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.
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.
The study of the optimal control problems governed by partial differential equations (PDEs) have been a significant research area in the 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 three parts.
In the first 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 second 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 new a posteriori error estimator for the control error. This estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems by adding terms at the contact boundary that address the non-linearity. We sketch out the proof of the estimator’s reliability and efficiency.
As a continuation of the first part, we extend our ideas to the linear parabolic equation in the third part of this 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 a sketch to demonstrate the existence and uniqueness of the solution; and the error estimates of state, adjoint state, and control.
We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on a suitable space $X$. The connection between extreme values and the indexing group $G$ is mediated by the action of $G$ on the limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth, which quantifies the distortion of measures on the boundary in comparison to the movement of points in the space $X$. We show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups. (Joint work with Jayadev Athreya and Mahan Mj, under review in Probability Theory and Related Fields.)
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.
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
In this talk, we shall focus on 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, whose proof will be sketched. If time permits, we will discuss bounds on the capacity and diameter of the Julia sets of such semigroups, for which we use the $F$-functional of Mhaskar and Saff.
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. If time permits, we shall also discuss a result on the invariance properties of the supports of the measures mentioned.
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.
In this talk, we focus on random graphs with a given degree sequence. In the first part, we look at uniformly chosen trees from the set of trees with a given child sequence. A non-negative sequence of integers $(c_1,c_2,\dots,c_l)$ with sum $l-1$ is a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1,v_2,\dots,v_l$ of the nodes of $t$, $v_i$ has exactly $c_i$ many children in $t$. We consider for each $n$, a child sequence $\mathbf{c}^{(n)}$, with sum $n-1$, and let $\mathbf{t}_n$ be the random tree having the uniform distribution on the set of all plane trees with $n$ vertices, which has $\mathbf{c}^{(n)}$ as their child sequence. Under the assumption that a finite number of vertices of $\mathbf{t}_n$ has large degrees, we show that the scaling limit of $\mathbf{t}_n$ is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff topology. This generalizes a result of Broutin and Marckert from 2012, where they show the scaling limit to be the Brownian Continuum Random Tree (BCRT), under the assumption that no vertex in $\mathbf{t}_n$ has large degree.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerný, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_*$, such that if a random walk is run up to time $un$ with $u<u_*$, $n$ being the total number of nodes in the graph, a giant component of linear size, in the subgraph spanned by the nodes yet unvisited by the random walk, emerges. Whereas, if the random walk is tun up to time un with $u>u_*$, the size of the largest component, of the subgraph spanned by nodes yet unvisited by the walk, is $\text{o}(n)$. With the help of simulations, we try to look for such a phase transition for supercritical configuration models, with heavy-tailed degrees.
A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a family of polynomial inequalities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although there are some obvious bounds).
In this talk, we will discuss some recent improvements on the previously known bounds, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.
Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\Omega_+=\Omega\cap C_+$ be the positive part of $\Omega$. We define a family ${-\Delta_{\eta}^+}$ of self-adjoint extensions of the Laplacian $-\Delta_{\Omega_+}$, labeled by homomorphisms $\eta\colon W\to {1,-1}$. In the construction of these $\eta$-Laplacians $\eta$-symmetrization of functions on $\Omega$ is involved. The Neumann Laplacian $-\Delta_{N,\Omega_+}$ is included and corresponds to $\eta\equiv 1$. If $H^{1}(\Omega)=H^{1}_0(\Omega)$, then the Dirichlet Laplacian $-\Delta_{D,\Omega_+}$ is either included and corresponds to $\eta={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\Psi(-\Delta_{N,\Omega})$ and $\Psi(-\Delta_{\eta}^+)$, or $\Psi(-\Delta_{D,\Omega})$ and $\Psi(-\Delta_{D,\Omega_+})$, where $\Psi$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.
In the talk, for simplicity, I will focus on the case $\Omega = \R^d$ (so $\Omega_+ = C_+$) and $\Psi = \Psi_t, t > 0$, where $\Psi_t(\lambda) = \exp(−t\lambda)$ for $\lambda > 0$. Then the integral kernels of $\Psi_t(-\Delta^{+}_{\eta})$, called the $\eta$-heat kernels, will be investigated in more detail.
In the last decades there have been many connections made between the analysis of a manifold M and the geometry of M. Said correctly, there are now many ways to make precise that well-behaved analysis on M is ’equivalent’ to the existence of lower bounds on Ricci curvature. Such ideas are the starting point for regularity theories and more abstract settings for analysis, including analysis on metric-measure spaces. We will begin this talk with an elementary review of these ideas. More recently it has become apparent analysis on the path space PM of a manifold is closely connected to two sided bounds on Ricci curvature. Again, said correctly one can make an equivalence that the analysis on PM is well behaved iff M has a two sided Ricci curvature bound. As a general phenomena, one see’s that analytic estimates on M lift to estimates on PM in the presence of two sided Ricci bounds. Our talk will mainly focus on explaining all the words in this abstract and giving some rough understanding of the broad ideas involved. Time allowing, we will briefly explain newer results with Haslhofer/Kopfer on differential harnack inequalities on path space.
