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.
The study of the optimal control problems governed by partial differential equations(PDEs) have been a significant research area in applied mathematics and its allied areas. The optimal control problem consists of finding a control variable that minimizes a cost functional subject to a PDE. In this talk, I will present finite element analysis of Dirichlet boundary optimal control problems governed by certain PDEs. This talk will be divided into four parts.
In the first part, we discuss the Dirichlet boundary control problem, its physical interpretation, mathematical formulation, and some approaches (numerical) to solve it.
In the second part, we study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for $L^2$ cost functional, but this analysis can also be extended to the gradient cost functional problem. A priori error estimates of optimal order in the energy norm are derived up to the regularity of the solution.
In the third part, we discuss the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using $\mathbf{P}_1$ elements (in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. We present a posteriori error estimators for the error in the state, co-state, and control variables. As a continuation of the second part, we extend our ideas to the linear parabolic equation in the last part of the presentation. The space discretization of the state and co-state variables is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. We use $H^1$-conforming 3D finite elements for the control variable. We present the error estimates of state, adjoint state, and control.
The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
theorem [Crelle 1911]
, which implies that absolutely monotonic
functions (i.e., power series with nonnegative coefficients) preserve
positivity on matrices of all dimensions. A famous result of Schoenberg
and of Rudin [Duke Math. J. 1942, 1959]
shows the converse: there are
no other such functions.
Motivated by modern applications, Guillot and Rajaratnam
[Trans. Amer. Math. Soc. 2015]
classified the entrywise positivity
preservers in all dimensions, which act only on the off-diagonal entries.
These two results are at “opposite ends”, and in both cases the preservers
have to be absolutely monotonic.
The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg’s motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the thesis. Finally, we present the (remaining) main results in the thesis, and conclude with some of the proofs.
Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of $\Theta$ positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of $\Theta$ positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.
Let $H$
be a subgroup of a group $G$
. For an irreducible representation $\sigma$
of $H$
, the triple $(G,H, \sigma)$
is called a Gelfand triple if $\sigma$
appears at most once in any irreducible representation of $G$
. Given a triple, it is usually difficult to determine whether a given triple is a Gelfand triple. One has a sufficient condition which is geometric in nature to determine if a given triple is a Gelfand triple, called Gelfand criterion. We will discuss some examples of the Gelfand triple which give us multiplicity one theorem for non-degenerate Whittaker models of ${\mathrm GL}_n$
over finite chain rings, such as $\mathbb{Z}/p^n\mathbb{Z}$
.
This is a joint work with Pooja Singla.
I will discuss some aspects of a singular version of the Donaldson-Uhlenbeck-Yau theorem for bundles and sheaves over normal complex varieties satisfying some conditions. Several applications follow, such as a characterization of the case of equality in the Bogomolov-Gieseker theorem. Such singular metrics also arise naturally under certain types of degenerations, and I will make some comments on the relationship between this result and the Mehta-Ramanathan restriction theorem.
The von Neumann inequality says the value of a polynomial at a contractive operator is bounded by the norm of the polynomial on the disk. The von Neumann inequality is often proven using the Sz.-Nagy dilation theorem, which essentially says that one can model a contraction by a unitary operator. We adapt a technique of Nelson for proving the von Neumann inequality: one considers the singular value decomposition and then replaces the singular values with automorphisms of the disk to obtain a matrix valued analytic function which must attain its maximum on the boundary. Moreover, the matrix valued function involved in fact gives a minimal unitary dilation. With McCullough, we adapt Nelson’s trick to various other classes of operators to obtain their dilation theory, including the quantum annulus, row contractions and doubly commuting contractions. We conjecture a geometric relationship between Ando’s inequality and Gerstenhaber’s theorem.
The video of this talk is available on the IISc Math Department channel.
Choose a homotopically non-trivial curve “at random” on a compact orientable surface- what properties is it likely to have? We address this when, by “choose at random”, one means running a random walk on the Cayley graph for the fundamental group with respect to the standard generating set. In particular, we focus on self-intersection number and the location of the metric in Teichmuller space minimizing the geodesic length of the curve. As an application, we show how to improve bounds (due to Dowdall) on dilatation of a point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of the defining curve, for a “random” point-pushing map. This represents joint work with Jonah Gaster.
