many solutions in $n\in\mathbb{N}$. Mahler also asked the characterization of all algebraic numbers satisfying the same property as the non-integral rational numbers. In 2004, Corvaja and Zannier proved a Thue-Roth-type inequality with moving targets and as consequence, they completely answered the above question of Mahler. In this talk, we will explore this theme and will present recent result, building on the earlier works of Corvaja and Zannier, establishing an inhomogeneous Thue-Roth’s type theorem with moving targets.
Let $q$ be a prime power and define $(n)_q:=1+q+q^2+\cdots+q^{n-1}$, for a non-negative integer $n$. Let $B_q(n)$ denote the set of all subspaces of $\mathbb{F}_q^n$, the $n$-dimensional $\mathbb{F}_q$-vector space of all column vectors with $n$ components.
Define a $B_q(n)\times B_q(n)$ complex matrix $M_{q,n}$ with entries given by
\begin{equation}
M_{q,n}(X,Y):=
\begin{cases}
1&\text{ if }Y\subseteq X, \dim(Y)=\dim(X)-1,\\
q^{\dim(X)}&\text{ if }X\subseteq Y, \dim(Y)=\dim(X)+1,\\
0&\text{ otherwise.}
\end{cases}
\end{equation}
We think of $M_{q,n}$ as a $q$-analog of the adjacency matrix of the
$n$-cube. We show that the eigenvalues of $M_{q,n}$ are
\begin{equation}
(n-k)_q - (k)_q\text{ with multiplicity }\binom{n}{k}_q,\quad k=0,1,\dots,n,
\end{equation}
and we write down an explicit canonical eigenbasis of $M_{q,n}$. We give
a weighted count of the number of rooted spanning trees in the $q$-analog
of the $n$-cube.
This talk is based on a joint work with M. K. Srinivasan.
In this talk, we will consider the issues of non-existence of solutions to a Yamabe type equation on bounded Euclidean domains (dim>2). The leading order terms of this equation are invariant under conformal transformations which leads to the classical Pohozaev identity. This in turn gives non-existence of solutions to the PDE when the domain is star-shaped with respect to the origin.
We show that this non-existence is surprisingly stable under perturbations, which includes situations not covered by the Pohozaev obstruction, if the boundary of the domain has a positive curvature. In particular, we show that there are no positive variational solutions to our PDE under $C^1$-perturbations of the potential when the domain is star-shaped with respect to the origin and the mean curvature of the boundary at the origin is positive. The proof of our result relies on sharp blow-up analysis. This is a joint work with Nassif Ghoussoub (UBC, Vancouver) and Frédéric Robert (Institut Élie Cartan, Nancy).
We bound a short second moment average of $\mathrm{GL}(3)$
and $\mathrm{GL}(3) \times \mathrm{GL}(1)$
$L$
-functions. These yield $t$
-aspect and depth aspect subconvexity bounds respectively, and improve upon the earlier subconvexity exponents. This moment estimate provides an analogue for cusp forms of Ivic’s bound for the sixth moment of the zeta function, and is the first time a short second moment has been used to obtain a subconvex bound in higher rank. This is a joint work with Ritabrata Munshi and Wing Hong Leung.
In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections as well as a qualitative strengthening of her theorem that describes what these curves (and their complements) actually look like. This is joint work with Francisco Arana-Herrera.
In the first part of the talk we will discuss the main statement of local class field theory and discuss the statement of Local Langlands correspondence for $GL_2(K)$, where $K$ is a non-archimedean local field. In the process, we will also introduce all the objects in the statement of correspondence. We will then discuss a brief sketch of the proof of the main statement of local class field theory.
We will discuss total mean curvatures, i.e., integrals of symmetric functions of the principle curvatures, of hypersurfaces in Riemannian manifolds. These quantities are fundamental in geometric variational problems as they appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in non positively curved spaces, which are obtained via Reilly’s identities, Chern’s formulas, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss the a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry. In this talk, we introduce the J-equation on holomorphic vector bundles over compact Kahler manifolds, as an extension of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental properties as well as examples. In particular, we give algebraic obstructions called the (asymptotic) J-stability in terms of subbundles on compact Kahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.
I will answer some questions (admissibility, dimensions of invariants by Moy-Prasad groups)
on representations of reductive $p$
-adic groups and on Hecke algebras modules raised in my paper for the 2022-I.C.M. Noether lecture.
This talk will be a continuation of my previous talk. In this talk, I will present the proof of a result stated in my earlier talk, which characterizes eigenfunctions of the Laplace–Beltrami operator through sphere averages as the radius of the sphere tends to infinity in a rank one symmetric space of noncompact type.
The video of this talk is available on the IISc Math Department channel.
If $\theta$
is an involution on a group $G$
with fixed points $H$
,
it is a question of considerable interest to classify irreducible representations of $G$
which carry an $H$
-invariant linear form. We will discuss some cases of this
question paying attention to finite dimensional representation of compact groups
where it is called the Cartan-Helgason theorem.
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macro-scale, we introduce the bivariate Modified-quadratic Hawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
We will present some recent work on the classification of shrinking gradient Kähler-Ricci solitons on complex surfaces. In particular, we classify all non-compact examples, which together with previous work of Tian, Wang, Zhu, and others in the compact case gives the complete classification. This is joint work with R. Bamler, R. Conlon, and A. Deruelle.
It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions “on average” for moduli $q$
in the range $q \le x^{1/2 -\epsilon }$
for any $\epsilon>0$
. In 1989, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli $q$
is extended to $q \le x/ (\log x)^B$
for any $B>1$
. We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
We prove existence of twisted Kähler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when -K_X is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. This is joint work with Kewei Zhang.
Hilbert modular forms are generalization of classical modular forms over totally real number fields. The Fourier coefficients of a modular form are of great importance owing to their rich arithmetic and algebraic properties. In the theory of modular forms one of the classical problems is to determine a modular form by a subset of all Fourier coefficient. In this talk, we discuss about to determination of a Hilbert modular form by the Fourier coefficients indexed by square-free integral ideals. In particular, we talk about the following result.
Given any $\epsilon>0$
, a non zero Hilbert cusp form $\mathbf{f}$
of weight $k=(k_1,k_2,\ldots, k_n)\in (\mathbb{Z}^{+})^n$
and square-free level $\mathfrak{n}$
with Fourier coefficients
$C(\mathbf{f},\mathfrak{m})$
, then there exists a square-free integral ideal $\mathfrak{m}$
with $N(\mathfrak{m})\ll k_0^{3n+\epsilon} N(\mathfrak{m})^{\frac{6n^2 +1}{2}+\epsilon}$
such that $C(\mathbf{f},\mathfrak{m})\neq 0$
. The implied constant depend on $\epsilon , F.$
Let $F$
be a global field and $\Gamma_F$
its absolute Galois group. Given
a continuous representation $\bar{\rho}: \Gamma_F \to G(k)$
, where $G$
is a split
reductive group and $k$
is a finite field, it is of interest to know when $\bar{\rho}$
lifts
to a representation $\rho: \Gamma_F \to G(O)$
, where $O$
is a complete discrete
valuation ring of characteristic zero with residue field $k$
. One would also like to control
the local behaviour of $\rho$
at places of $F$
, especially at primes dividing $p = \mathrm{char}(k)$
(if $F$
is a number field). In this talk I will give an overview of a method developed in joint work with
Chandrashekhar Khare and Stefan Patrikis which allows one to construct such lifts in many cases.