A profound mathematical mystery of our times is to be able to explain the phenomenon of training neural nets i.e “deep-learning”. The dramatic progress of this approach in the last decade has gotten us the closest we have ever been to achieving “artificial intelligence”. But trying to reason about these successes - for even the simplest of nets - immediately lands us into a plethora of extremely challenging mathematical questions, typically about discrete stochastic processes. In this talk we will describe the various themes of our work in provable deep-learning.
We will start with a brief introduction to neural nets and then see glimpses of our initial work on understanding neural functions, loss functions for autoencoders and algorithms for exact neural training. Next, we will explain our recent result about how under mild distributional conditions we can construct an iterative algorithm which can be guaranteed to train a ReLU gate in the realizable setting in linear time while also keeping track of mini-batching - and its provable graceful degradation of performance under a data-poisoning attack. We will show via experiments the intriguing property that our algorithm very closely mimics the behaviour of Stochastic Gradient Descent (S.G.D.), for which similar convergence guarantees are still unknown.
Lastly, we will review this very new concept of “local elasticity” of a learning process and demonstrate how it appears to reveal certain universal phase transitions during neural training. Then we will introduce a mathematical model which reproduces some of these key properties in a semi-analytic way. We will end by delineating various exciting future research programs in this theme of macroscopic phenomenology with neural nets.
The Hitchin-Simpson equations defined over a Kaehler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin–Simpson equations with norms of these 1-forms unbounded. We will talk about two applications of this compactness theorem, one is the realization problem of the Taubes’ Z2 harmonic 1-form and another is the Hitchin’s WKB problem in higher dimensional. We will also discuss some open questions related to this question.
Let $(K\ltimes G,K)$ be a Gelfand pair, where $K\ltimes G$ is the semidirect product of a Lie group $G$ with polynomial growth and $K$ a compact group of automorphisms of $G$. Then the Gelfand spectrum $\Sigma$ of the commutative convolution algebra of $K$-invariant integrable functions on $G$ admits natural embeddings into $\mathbb{R}^n$ spaces as a closed subset. Let $\mathcal{S}(G)^K$ be the space of $K$-invariant Schwartz functions on $G$. Defining $\mathcal{S}(\Sigma)$ as the space of restrictions to $\Sigma$ of Schwartz functions on $\mathbb R^n$, we call Schwartz correspondence for $(K\ltimes G,K)$ the property that the spherical transform is an isomorphism of $\mathcal{S}(G)^K$ onto $\mathcal{S}(\Sigma)$. In all the cases studied so far, the Schwartz correspondence has been proved to hold true. These include all pairs $(K\ltimes G,K)$ with $K$ abelian and a large number of pairs with $G$ nilpotent. In this talk we show that the Schwartz correspondence holds for the pair $(K\ltimes G,K)$, where $G=U_2\ltimes \mathbb{C}^2$ is the complex motion group and $K={\rm Int}(U_2) $ is the group of inner automorphisms of $G$ induced by elements of $U_2$. This is one of the simplest pairs with $G$ non-nilpotent and $K$ non-abelian. This work arises from a collaboration with Francesca Astengo and Fulvio Ricci.
A result due to Hulanicki (and refined by Veneruso) states that if $m$ is a Schwartz function on $\mathbb{R}^2$ and $L, T$ are the Heisenberg sublaplacian and the central derivative, then the operator $m(L,i^{-1}T)$ has a Schwartz radial convolution kernel $k$. It is therefore natural to ask whether all Schwartz convolution kernels arise in this way. In collaboration with Bianca Di Blasio and Fulvio Ricci we are considering this kind of problem in the context of Gelfand Pairs of polynomial growth. In this talk I will discuss some old and new results.
It is well known that the system of translates $\{T_k\phi:k\in\mathbb{Z}\}$ is a Riesz sequence in $L^2(\mathbb{R})$ if and only if there exist $A,B>0$ such that \begin{equation} A\leq\sum_{k\in\mathbb{Z}}|\widehat{\phi}(\xi+k)|^2\leq B\hspace{.5 cm}a.e.\ \xi\in[0,1], \end{equation} where $\widehat{\phi}$ denotes the Fourier transform of $\phi$. This result is very important in time-frequency analysis especially in constructing wavelet basis for $L^2(\mathbb{R})$ using multiresolution analysis technique and also in studying sampling problems in a shift-invariant space.