In 1998 Shuzhou Wang, in his pioneering work, introduced quantum symmetry groups of finite spaces motivated by a general question posed by Alain Connes: what is the quantum automorphism group of a space? By finite spaces, here we mean finite-dimensional C*-algebras. Wang’s results have initiated several fundamental developments in operator algebras, quantum groups and noncommutative geometry. Let us consider a generalised situation where we shall equip the finite spaces with a continuous action of the circle group. This talk aims to understand the object that captures the quantum symmetries of these systems and their applications.
The video of this talk is available on the IISc Math Department channel.
In this talk, we will see an interplay between hermitian metrics and singular Riemann surface foliations. It will be divided into three parts. The first part of the talk is about the study of curvature properties of complete Kahler metrics on non-pseudoconvex domains. Examples of such metrics were constructed by Grauert in 1956, who showed that it is possible to construct complete Kahler metrics on the complement of complex analytic sets in a domain of holomorphy. In particular, he gave an explicit example of a complete Kahler metric (the Grauert metric) on $\mathbb{C}^n \setminus {0}$. We will confine ourselves to the study of such complete Kahler metrics. We will make some observations about the holomorphic sectional curvature of such metrics in two prototype cases, namely (i) $\mathbb{C}^n \setminus {0}$, $n>1$, and (ii) $(B^N)\setminus A$, where $A$ is an affine subspace. We will also study complete Kahler metrics using Grauert’s construction on the complement of a principal divisor in a domain of holomorphy and show that there is an intrinsic continuity in the construction of this metric, i.e., we can choose this metric in a continuous fashion if the corresponding principal divisors vary continuously in an appropriate topology.
The second part of the talk deals with Verjovsky’s modulus of uniformization that arises in the study of the leaf-wise Poincare metric on a hyperbolic singular Riemann surface lamination. This is a function defined away from the singular locus. One viewpoint is to think of this as a domain functional. Adopting this view, we will show that it varies continuously when the domains vary continuously in the Hausdorff sense. We will also give an analogue of the classical Domain Bloch constant by D. Minda for hyperbolic singular Riemann surface laminations.
In the last part of the talk, we will discuss a parametrized version of the Mattei-Moussu theorem, namely a holomorphic family of holomorphic foliations in $\mathbb{C}^2$ with an isolated singular point at the origin in the Siegel domain are holomorphically equivalent if and only if the holonomy maps of the horizontal separatrix of the corresponding foliations are holomorphically conjugate.
In this talk, I shall give a panoramic view of my research work. I shall introduce the notion of hyperbolic polynomials and discuss an algebraic method to test hyperbolicity of a multivariate polynomial w.r.t. some fixed point via sum-of-squares relaxation, proposed in my research work. An important class of hyperbolic polynomials are definite determinantal polynomials. Helton–Vinnikov curves in the projective plane are cut-out by hyperbolic polynomials in three variables. This leads to the computational problem of explicitly producing a symmetric positive definite linear determinantal representation for a given curve. I shall focus on two approaches to this problem proposed in my research work: an algebraic approach via solving polynomial equations, and a geometric-combinatorial approach via scalar product representations of the coefficients and its connection with polytope membership problem. The algorithms to solve determinantal representation problems are implemented in Macaulay2 as a software package DeterminantalRepresentations.m2. Then I shall briefly address the methodologies to find the degree and the defining equations of certain varieties which are obtained as the image of some given varieties of $\mathbb{P}_n$ under coordinate-wise power map, for example the $4 \times 4$ orthostochastic variety. Finally, I shall demonstrate a connection of symmetroids with the real degeneracy loci of matrices.
The theory of $\delta$
-geometry was developed by A. Buium based on the analogy with differential algebra where the analogue of a differential operator is played by a $\pi$
-derivation $\delta$
. A $\pi$
-derivation $\delta$
arises from the $\pi$
-typical Witt vectors and naturally associates with a lift of Frobenius $\phi$
. In this talk, we will discuss the theory of $\delta$
-geometry for Anderson modules. Anderson modules are higher dimensional generalizations of Drinfeld modules.