In rank one symmetric space of noncompact type, we shall talk about the characterization of all eigenfunctions of the Laplace–Beltrami operator through sphere and ball averages as the radius of the sphere or ball tends to infinity.
The video of this talk is available on the IISc Math Department channel.
The modularity lifting theorem of Boxer-Calegari-Gee-Pilloni established for the first time the existence of infinitely many modular abelian surfaces $A / \mathbb{Q}$
upto twist with $\text{End}_{\mathbb{C}}(A) = \mathbb{Z}$
. We render this explicit by first finding some abelian surfaces whose associated mod-$p$
representation is residually modular and for which the modularity lifting theorem is applicable, and then transferring modularity in a family of abelian surfaces with fixed $3$
-torsion representation. Let $\rho: G_{\mathbb{Q}} \rightarrow GSp(4,\mathbb{F}_3)$
be a Galois representation with cyclotomic similitude character. Then, the transfer of modularity happens in the moduli space of genus $2$
curves $C$
such that $C$
has a rational Weierstrass point and $\mathrm{Jac}(C)[3] \simeq \rho$
. Using invariant theory, we find explicit parametrization of the universal curve over this space. The talk will feature demos of relevant code in Magma.
In 2006, Labourie defined a map from a bundle over Teichmuller space to the Hitchin component of the representation variety $Rep(\pi_1(S),PSL(n,R))$, and conjectured that it is a homeomorphism for every $n$ (it was known for $n =2,3$). I will describe some of the background to the Labourie conjecture, and then show that it does not hold for any $n >3$. Having shown that Labourie’s map is more interesting than a mere homeomorphism, I will describe some new questions and conjectures about how it might look.
This talk is based on the work of Stark and Terras (Zeta functions of Finite graphs and Coverings I, II, III). In this talk we start with an introduction to zeta functions in various branches of mathematics. Our focus is mainly on zeta functions on finite undirected connected graphs. We obtain an analogue of the prime number theorem, but for graphs, using the Ihara Zeta Function. We also introduce edge and path zeta functions and show interesting results.
The preservation of positive curvature conditions under the Ricci flow has been an important ingredient in applications of the flow to solving problems in geometry and topology. Works by Hamilton and others established that certain positive curvature conditions are preserved under the flow, culminating in Wilking’s unified, Lie algebraic approach to proving invariance of positive curvature conditions. Yet, some questions remain. In this talk, we describe positive sectional curvature metrics on $\mathbb{S}^4$ and $\mathbb{C}P^2$, which evolve under the Ricci flow to metrics with sectional curvature of mixed sign. This is joint work with Renato Bettiol.
Let $k$
be a nonarchimedian local field, $\widetilde{G}$
a connected reductive $k$
-group, $\Gamma$
a finite group of automorphisms of $\widetilde{G}$,
and $G:= (\widetilde{G}^\Gamma)^\circ$
the connected part
of the group of $\Gamma$
-fixed points of $\widetilde{G}$
.
The first half of my talk will concern motivation: a desire for a more explicit understanding of base change and other liftings of representations. Toward this end, we adapt some results of Kaletha-Prasad-Yu. Namely, if one assumes that the residual characteristic of $k$
does not divide the order of $\Gamma$
, then they show, roughly speaking, that $G$
is reductive, the building $\mathcal{B}(G)$
of $G$
embeds in the set of $\Gamma$
-fixed points of $\mathcal{B}(\widetilde{G})$
, and similarly for reductive quotients of parahoric subgroups.
We prove similar statements, but under a different hypothesis on $\Gamma$
. Our hypothesis does not imply that of K-P-Y, nor vice versa. I will include some comments on how to resolve such a totally unacceptable situation.
(This is joint work with Joshua Lansky and Loren Spice.)
The polynomial method is an ever-expanding set of algebraic techniques, which broadly entails capturing combinatorial objects by algebraic means, specifically using polynomials, and then employing algebraic tools to infer their combinatorial features. While several instances of the polynomial method have been part of the combinatorialist’s toolkit for decades, development of this method has received more traction in recent times, owing to several breakthroughs like (i) Dvir’s solution (2009) to the finite-field Kakeya problem, followed by an improvement by Dvir, Kopparty, Saraf, and Sudan (2013), (ii) Guth and Katz (2015) proving a conjecture by Erdös on the distinct distances problem, (iii) solutions to the capset problem by Croot, Lev, and Pach (2017), and Ellenberg and Gijswijt (2017), to name a few.
One of the ways to employ the polynomial method is via the classical algebraic objects – (affine) Zariski closure, (affine) Hilbert function, and Gröbner basis. Owing to their applicability in several areas like computational complexity, combinatorial geometry, and coding theory, an important line of enquiry is to understand these objects for ‘structured’ sets of points in the affine space. In this talk, we will be mainly concerned with Zariski closures of symmetric sets of points in the Boolean cube.
Firstly, we will look at a combinatorial characterization of Zariski closures of all symmetric sets, and its application to some hyperplane and polynomial covering problems for the Boolean cube, over any field of characteristic zero. We will also briefly look at Zariski closures over fields of positive characteristic, although much less is known in this setting. Secondly, we will see a simple illustration of a ‘closure statement’ being used as a technique for proving bounds on the complexity of approximating Boolean functions by polynomials. We will conclude with some open questions on Zariski closures motivated by problems on these two fronts.
Some parts of this talk will be based on the works: https://arxiv.org/abs/2107.10385, https://arxiv.org/abs/2111.05445, https://arxiv.org/abs/1910.02465.
I will explain a generalisation of the constructions Quillen used to prove that the $K$-groups of rings of integers are finitely generated. It takes the form of a ‘rank’ spectral sequence, converging to the homology of Quillen’s $Q$-construction on the category of coherent sheaves over a Noetherian integral scheme, and whose $E^1$ terms are given by homology of Steinberg modules. Computing its $d^1$ differentials is a challenge, which can be approached through the universal modular symbols of Ash-Rudolph.
The Thomas-Yau conjecture is an open-ended program to relate special Lagrangians to stability conditions in Floer theory, but the precise notion of stability is subject to many interpretations. I will focus on the exact case (Stein Calabi-Yau manifolds), and deal only with almost calibrated Lagrangians. We will discuss how the existence of destabilising exact triangles obstructs special Lagrangians, under some additional assumptions, using the technique of integration over moduli spaces.