As an application of the above, we will construct a canonical $z$
-isocrystal $\mathbb{H}(E)$
with a Hodge- Pink structure associated to an Anderson module $E$
defined over a $\pi$
-adically complete ring $R$
with a fixed $\pi$
-derivation $\delta$
on it. Depending on a $\delta$
-modular parameter, we show that the $z$
-isocrystal $\mathbb{H}(E)$
is weakly admissible in the case of Drinfeld modules of rank $2$
. Hence, by the analogue of Fontaine’s mysterious functor in the positive characteristic case (as constructed by Hartl), one associates a Galois representation to such an $\mathbb{H}(E)$
. The relation of our construction with the usual Galois representation arising from the Tate module of $E$
is currently not clear. This is a joint work with Sudip Pandit.
A finitely generated group can be viewed as the group of symmetries of a metric space, for example its Cayley graph. When the metric space has non-positive curvature, then the group satisfies some exceptional properties. In this talk, I will introduce two notions of non-positive curvature – CAT(0) and delta hyperbolic. I will present some results comparing groups acting on such spaces. I will also talk about the group of outer automorphisms of a free group, which itself is neither CAT(0) nor delta-hyperbolic, but still benefits a lot from the presence of non-positive curvature.
Machine Learning, particularly Deep Learning, algorithms are being increasingly used to approximate solutions of partial differential equations (PDEs). We survey recent results on different aspects of deep learning in the context of PDEs namely, 1) Supervised learning for high-dimensional parametrized PDEs 2) Operator learning for approximating infinite-dimensional operators which arise in PDEs and 3) Physics informed Neural Networks for approximating both forward and inverse problems for PDEs. We will highlight open questions in the analysis of deep learning algorithms for PDEs.
The video of this talk is available on the IISc Math Department channel.
A celebrated theorem of Gromov-Lawson and Schoen-Yau states that a n-torus cannot admit metrics with positive scalar curvature. Thus, the torus is of vanishing Yamabe type. In this talk, we will discuss its extension to metrics with some singularity. This is a joint work with L.-F. Tam.
In this thesis, we analyse certain dynamically interesting measures arising in holomorphic dynamics beyond the classical framework of maps. We will consider measures associated with semigroups and, more generally, with meromorphic correspondences, that are invariant in a specific sense. Our results are of two different flavours. The first type of results deal with potential-theoretic properties of the measures associated with certain polynomial semigroups, while the second type of results are about recurrence phenomena in the dynamics of meromorphic correspondences. The unifying features of these results are the use of the formalism of correspondences in their proofs, and the fact that the measures that we consider are those that describe the asymptotic distribution of the iterated inverse images of a generic point.
The first class of results involve giving a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh–Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which we can describe explicitly given a choice of a set of generators. In particular, we generalize the classical result of Brolin to certain finitely generated polynomial semigroups. To do so, we establish the continuity of the logarithmic potential for the Dinh–Sibony measure, which might also be of independent interest. Thereafter, we use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.
The second class of results involves meromorphic correspondences. These are, loosely speaking, multi-valued analogues of meromorphic maps. We shall present an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to the measures alluded to above. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result on the invariance properties of the supports of the measures mentioned, and, as a corollary, give a geometric description of the support of such a measure.
In the 1980’s, Greene defined hypergeometric functions over finite fields using
Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric
series studied by Gauss, Kummer and others. These functions have played important roles in the study of Apery-style supercongruences, the Eichler-Selberg trace
formula, Galois representations, and zeta-functions of arithmetic varieties. In this
talk we discuss the distributions (over large finite fields) of natural families of these
functions. For the $_2F_1$
functions, the limiting distribution is semicircular, whereas
the distribution for the $_3F_2$
functions is Batman distribution.
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
The delta symbol is the key in solving many different problems in the analytic theory of numbers. In recent years this has been used to solve various sub-convexity problems for higher rank $L$
-functions. This talk will be a brief report on some new progresses. In particular, I will mention the results obtained in recent joint works with Roman Holowinsky & Zhi Qi and Sumit Kumar & Saurabh Singh.
We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk. Joint work with Bickel, Pascoe, Sola.
The video of this talk is available on the IISc Math Department channel.
In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe work in progress on a generalization of their result. I will review the definition of the “enhanced Teichmüller space” which has been widely studied in the mathematical physics and cluster algebra literature. I will then describe a version of the result of Hitchin and Wolf which relates meromorphic quadratic differentials to the enhanced Teichmüller space. This builds on earlier work by a number of authors, including Wolf, Lohkamp, Gupta, and Biswas-Gastesi-Govindarajan.
We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed with respect to the Haar measure. However, for representations of $PGL_2(\mathbb{F}_q)$
, they are clustered around $1$
and $-1$
for odd $p$
and around $1$
for $p=2$
. This is joint work with Sameer Kulkarni.
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First, we have focused on the geometry of these two domains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a collection of orbits under the action of its automorphism group. We investigate the properties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some properties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence.
Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson–Sjölin–Hörmander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
The video of this talk is available on the IISc Math Department channel.
We will discuss the $L^\infty$ estimates for a class of fully nonlinear partial differential equations on a compact Kahler manifold, which includes the complex Monge-Ampere and Hessian equations. Our approach is purely based on PDE methods, and is free of pluripotential theory. We will also talk about some generalizations to the stability of MA and Hessian equations. This is based on joint works with D.H. Phong and F. Tong.
The k-differentials are sections of the tensorial product of the canonical bundle of a complex algebraic curves. Fixing a partition (m_1,…,m_n) of k(2g-2), we can define the strata of k-differentials of type (m_1,…,m_n) to be the space of k-differentials on genus g curves with zeroes of orders m_i. After checking that the strata or not empty, the first interesting topological question about these strata is the classification of their connected component. In the case k=1, this was settled in an important paper of Kontsevich and Zorich. This result was extend to k=2 by Lanneau, with corrections of Chen-Möller. The classification is unknown for k greater or equal to 3 as soon as g is greater or equal to 2. In this talk, I will present partial results on this classification problem obtained together with Dawei Chen (arXiv:2101.01650) and in progress with Andrei Bogatyrev. In particular, I will highlight the way Pell-Abel equation appears in this problem.
We consider three different spherical means on a Heisenberg type group. First, the standard spherical means, which is the average of a function over the spheres in the complement of the center of the group, second is the average over product of spheres in the center and its complement and the third one over spheres defined by a homogeneous norm on the group. We establish injectivity results for these means on $L^p$ spaces for the range $1 \leq p \leq 2m/(m-1)$ where $m$ is the dimension of the center. Our results extend and generalize S. Thangavelu’s results for spherical means on the Heisenberg group. (Joint work with P. K. Sanjay and K. T. Yasser)
The video of this talk is available on the IISc Math Department channel.
Let $K$
be a finite extension of $\mathbb{Q}_p$
. The theory of $(\varphi, \Gamma)$
-modules constructed by Fontaine provides a good category to study $p$
-adic representations of the absolute Galois group $Gal(\bar{K}/K)$
. This theory arises from a ‘‘devissage’’ of the extension $\bar{K}/K$
through an intermediate extension $K_{\infty}/K$
which is the cyclotomic extension of $K$
. The notion of $(\varphi, \tau)$
-modules generalizes Fontaine’s constructions by using Kummer extensions other than the cyclotomic one. It encapsulates the important notion of Breuil-Kisin modules among others. It is thus desirable to establish properties of $(\varphi, \tau)$
-modules parallel to the cyclotomic case. In this talk, we explain the construction of a functor that associates to a family of $p$
-adic Galois representations a family of $(\varphi, \tau)$
-modules. The analogous functor in the $(\varphi, \Gamma)$
-modules case was constructed by Berger and Colmez . This is joint work with Leo Poyeton.
A theorem attributed to Beurling for the Fourier transform pairs asserts that for any nontrivial function $f$ on $\mathbb{R}$ the bivariate function $ f(x) \hat{f}(y) e^{|xy|} $ is never integrable over $ \mathbb{R}^2.$ Well known uncertainty principles such as theorems of Hardy, Cowling–Price etc. follow from this interesting result. In this talk we explore the possibility of formulating (and proving!) an analogue of Beurling’s theorem for the operator valued Fourier transform on the Heisenberg group.
The video of this talk is available on the IISc Math Department channel.
Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.
In this talk, we will discuss genericity of cuspidal representations of $p$
-adic unitary groups. Generic representations play a central role in the local Langlands correspondences and explicit knowledge of such representations will be useful in understanding the local Langlands correspondence in a more explicit way. After a brief review of $p$
-adic unitary groups, their unipotent subgroups, Whittaker functionals and genericity of cuspidal representations in this context, we will discuss the arithmetic nature of the problem.