In this talk, we discuss the problem of obtaining sharp $L^p\to L^q$ estimates for the local maximal operator associated with averaging over dilates of the Koranyi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Koranyi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.
Euler systems are cohomological tools that play a crucial role in the study of special values of $L$
-functions; for instance, they have been used to prove cases of the Birch–Swinnerton-Dyer conjecture and have recently been used to prove cases of the more general Bloch–Kato conjecture. A fundamental technique in these recent advances is to show that Euler systems vary in $p$
-adic families. In this talk, we will first give a general introduction to the theme of $p$
-adic variation in number theory and introduce the necessary background from the theory of Euler systems; we will then explain the idea and importance of $p$
-adically varying Euler systems, and finally discuss current work in progress on $p$
-adically varying the Asai–Flach Euler system, which is an Euler system arising from quadratic Hilbert modular eigenforms.
I will talk about recent work pertaining to the existence of abelian varieties not isogenous to Jacobians over fields of both characteristic zero and p. This is joint work with Jacob Tsimerman.
For a natural number $n$ and $1 \leq p < \infty$, consider the Hardy space $H^p(D^n)$ on the unit polydisk. Beurling’s theorem characterizes all shift cyclic functions in $H^p(D^n)$ when $n = 1$. Such a theorem is not known to exist in most other analytic function spaces, even in the one variable case. Therefore, it becomes natural to ask what properties these functions satisfy to understand them better. The goal of this talk is to showcase some important properties of cyclic functions in two different settings.
Fix $1 \leq p,q < \infty$ and natural numbers $m, n$. Let $T : H^p(D^n) \to H^q(D^m)$ be a bounded linear operator. Then $T$ preserves cyclic functions i.e., $Tf$ is cyclic whenever $f$ is, if and only if $T$ is a weighted composition operator.
Let $H$ be a normalized complete Nevanlinna-Pick (NCNP) space, and let $f, g$ be functions in $H$ such that $fg$ also lies in $H$. Then, $f$ and $g$ are multiplier cyclic if and only if $fg$ is multiplier cyclic.
We also extend (1) to a large class of analytic function spaces. Both properties generalize all previously known results of this type.
In this talk, we consider the optimal control problem (OCP) governed by the steady Stokes system in a two-dimensional domain $\Omega_{\epsilon}$ with a rapidly oscillating boundary prescribed with Neumann boundary condition and Dirichlet boundary conditions on the rest of the boundary. We aim to study the convergences analysis of the optimal solution (as $\epsilon\to 0$) and identify the limit OCP problem in a fixed domain.
The primary goal of this dissertation is to establish bounds for the sup-norm of the Bergman kernel of Siegel modular forms. Upper and lower bounds for them are studied in the weight as well as level aspect. We get the optimal bound in the weight aspect for degree 2 Siegel modular forms of weight $k$ and show that the maximum size of the sup-norm $k^{9/2}$. For higher degrees, a somewhat weaker result is provided. Under the Resnikoff-Saldana conjecture (refined with dependence on the weight), which provides the best possible bounds on Fourier coefficients of Siegel cusp forms, our bounds become optimal. Further, the amplification technique is employed to improve the generic sup-norm bound for an individual Hecke eigen-forms however, with the sup-norm being taken over a compact set of the Siegel’s fundamental domain instead. In the level aspect, the variation in sup-norm of the Bergman kernel for congruent subgroups $\Gamma_0^2(p)$ are studied and bounds for them are provided. We further consider this problem for the case of Saito-Kurokawa lifts and obtain suitable results.
I will describe the construction of an integer-valued symplectic invariant counting embedded pseudo-holomorphic curves in a Calabi–Yau 3-fold in certain cases. This may be seen as an analogue of the Gromov invariant defined by Taubes for symplectic 4-manifolds. The construction depends on a detailed bifurcation analysis of the moduli space of embedded curves along generic paths of almost complex structures. This is based on joint work with Shaoyun Bai.
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
In the first part of the talk we will discuss the main statement of local class field theory and sketch a proof of it. We will then discuss the statement of local Langlands correspondence for $GL_2(K)$, where $K$ is a non archimedian local field. In the process we will also introduce all the objects that go in the statement of the correspondence.
In this talk, we will explain the existence of a universal braided compact quantum group acting on a graph $C^*$-algebra in the twisted monoidal category of $C^*$-algebras equipped with an action of the circle group. To achieve this we construct a braided version of the free unitary quantum group. Finally, we will compute this universal braided compact quantum group for the Cuntz algebra. This is a joint work in progress with Suvrajit Bhattacharjee and Soumalya Joardar.
(Joint work with Andy O’Desky) There is a very classical formula counting the number of irreducible polynomials in one variable over a finite field. We study the analogous question in many variables and generalize Gauss’ formula. Our techniques can be used to answer many other questions about the space of irreducible polynomials in many variables such as it’s euler characteristic or euler hodge-deligne polynomial. To prove these results, we define a generalization of the classical ring of symmetric functions and use natural basis in it to help us compute the answer to the above questions.
We start by considering analogies between graphs and Riemann surfaces. Taking cue from this, we formulate an analogue of Brill–Noether theory on a finite, undirected, connected graph. We then investigate related conjectures from the perspective of polyhedral geometry.
Random fields indexed by amenable groups arise naturally in machine learning algorithms for structured and dependent data. On the other hand, mixing properties of such fields are extremely important tools for investigating asymptotic properties of any method/algorithm in the context of space-time statistical inference. In this work, we find a necessary and sufficient condition for weak mixing of a left-stationary symmetric stable random field indexed by an amenable group in terms of its Rosinski representation. The main challenge is ergodic theoretic - more precisely, the unavailability of an ergodic theorem for nonsingular (but not necessarily measure preserving) actions of amenable groups even along a tempered Følner sequence. We remove this obstacle with the help of a truncation argument along with the seminal work of Lindenstrauss (2001) and Tempelman (2015), and finally applying the Maharam skew-product. This work extends the domain of application of the speaker’s previous paper connecting stable random fields with von Neumann algebras via the group measure space construction of Murray and von Neumann (1936). In particular, weak mixing has now become $W^*$-rigid properties for stable random fields indexed by any amenable group, not just $\mathbb{Z}^d$. We have also shown that many stable random fields generated by natural geometric actions of hyperbolic groups on various negatively curved spaces are actually mixing and hence weakly mixing.
This talk is based on an ongoing joint work with Mahan Mj (TIFR Mumbai) and Sourav Sarkar (University of Cambridge).
The video of this talk is available on the IISc Math Department channel. Here are the slides.