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodal set that we study fall into two broad categories: nodal component count of Gaussian Laplace eigenfunctions and volume of the nodal set of centered stationary Gaussian processes (SGPs) on $\mathbb{R}^d$, $d \geq 1$.
Gaussian Laplace eigenfunctions: Nazarov–Sodin pioneered the study of nodal component count for Gaussian Laplace eigenfunctions; they investigated this for random spherical harmonics (RSH) on the two-dimensional sphere $S^2$ and established exponential concentration for their nodal component count. An analogous result for arithmetic random waves (ARW) on the $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$, was established soon after by Rozenshein.
We establish concentration results for the nodal component count in the following three instances: monochromatic random waves (MRW) on growing Euclidean balls in $\R^2$; RSH and ARW, on geodesic balls whose radius is slightly larger than the Planck scale, in $S^2$ and $\mathbb{T}^2$ respectively. While the works of Nazarov–Sodin heavily inspire our results and their proofs, some effort and a subtler treatment are required to adapt and execute their ideas in our situation.
Stationary Gaussian processes: The study of the volume of nodal sets of centered SGPs on $\mathbb{R}^d$ is classical; starting with Kac and Rice’s works, several studies were devoted to understanding the nodal volume of Gaussian processes. When $d = 1$, under somewhat strong regularity assumptions on the spectral measure, the following results were established for the zero count on growing intervals: variance asymptotics, central limit theorem and exponential concentration.
For smooth centered SGPs on $\mathbb{R}^d$, we study the unlikely event of overcrowding of the nodal set in a region; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for probability of the overcrowding event. We first obtain overcrowding estimates for the zero count of SGPs on $\mathbb{R}$, we then deal with the overcrowding question in higher dimensions in the following way. Crofton’s formula gives the nodal set’s volume in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^d$. We discretize this formula to get a more workable version of it and, in a sense, reduce this higher dimensional overcrowding problem to the one-dimensional case.
Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of random triangulated surfaces in the large genus setting, and proved results about the systole, diameter and Cheeger constant of random triangulated surfaces. Subsequently Mirzakhani proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of random triangulated surfaces mirrors the geometry of random hyperbolic surfaces in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space.
Let $X$ be your favorite Banach space of continuous functions on $\mathbb{R}^n$. Given a real-valued function $f$ defined on some (possibly awful) set $E$ in $\mathbb{R}^n$, how can we decide whether $f$ extends to a function $F$ in $X$? If such an $F$ exists, then how small can we take its norm? Can we make $F$ depend linearly on $f$? What can we say about the derivatives of $F$ at or near points of $E$ (assuming $X$ consists of differentiable functions)?
Suppose $E$ is finite. Can we compute a nearly optimal $F$? How many computer operations does it take? What if we demand merely that $F$ agree approximately with $f$? Suppose we allow ourselves to discard a few data points as “outliers”. Which points should we discard?
The video of this talk is available on the IISc Math Department channel.
In this talk we will discuss the geometry of Strominger connection of Hermitian manifolds, based on recent joint works with Quanting Zhao. We will focus on two special types of Hermitian manifolds: Strominger Kaehler-like (SKL) manifolds, and Strominger parallel torsion (SPT) manifolds. The first class means Hermitian manifolds whose Strominger connection (also known as Bismut connection) has curvature tensor obeying all Kaehler symmetries, and the second class means Hermitian manifolds whose Strominger conneciton has parallel torsion. We showed that any SKL manifold is SPT, which is known as (an equivalent form of) the AOUV Conjecture (namely, SKL implies pluriclosedness). We obtained a characterization theorem for SPT condition in terms of Strominger curvature, which generalizes the previous theorem. We will also discuss examples and some structural results for SKL and SPT manifolds.
Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincare inequalities on $(X,d,\mu)$ if it satisfies a local Poincare inequality ($P_{loc}$), and a condition on the growth of volume. Consequently, if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a uniform $(\sigma,\beta,\sigma)$-Poincare inequality. If $(X,d,\mu)$ is a Gromov-hyperbolic space, then using the volume comparison theorem introduced by Besson, Courtoise, Gallot, and Sambusetti, we obtain a uniform Poincare inequality with the exponential growth of the Poincare constant. Next, we relate the growth of Poincare constants to the growth of discrete subgroups of isometries of $X$, which act on it properly. This is Joint work with Gautam Nilakantan.