A conjecture of Katz and Sarnak predicts that the distribution of spacings between ``straightened” Hecke angles (corresponding to Fourier coefficients of Hecke newforms) matches that of a uniformly distributed, random sequence in the unit interval. This comparison is made with the help of local spacing statistics, such as the level spacing distribution and various types of correlations of the Hecke angles. In previous joint work with Baskar Balasubramanyam and ongoing joint work with my PhD student Jewel Mahajan, we have provided evidence in favour of this conjecture, by showing that the pair correlation function of the Hecke angles, averaged over families of Hecke newforms, is expected to be Poissonnian, with variance converging to zero as we take larger and larger families. In this talk, we will explore various types of questions arising in the study of the local behaviour of sequences of Hecke angles, and explain the above-mentioned results.
Let $T$ be a linear endomorphism of a $2m$-dimensional vector space. An $m$-dimensional subspace $W$ is said to be $T$-splitting if $W$ intersects $TW$ trivially.
When the underlying field is finite of order $q$ and $T$ is diagonal with distinct eigenvalues, the number of splitting subspaces is essentially the the generating function of chord diagrams weighted by their number of crossings with variable $q$. This generating function was studied by Touchard in the context of the stamp folding problem. Touchard obtained a compact form for this generating function, which was explained more clearly by Riordan.
We provide a formula for the number of splitting subspaces for a general operator $T$ in terms of the number of $T$-invariant subspaces of various dimensions. Specializing to diagonal matrices with distinct eigenvalues gives an unexpected and new proof of the Touchard–Riordan formula.
This is based on joint work with Samrith Ram.
We survey the recent progress on the fundamental group of open manifolds with nonnegative Ricci curvature. This includes finite generation and virtual abelianness/nilpotency of the fundamental groups.
For commuting contractions $T_1,\ldots ,T_n$ acting on a Hilbert space $\mathcal{H}$ with $T=\prod_{i=1}^{n}T_i$, we find a necessary and sufficient condition under which $(T_1,\ldots ,T_n)$ dilates to commuting isometries $(V_1,\ldots ,V_n)$ or commuting unitaries $(U_1,\ldots ,U_n)$ acting on the minimal isometric dilation space or the minimal unitary dilation space of $T$ respectively, where $V=\prod_{i=1}^{n}V_i$ and $U=\prod_{i=1}^{n}U_i$ are the minimal isometric and the minimal unitary dilations of $T$ respectively. We construct both Schäffer and Sz. Nagy-Foias type isometric and unitary dilations for $(T_1,\ldots ,T_n)$. Also, a special minimal isometric dilation is constructed where the product $T$ is a $C_0$ contraction, that is $T^{*n}\to 0$ strongly as $n\to \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\ldots ,T_n)$. When the product $T$ is a $C_0$ contraction, the dilation of $(T_1,\ldots ,T_n)$ leads to a natural factorization of $T$ in terms of compression of Toeplitz operators with linear analytic symbols.
Lambert series lie at the heart of modular forms and the theory of the Riemann zeta function. The early pioneers in the subject were Ramanujan and Wigert. We discuss Ramanujan’s formula for odd zeta values and its generalizations and analogues obtained by the speaker with his co-authors culminating into a recent transformation for $\sum_{n=1}^{\infty}\sigma_a(n)e^{-ny}$
for $a\in\mathbb{C}$
and Re$(y)>0$
. We will discuss several applications of this result. A formula of Wigert and its recent analogue found by Soumyarup Banerjee, Shivajee Gupta and the author will be discussed and its application in the zeta-function theory will be given. This talk is an amalagamation of results of the author on this topic from various papers co-authored with Bibekananda Maji, Rahul Kumar, Rajat Gupta, Soumyarup Banerjee and Shivajee Gupta.
Given a group $G$
and two Gelfand subgroups $H$
and $K$
of $G$
, associated to an irreducible representation $\pi$
of $G$
, there is a notion of $H$
and $K$
being correlated with respect to $\pi$
in $G$
. This notion is defined by Benedict Gross in 1991. We discuss this theme and give some details in a specific example (which is joint work with Arindam Jana).
Hirschman–Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, that is, integrable functions which give rise to totally positive Toeplitz kernels. This talk will introduce this class of densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.
This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).
The video of this talk is available on the IISc Math Department channel.
This thesis has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. In contrast, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this thesis, we aim to prove various analogues of theorems of Ingham and Chernoff in different contexts such as the Heisenberg group, Hermite and special Hermite expansions, rank one Riemannian symmetric spaces, and Euclidean space with Dunkl setting. More precisely, we prove various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. The main idea is to reduce the situation to the radial case by employing appropriate spherical means or spherical harmonics and then to apply Chernoff type theorems to the radial parts of the operators indicated above. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In the second part of this thesis, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
If $m$ is a function on a commutative group $G$, one may define an associated Fourier multiplier $T_m$, which acts on functions on the dual group. If this $T_m$ is a bounded linear map on the $L_p$ space of the dual group, is the restriction of $m$ to a subgroup $H$ also the symbol of a bounded multiplier on the $L_p$ space of the dual group of $H$? De Leeuw showed that this is indeed the case when $G=\mathbb{R}^n$, and others later extended this to all locally compact commutative groups. Moreover, the norm of the multiplier corresponding to the restricted symbol is bounded above by the norm of the original multiplier. For non-commutative groups, one may ask the same question by replacing “$L_p$ spaces of the dual group” with the non-commutative $L_p$ space of the group von Neumann algebra. Caspers, Parcet, Perrin and Ricard showed that the answer is still yes in the non-commutative case, provided $G$ has something called the “small-almost invariant neighbourhood property with respect to the subgroup $H$”.
In recent joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi, we prove a local version of this result, which removes this restriction (for a price). We show that the norm of the $L_p$ Fourier multiplier for the subgroup is bounded by some constant depending only on the support of the symbol $m$. This constant measures the failure of the small invariant neighbourhood property, and can be explicitly estimated for real reductive Lie groups. We also prove non-commutative multilinear versions of the De Leeuw theorems, and use these to construct examples of multilinear multipliers on the Heisenberg group. I will outline these results in my talk, and if time permits, describe some possible extensions.
The video of this talk is available on the IISc Math Department channel.
The problem of locating the poles and zeros of complex functions in a finite domain of the complex plane, occurs in many scientific disciplines e.g., dispersion relations in plasma physics, the singularity expansion method in electro-magnetic scattering or antenna problems.
The principle of the argument or the winding number is useful in finding the number of zeros of an analytic function in a given contour. A simple extension of this theorem yields relationships involving the locations of these zeros! The resulting equations can be solved very accurately for the zero locations, thus avoiding initial, guess values, which are required by many other techniques. Examples such as a 20th order polynomial, natural frequencies of a thin wire will be discussed.
This method has been extended to the problem of locating the zeros and poles of a complex meromorphic function $M(s)$ in a specified rectangular or square region of the complex plane. It is assumed that $M(s)$ has to be numerically computed. It is interesting to note that the word “meromorphic” is derived from the Greek meros $(\mu \varepsilon \rho \omicron \zeta)$ = fraction and morph $(\mu \omicron \rho \varphi \eta)$ = form, and means “like a fraction.” In keeping with the origin of the word “meromorphic,” the complex function $M(s)$ considered in this paper will be a ratio of two entire functions of the complex variables. The procedure developed here eliminates the usual 2-dimensional search and replaces it with a direct constructive method for determining the poles of $M(s)$ based on an application of Cauchy’s residue theorem. Two examples, i.e., 1) ratios of polynomials and 2) input impedance of a biconical antenna, are numerically illustrated.
The ($p^{\infty}$
) fine Selmer group (also called the $0$
-Selmer group) of an elliptic curve is a subgroup of the usual $p^{\infty}$
Selmer group of an elliptic curve and is related to the first and the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$
-extension of a number field $K$
is intricately related to Iwasawa’s $\mu$
-invariant vanishing conjecture on the growth of $p$
-part of the ideal class group of $K$
in the cyclotomic tower. In this talk, we will discuss the structure and properties of the fine Selmer group over certain $p$
-adic Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.
Hausdorff dimension is a notion of size ubiquitous in geometric measure theory. A set of large Hausdorff dimension contains many points, so it is natural to expect that it should contain specific configurations of interest. Yet many existing results in the literature point to the contrary. In particular, there exist full-dimensional sets $K$ in the plane with the property that if a point $(x_1, x_2)$ is in $K$, then no point of the form $(x_1, x_2 + t)$ lies in $K$, for any $t \neq 0$.
A recent result of Kuca, Orponen and Sahlsten shows that every planar set of Hausdorff dimension sufficiently close to 2 contains a two-point configuration of the form $(x1, x2)+\{(0, 0), (t, t^2)\}$ for some $t \neq 0$. This suggests that sets of sufficiently large Hausdorff dimension may contain patterns with “curvature”, suitably interpreted. In joint work with Benjamin Bruce, we obtain a characterization of smooth functions $\Phi : \mathbb{R} \to \mathbb{R}^d$ such that every set of sufficiently high Hausdorff dimension in $d$-dimensional Euclidean space contains a two point configuration of the form $\{x, x + \Phi(t)\}$, for some $t$ with $\Phi(t) \neq 0$.
The video of this talk is available on the IISc Math Department channel.
The horofunction compactification of a metric space keeps track of the possible limits of balls whose centers go off to infinity. This construction was introduced by Gromov, and although it is usually hard to visualize, it has proved to be a useful tool for studying negatively curved spaces. In this talk I will explain how, under some metric assumptions, the horofunction compactification is a refinement of the significantly simpler visual compactification. I will then go over how this relation allows us to use the simplicity of the visual compactification to get geometric and topological properties of the horofunction compactification. Most of these applications will be in the context of Teichmüller spaces with respect to the Teichmüller metric, where the relation allows us to prove, among other things, that Busemann points are not dense within the horoboundary and that the horoboundary is path connected.
In this talk I will explain new research on $L$
-invariants of modular forms, including ongoing joint work with Robert Pollack. $L$
-invariants, which are $p$
-adic invariants of modular forms, were discovered in the 1980’s, by Mazur, Tate, and Teitelbaum. They were formulating a $p$
-adic analogue of Birch and Swinnerton-Dyer’s conjecture on elliptic curves. In the decades since, $L$
-invariants have shown up in a ton of places: $p$
-adic $L$
-series for higher weight modular forms or higher rank automorphic forms, the Banach space representation theory of $\mathrm{GL}(2,\mathbb{Q}_p)$
, $p$
-adic families of modular forms, Coleman integration on the $p$
-adic upper half-plane, and Fontaine’s $p$
-adic Hodge theory for Galois representations. In this talk I will focus on recent numerical and statistical investigations of these $L$
-invariants, which touch on many of the theories just mentioned. I will try to put everything into the context of practical questions in the theory of automorphic forms and Galois representations and explain what the future holds.
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes (hereafter QHawkes) process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macroscale, we introduce the bivariate Modified-QHawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metrics with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a folklore conjecture involving the Tian-Yau metric.
A fundamental and widely used mathematical fact states that the arithmetic mean of a collection of non-negative real numbers is at least as large as its geometric mean. This is the most basic example of a large family of inequalities between symmetric functions that have attracted the interest of combinatorialists in recent years. This talk will present recent joint work with Jon Novak at UC San Diego, which unifies many such inequalities as corollaries of a fundamental monotonicity property of spherical functions on symmetric spaces. We will also discuss conjectural extensions of these results to even more general objects such as Heckman-Opdam hypergeometric functions and Macdonald polynomials.
The talk will be accessible to a broad mathematical audience and will not assume any knowledge of symmetric spaces or symmetric functions. However, the second half of the talk will assume familiarity with basic constructions of Lie theory, such as root systems and the Iwasawa decomposition. Details of the relevant work can be found in this pre-print.
The video of this talk is available on the IISc Math Department channel.
Postnikov defined the totally nonnegative Grassmannian as the part of the Grassmannian where all Plücker coordinates are nonnegative. This space can be described by the combinatorics of planar bipartite graphs in a disk, by affine Bruhat order, and by a host of other combinatorial objects. In this talk, I will recall some of this story, then talk about in progress joint work, together with Chris Fraser and Jacob Matherne, which hopes to extend this combinatorial description to more general partial flag varieties.
Here are two problems about hyperplane arrangements.
Problem 1: If you take a collection of planes in $\mathbb{R}^3$, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.
Problem 2: Given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it. These polynomials should have nonnegative coefficients.
Both of these problems were formulated for all matroids, and in the case of hyperplane arrangements they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a solution to both problems for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.
We will talk on the an analogue of the Tamagawa Number conjecture, with coefficients over varieties over finite fields. This a joint work with O. Brinon (Bordeaux) and a work in progress.
In the analysis on symmetric cones and the classical theory of hypergeometric functions of matrix argument, the Laplace transform plays an essential role. In an unpublished manuscript dating back to the 1980ies, I.G. Macdonald proposed a generalization of this theory, where the spherical polynomials of the underlying symmetric cone - such as the cone of positive definite matrices - are replaced by Jack polynomials with arbitrary index. He also introduced a Laplace transform in this context, but many of the statements in his manuscript remained conjectural. In the late 1990ies, Baker and Forrester took up these matters in their study of Calogero-Moser models, still at a rather formal level, and they noticed that they were closely related to Dunkl theory.
In this talk, we explain how Macdonald’s Laplace transform can be rigorously established within Dunkl theory, and we discuss several of its applications, including Riesz distributions and Laplace transform identities for the Cherednik kernel and for Macdonald’s hypergeometric series in terms of Jack polynomials.