Homological stability is an interesting phenomenon exhibited by many natural sequences of classifying spaces and moduli spaces like the moduli spaces of curves M_g and the moduli spaces of principally polarized abelian varieties A_g. In this talk I will explain some efforts to find similar phenomena in the cohomology of discrimination complements.
Let $F$
be a non-archimedean local field of residue characteristic $p$
. The classical local Langlands correspondence is a 1-1 correspondence between 2-dimensional irreducible complex representations of the Weil group of $F$
and certain smooth irreducible complex representations of $GL_2(F)$
. The number-theoretic applications made it necessary to seek such correspondence of representations on vector spaces over a field of characteristic $p$
. In this talk, however, I will show that for $F$
of residue degree $> 1$
, unfortunately, there is no such 1-1 mod $p$
correspondence. This result is an elaboration of the arguments of Breuil and Paskunas to an arbitrary local field of residue degree $> 1$
.
We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operators. We show that these solitons are non-collapsed.
In Geostatistics one examines measurements depending on the location on the earth and on time. This leads to Random Fields of stochastic variables $Z(\xi,u)$ indexed by $(\xi,u)$ belonging to $\mathbb{S}^2\times \mathbb{R}$, where $\mathbb{S}^2$–the 2-dimensional unit sphere–is a model for the earth, and $\mathbb{R}$ is a model for time.
If the variables are real-valued, one considers a basic probability space $(\Omega,\mathcal F,P)$, where all the random variables $Z(\xi,u)$ are defined as measurable mappings from $\Omega$ to $\mathbb{R}$.
One is interested in isotropic and stationary random fields $Z(\xi,u),\;(\xi,u)\in\mathbb{S}^2 \times\mathbb{R}$, i.e., the situation where there exists a continuous function $f:[-1,1] \times \mathbb{R} \to \mathbb{R}$ such that the covariance kernel is given as
\begin{equation} \mbox{cov}(Z(\xi,u),Z(\eta,v))=f(\xi\cdot\eta,v-u),\quad \xi,\eta\in\mathbb{S}^2,\;u,v\in\mathbb{R}. \end{equation}
Here $\xi\cdot\eta=\cos(\theta(\xi,\eta))$ is the scalar product equal to cosine of the length of the geodesic arc (=angle) between $\xi$ and $\eta$.
We require with other words that the covariance kernel only depends on the geodesic distance between the points on the sphere and on the time difference.
Porcu and Berg (2017) gave a characterization of such kernels by having uniformly convergent expansions
\begin{equation} f(x,u)=\sum_{n=0}^\infty b_n(u)P_n(x), \quad \sum_{n=0}^\infty b_n(0)<\infty, \end{equation}
where $(b_n)$ is a sequence of real-valued characteristic (=continuous positive definite) functions on $\mathbb{R}$ and $P_n$ are the Legendre polynomials on $[-1,1]$ normalized as $P_n(1)=1$. The result can be generalized to spheres $\mathbb{S}^d$ of any dimension $d$ and $\mathbb{R}$ can be replaced by an arbitrary locally compact group.
In work of Peron, Porcu and Berg (2018) it was pointed out that the spheres can be replaced by compact homogeneous spaces $G/K$, where $(G,K)$ is a Gelfand pair.
We shall explain the theory of Gelfand pairs and also show how recent work of several people can be extended to this framework.
The presentation is largely based on the recent paper of the speaker with the same title as the talk published in Journal Fourier Analysis and Applications 26 (2020).
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.
In this thesis colloquium, we 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. Due to time constraints, we may not discuss other chapters of the thesis.
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 the talk will be homogenizing optimal control problems subject to the considered PDEs. 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 do obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
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. 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. The research goal of the thesis is to improve the state of art on rates of these schemes and get near-optimal/optimal rates.
In this talk, I will specifically focus on leakage resilient secret sharing schemes, describe the leakage model, and take you through the state of the art on their rates. Finally, I will present a recent construction of an optimal (constant) rate, leakage resilient secret sharing scheme in the so-called “joint and adaptive leakage model” where leakage queries can be made adaptively and jointly on multiple shares.