Part of the talk is based on joint work with Dominik Brennecken.
The video of this talk is available on the IISc Math Department channel.
We present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$. The obstructions arise from studying moduli spaces of holomorphic disks with corners and boundaries on immersed objects called Lagrangian tangles. The obstructions boil down to area relations and sign conditions on disks bound by knot diagrams of the boundaries of the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots that cannot bound Lagrangian disks.
We consider certain degenerating families of complex manifolds, each carrying a canonical measure (for example, the Bergman measure on a compact Riemann surface of genus at least one). We show that the measure converges, in a suitable sense, to a measure on a non-Archimedean space, in the sense of Berkovich. No knowledge of non-Archimedean geometry will be assumed.
Non-malleable codes (NMCs) are coding schemes that help in protecting crypto-systems under tampering attacks, where the adversary tampers the device storing the secret and observes additional input-output behavior on the crypto-system. NMCs give a guarantee that such adversarial tampering of the encoding of the secret will lead to a tampered secret, which is either same as the original or completely independent of it, thus giving no additional information to the adversary. The specific tampering model that we consider in this work, called the “split-state tampering model”, allows the adversary to tamper two parts of the codeword arbitrarily, but independent of each other. Leakage resilient secret sharing schemes help a party, called a dealer, to share his secret message amongst n parties in such a way that any $t$ of these parties can combine their shares to recover the secret, but the secret remains hidden from an adversary corrupting $< t$ parties to get their complete shares and additionally getting some bounded bits of leakage from the shares of the remaining parties.
For both these primitives, whether you store the non-malleable encoding of a message on some tamper-prone system or the parties store shares of the secret on a leakage-prone system, it is important to build schemes that output codewords/shares that are of optimal length and do not introduce too much redundancy into the codewords/shares. This is, in particular, captured by the rate of the schemes, which is the ratio of the message length to the codeword length/largest share length. This thesis explores the question of building these primitives with optimal rates.
The focus of this talk will be on taking you through the journey of non-malleable codes culminating in our near-optimal NMCs with a rate of 1/3.
Euler solved the famous Basel problem and discovered that Riemann zeta functions at positive even integers are rational multiples of powers of $\pi$
. Multiple zeta values (MSVs) are a multi-dimensional generalization of the Riemann zeta values, and MZVs which are rational multiples of powers of $\pi$
is called Eulerian MZVs. In 1996, Borwein-Bradley-Broadhurst discovered a series of conjecturally Eulerian MZVs which together with the known Eulerian family seems to exhaust all Eulerian MZVs at least numerically. A few years later, Borwein-Bradley-Broadhurst-Lisonek discovered two families of interesting conjectural relations among MZVs generalizing the previous conjecture of Eulerian MZVs, which were later extended further by Charlton in light of alternating block structure. In this talk, I would like to present my recent joint work with Minoru Hirose concerning block shuffle relations that simultaneously resolve and generalize the conjectures of Charlton.
Recently, Markovic proved that there exists a maximal representation into (PSL(2,R))^3 such that the associated energy functional on Teichmuller space admits multiple critical points. In geometric terms, there is more than one minimal surface in the relevant homotopy class in the corresponding product of closed Riemann surfaces. This is related to an important question in Higher Teichmuller theory. In this talk, we explain that this non-uniqueness arises from non-uniqueness of minimal surfaces in products of trees. We plan to discuss energy minimizing properties for minimal maps into trees, as well as the geometry of the surfaces found in Markovic’s work. This is work in progress, joint with Vladimir Markovic.
This talk is motivated by interest in Crouzeix’s conjecture for compressions of the shift with finite Blaschke products as symbols. Specifically, in this setting, Crouzeix’s conjecture suggests a related, weaker conjecture about the behavior of level sets of finite Blaschke products. I’ll discuss this level set conjecture in several cases, though the main case of interest will involve uncritical finite Blaschke products. Here, the geometry of the numerical ranges of their associated compressions of the shift has allowed us to establish the conjecture in low degree situations (n=3, n=4, n =5 with a caveat). Time permitting, I’ll explain how these geometric results also give insights into Crouzeix’s conjecture for the associated compressed shifts. This talk is based on joint work with Pam Gorkin.
The video of this talk is available on the IISc Math Department channel.
We reprove the main equidistribution instance in the Ferrero–Washington proof of the vanishing of cyclotomic Iwasawa $\mu$
-invariant, based on the ergodicity of a certain $p$
-adic skew extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).
In his 1976 proof of the converse to Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-$p$
extensions of the $p$
-th cyclotomic field when $p$
is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-$p$
extensions of $\mathbb{Q}(N^{1/p})$
when $N$
is a prime that is congruent to $-1$
mod $p$
. This answers a question posted on Frank Calegari’s blog.
A conjectural correspondence due to Yau, Tian and Donaldson relates the existence of certain canonical Kähler metrics (“constant scalar curvature Kähler metrics”) to an algebro-geometric notion of stability (“K-stability”). I will describe a general framework linking geometric PDEs (“Z-critical Kähler metrics”) to algebro-geometric stability conditions (“Z-stability”), in such a way that the Yau-Tian-Donaldson conjecture is the classical limit of these new broader conjectures. The main result will prove that a special case of the main conjecture: the existence of Z-critical Kähler metrics is equivalent to Z-stability.
Let $D\subset\mathbb{C}^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani & E. M. Stein states that the Cauchy–Szegö projection $\mathcal{S}_\omega$ maps $L^p(bD, \omega)$ to $L^p(bD, \omega)$ continuously for any $1<p<\infty$ whenever the reference measure $\omega$ is a bounded, positive continuous multiple of induced Lebesgue measure. Here we show that $\mathcal{S}_\omega$ (defined with respect to any measure $\omega$ as above) satisfies explicit, optimal bounds in $L^p(bD, \Omega_p)$, for any $1<p<\infty$ and for any $\Omega_p$ in the maximal class of $A_p$-measures, that is $\Omega_p = \psi_p\sigma$ where $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is the induced Lebesgue measure. As an application, we characterize boundedness in $L^p(bD, \Omega_p)$ with explicit bounds, and compactness, of the commutator $[b, \mathcal{S}_\omega]$ for any $A_p$-measure $\Omega_p$, $1<p<\infty$. We next introduce the notion of holomorphic Hardy spaces for $A_p$-measures, and we characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $[b,\mathcal{S}_{\Omega_2}]$ where $\mathcal{S}_{\Omega_2}$ is the Cauchy–Szegö projection defined with respect to any given $A_2$-measure $\Omega_2$. Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy–Szegö kernel, but these are unavailable in our setting of minimal regularity of $bD$; at the same time, recent techniques that allow to handle domains with minimal regularity, are not applicable to $A_p$-measures. It turns out that the method of quantitative extrapolation is an appropriate replacement for the missing tools.
This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis).
The video of this talk is available on the IISc Math Department channel.
In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components, the spaces of maximal representations and the other spaces of positive representations. Fock-Goncharov generalized Thurston’s shear coordinates and Penner’s Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety. We applied a similar strategy to Maximal Representations and we found new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This was joint work with Guichard, Rogozinnikov and Wienhard. In a project in progress we are generalizing these coordinates to the other sets of positive representations.
Following a joint work with Sara Arias-de-Reyna and François Legrand, we present a new kind of families of modular forms. They come from representations of the absolute Galois group of rational function fields over $\mathbb{Q}$
. As a motivation and illustration, we discuss in some details one example: an infinite Galois family of Katz modular forms of weight one in characteristic $7$
, all members of which are non-liftable. This may be surprising because non-liftability is a feature that one might expect to occur only occasionally.
Up to biholomorphic change of variable, local invariants of a quadratic differential at some point of a Riemann surface are the order and the residue if the point is a pole of even order. Using the geometric interpretation in terms of flat surfaces, we solve the Riemann-Hilbert type problem of characterizing the sets of local invariants that can be realized by a pair (X,q) where X is a compact Riemann surface and q is a meromorphic quadratic differential. As an application to geometry of surfaces with positive curvature, we give a complete characterization of the distributions of conical angles that can be realized by a cone spherical metric with dihedral monodromy.
This is a continuation of a talk I gave at the University of Delhi in $2015.$ Let $G$ be a separable locally compact unimodular group of type I, and $\widehat G$ the unitary dual of $G$ endowed with the Mackey Borel structure. We regard the Fourier transform $\mathcal F$ as a mapping of $L^1(G)$ to a space of $\mu$-measurable field of bounded operators on $\widehat G$ defined for $\pi\in\widehat G$ by $ L^1(G)\ni f\mapsto \mathcal Ff : \mathcal Ff(\pi)=\pi(f), $ where $\mu$ denotes the Plancherel measure of $G$. The mapping $f \mapsto \mathcal F f$ extends to a continuous operator $\mathcal F^p : L^p(G) \to L^q(\widehat G)$, where $p\geq 1$ is real number and $q$ its conjugate. We are concerned in this talk with the norm of the linear map $\mathcal F^p$. We first record some results on the estimate of this norm for some classes of solvable Lie groups and their compact extensions and discuss the sharpness problem. We look then at the case where $G$ is a separable unimodular locally compact group of type I. Let $N$ be a unimodular closed normal subgroup of $G$ of type I, such that $G/N$ is compact. We show that $\Vert \mathscr F^p(G)\Vert \leq \Vert \mathscr F^p(N )\Vert$. In the particular case where $G=K\ltimes N$ is defined by a semi-direct product of a separable unimodular locally compact group $N$ of type I and a compact subgroup $K$ of the automorphism group of $N$, we show that equality holds if $N$ has a $K$-invariant sequence $(\varphi_j)_j$ of functions in $L^1(N)\cap L^p(N)$ such that ${\Vert \mathscr F\varphi_j \Vert_q}/{\Vert \varphi_j \Vert_p}$ tends to $\Vert \mathscr F^p(N )\Vert$ when $j$ goes to infinity.
The video of this talk is available on the IISc Math Department channel.
Let $R$
be the Iwasawa algebra over a compact, $p$
-adic, pro-$p$
group
$G$
, where $G$
arises as a Galois group of number fields from Galois representations.
Suppose $M$
is a finitely generated $R$
-module. In the late 1970’s , Harris studied the
asymptotic growth of the ranks of certain coinvariants of $M$
arising from the action
of open subgroups of $G$
and related them to the codimension of $M$
. In this talk, we
explain how Harris’ proofs can be simplified and improved upon, with possible
applications to studying some natural subquotients of the Galois groups of number fields.
In this talk, I shall talk about analogues of pseudo-differential operators (pseudo-multipliers) associated with the joint functional calculus for the Grushin operator. In particular, we shall discuss some sufficient conditions on a symbol function which imply $L^2$-boundedness of the associated Grushin pseudo-multiplier. This talk is based on a joint work (arXiv:2111.10098) with Sayan Bagchi.
The video of this talk is available on the IISc Math Department channel.
This talk will be a report of work in progress with Ming-Lun Hsieh. Just as in classical Iwasawa theory where one studies congruences involving Hecke eigenvalues associated to Eisenstein series, we study congruences involving $p$
-adic families of Hecke eigensystems associated to the space of Yoshida lifts of two Hida families. Our goal is to show that under suitable assumptions, the characteristic ideal of a dual Selmer group is contained inside the congruence ideal.
Multiple zeta values are the real numbers \begin{equation} \zeta({\bf a})= \sum_{n_1>\cdots>n_r>0}n_1^{-a_1}\cdots n_r^{-a_r}, \end{equation} where ${\bf a}=(a_1, \ldots ,a_r) $ is an admissible composition, i.e. a finite sequence of positive integers, with $a_1 \geqslant 2$ when $r\neq 0$.
The multiple Apéry-like sums defined by \begin{equation} \sigma({\bf a})=\sum_{n_1>\cdots>n_r>0}\left({2 n_1 \atop n_1}\right)^{-1}n_1^{-a_1}\cdots n_r^{-a_r} \end{equation} when ${\bf a}\neq\varnothing$ and by $\sigma(\varnothing)=1$. We show that for any admissible composition ${\bf a}$, there exists a finite formal $\bf Z$-linear combination $\sum \lambda_{\bf b} {\bf b}$ of admissible compositions such that \begin{equation} \zeta({\bf a})=\sum \lambda_{\bf b}\, \sigma({\bf b}). \end{equation} The simplest instance of this fact is the identity \begin{equation} \sum_{n=1}^{\infty}\frac{1}{n^2}=3\sum_{n=1}^{\infty}\frac{1}{\left({2n \atop n}\right)n^2} \end{equation} discovered by Euler, which expresses that $\zeta(2)=3\,\sigma(2)$. Note that multiple Apéry-like sums have the advantage on multiple zeta values to be exponentially quickly convergent.
This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier’s formulas for $\zeta(2,\ldots,2,3,2,\ldots,2)$ (D. Zagier, Evaluation of the multiple zeta values $\zeta(2,\ldots,2,3,2,\ldots,2)$, Annals of Math. 175 (2012), 977–1000) as well as of similar ones in the context of Apéry-like sums.
There many operators in Harmonic Analysis which can be described as an average of a family of operators $\{T_j\}_j$ for which some boundedness properties are known. In particular, if $T_j$ are uniformly bounded on $L^p$, then the Minkowski integral inequality tells us that $T$ also satisfies this property. But things change completely if the information that we have is that $T_j$ are of weak type (1,1).
However, under certain condition on the operators $T_j$, the weak type boundedness of $T$ can be reached.
This is a joint work with my student Sergi Baena.
The video of this talk is available on the IISc Math Department channel.
One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher’s famous work on Smale conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some “exotic” phenomena and if time permits, I will talk a few words on my work with Jianfeng Lin where we used an idea motivated from this to disprove a long standing open problem about stabilizations of 4-manifolds.
This talk has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. On the other hand, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this talk, we plan to discuss various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite, and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In this second part of this talk, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
Classical groups and their generalizations are central objects in Algebraic $K$-theory. Orthogonal groups are one type of classical groups. We shall discuss a generalized version of elementary orthogonal groups.
Let $R$ be a commutative ring in which $2$ is invertible. Let $Q$ be a non-degenerate quadratic space over $R$ of rank $n$ and let $\mathbb{H}(R)^m$ denote the hyperbolic space of rank $m$. We consider the elementary orthogonal transformations of the quadratic space $Q \perp \mathbb{H}(R)^m$. These transformations were introduced by Amit Roy in $1968$. Earlier forms of these transformations over fields were considered by Dickson, Siegel, Eichler and Dieudonné. We call the elementary orthogonal transformations as Dickson–Siegel–Eichler–Roy elementary orthogonal transformations or Roy’s elementary orthogonal transformations. The group generated by these transformations is called DSER elementary orthogonal group. We shall discuss the structure of this group.
As part of the solution to the famous Serre’s problem on projective modules, D. Quillen had proved the remarkable Local-Global criterion for a module $M$ to be extended. This result is known as Quillen’s Patching Theorem or Quillen’s Local-Global Principle. The Bass–Quillen conjecture is a natural generalization of Serre’s problem. In this talk, we shall see the solution of the quadratic version of the Bass–Quillen conjecture over an equicharacteristic regular local ring.
The DSER elementary orthogonal group is a normal subgroup of the orthogonal group. We shall also discuss some generalizations of classical groups over form rings and their comparison with the DSER elementary orthogonal group.
Some recent improvements of Wigner’s unitary-antiunitary theorem will be presented. A connection with Gleason’s theorem will be explained.
The video of this talk is available on the IISc Math Department channel.
Advances in various fields of modern studies have shown the limitations of traditional probabilistic models. The one such example is that of the Poisson process which fails to model the data traffic of bursty nature, especially on multiple time scales. The empirical studies have shown that the power law decay of inter-arrival times in the network connection session offers a better model than exponential decay. The quest to improve Poisson model led to the formulations of new processes such as non-homogeneous Poisson process, Cox point process, higher dimensional Poisson process, etc. The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some recently introduced generalized counting processes and their fractional variants. The system of differential equations that governs their state probabilities are discussed.
Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. This is the starting point for modeling human organs in computational medicine and for modeling the mechanics of permeable rock in geophysics. Finite element methods for Biot’s consolidation model have been widely studied over the past four decades.
In the first part of the talk, we discuss a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. The simplest of these is a conventional residual-based estimator. We establish bounds relating the estimated and true errors, and show that these are independent of the physical parameters. The other two estimators require the solution of local problems. These local problem estimators are also shown to be reliable, efficient and robust. Numerical results are presented that validate the theoretical estimates, and illustrate the effectiveness of the estimators in guiding adaptive solution algorithms.
In the second part of the talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young’s modulus and hydraulic conductivity field. After introducing a five-field mixed variational formulation of the standard Biot consolidation model, we discuss stochastic Galerkin mixed finite element approximation, focusing on the issue of well-posedness and efficient linear algebra for the discretized system. We introduce a new preconditioner for use with MINRES and establish eigenvalue bounds. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach.
Finally, we discuss some remarks related to non-conforming approximation of Biot’s consolidation model.
Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominators of $\mathfrak g$, and this gives a Lie theoretic proof of Stanley’s expression for chromatic symmetric function in terms of power sum symmetric functions. Also, this gives an expression for the chromatic symmetric function of $G$ in terms of root multiplicities of $\mathfrak g$. We prove a modified Weyl denominator identity for Borcherds algebras which is an extension of the celebrated classical Weyl denominator identity, and this plays an important role in the proof our results. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem, we prove that certain coefficients appearing in the above said expression of chromatic symmetric function is equal to the chromatic discriminant of $G$. Also, we find a connection between the Weyl denominators and the $G$-elementary symmetric functions. Using this connection, we give a Lie-theoretic proof of non-negativity of coefficients of $G$-power sum symmetric functions. I will also talk about the plethysms of chromatic symmetric functions.
Intersection cohomology is a cohomology theory for describing the topology of singular algebraic varieties. We are interested in studying intersection cohomology of complete complex algebraic varieties endowed with an action of an algebraic torus. An important invariant in the classification of torus actions is the complexity. It is defined as the codimension of a general torus orbit. Classification of torus actions is intimately related to questions of convex geometry.
In this talk, we focus on the calculation of the (rational) intersection cohomology Betti numbers of complex complete normal algebraic varieties with a torus action of complexity one. Intersection cohomology for the surface and toric cases was studied by Stanley, Fieseler–Kaup, Braden–MacPherson and many others. We suggest a natural generalisation using the geometric and combinatorial approach of Altmann, Hausen, and Süß for normal varieties with a torus action in terms of the language of divisorial fans.
Several critical physical properties of a material are controlled by its geometric construction. Therefore, analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex domains such as perforated domain, thin domain, junctions of the thin domain of different configuration, domain with rapidly oscillating boundary, networks domain, etc.
This talk will discuss various homogenization problems posed on high oscillating domains. We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast material acts as near insulation in-between the conducting materials. Mathematically this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the curved interface with general cost functional.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as it is the main tool in our analysis.
The second part of this talk will briefly discuss the boundary optimal control problems subject to Laplacian and Stokes systems.
In the third part of the talk, we will discuss the homogenization of optimal control problems subject to a elliptic variational form with high contrast diffusivity coefficients. The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive the \two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
Let $H^2$ denote the Hardy space on the open unit disk $\mathbb{D}$. For a given holomorphic self map $\varphi$ of $\mathbb{D}$, the composition operator $C_\varphi$ on $H^2$ is defined by $C_\varphi(f) = f \circ \varphi$. In this talk, we discuss about Beurling type invariant subspace of composition operators, that is common invariant subspaces of shift (multiplication) and composition operators. We will also consider the model spaces that are invariant under composition operators.
Since the Calabi conjecture was proved in 1978 by S.T. Yau, there has been extensive studies into nonlinear PDEs on complex manifolds. In this talk, we consider a class of fully nonlinear elliptic PDEs involving symmetric functions of partial Laplacians on Hermitian manifolds. This is closely related to the equation considered by Székelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions, we derive apriori estimates and show the existence of solutions. In addition, we also consider the parabolic counterpart of this equation and prove the long-time existence and convergence of solutions.