In this talk, we ask a similar question for the system of left translates $\{ L_\gamma\phi:\gamma\in\Gamma\}$ on the Heisenberg group $\mathbb{H}^n$, where $\phi\in L^2(\mathbb{H}^n)$ and $\Gamma$ is a lattice in $\mathbb{H}^n$. We take $\Gamma= \{(2k,l,m):k,l\in\mathbb{Z}^n,m\in\mathbb{Z}\}$ as the standard lattice in order to avoid computational complexity. Recently it has been proved that if $\phi\in L^2(\mathbb{H}^n)$ is such that \begin{equation} \sum_{r\in\mathbb{Z}}\left\langle \widehat{\phi}(\lambda+r),\widehat{L_{(2k,l,0)}\phi}(\lambda+r) \right\rangle_{\mathcal{B}_2}|\lambda+r|^n=0\ a.e.\ \lambda\in(0,1], \end{equation} for all $(k,l)\in\mathbb{Z}^{2n}\setminus\{(0,0)\}$, then $\{L_{(2k,l,m)}\phi:k,l\in\mathbb{Z}^n, m\in\mathbb{Z}\}$ is a Riesz sequence if and only if there exist $A,B>0$ such that \begin{equation} A\leq \sum_{r\in\mathbb{Z}}\left|\widehat{\phi}(\lambda+r)\right|_{\mathcal{B}_2}^2|\lambda+r|^n\leq B\ \ a.e.\ \lambda\in(0,1]. \end{equation} Here $\widehat{\phi}$ denotes the group Fourier transform of $\phi$ and $\mathcal{B}_2$ denotes the Hilbert space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^n)$. In the absence of the above condition, the requirement of Riesz sequence is given in terms of the Gramian of the system $\{\tau\left(L_{(2k,l,0)}\phi\right)(\lambda):k,l\in\mathbb{Z}^n\}$ for $\lambda\in(0,1]$, where $\tau$ is the fiber map. We shall discuss these results in the talk along with the computational issues.
Let K(n, V) be the space of n-dimensional compact Kahler-Einstein manifolds with negative scalar curvature and volume bounded above by V. We prove that any sequence in K(n, V) converges in pointed Gromov-Hausdorff topology to a finite union of complete Kahler-Einstein metric spaces without loss of volume, which is biholomorphic to an algebraic semi-log canonical model with its non-log terminal locus removed. We further show that the Weil-Petersson metric extends uniquely to a Kahler current with continuous local potentials on the KSB compactification of the moduli space of canonically polarized manifolds. In particular, the Weil-Petersson volume of the KSB moduli space is finite.
In this talk, we discuss a basic (and somewhat classical) problem of Laplace eigenfunction mass concentration on convex polyhedra. We show quantitative mass concentration in a neighbourhood of the non-smooth part of the boundary, or the “pockets” of the billiard. On the way, we discuss several new dynamical properties of the billiard flow which are required for the proof.
I will discuss the geometry of Kaehler manifolds with a lower bound on the holomorphic bisectional curvature, along with their pointed Gromov-Hausdorff limits. Some of the proofs use Ricci flow.
In this talk we give a survey on a certain number of multi-parameter structures, on $\mathbb{R}^n$ and on nilpotent groups, that have first appeared in joint work of mine with A. Nagel, E. Stein and S. Wainger. They include flag and multi-norm structures.
These structures are intermediate between the one-parameter dilation structures of standard Calderón-Zygmund theory and the full n-parameter product structure. Each structure has its own type of maximal functions, singular integral operators, square functions, Hardy spaces.
Projective geometry provides a common framework for the study of classical Euclidean, spherical, and hyperbolic geometry. A major difference with the classical case is that a projective structure is not completely determined by its holonomy representation. In general, a complete description of the space of structures with the same holonomy is still missing. We will consider certain structures on punctured surfaces, and we will discuss how to describe all of those with a given holonomy in the case of the thrice-punctured sphere. This is done in terms of a certain geometric surgery known as grafting. Our approach involves a study of the Möbius completion, and of certain meromorphic differentials on Riemann surfaces. This is joint work with Sam Ballas, Phil Bowers, and Alex Casella.
The celebrated Wiener Tauberian theorem asserts that for $ f \in L^1(\mathbb{R})$, the closed ideal generated by the function $f$ is equal to the whole of $ L^1(\mathbb{R})$ if and only if its Fourier transform $\hat f $ is nowhere vanishing on $\mathbb{R}$. The analogous result holds for locally compact abelian groups.
However in 1955, L. Ehrenpreis and F. I. Mautner observed that the corresponding result is not true for the commutative Banach algebra $L^1(G//K)$ of $K$-biinvariant functions on $G$ and proved Wiener Tauberian theorem with additional conditions, for $G= \mathrm{SL(2,\mathbb{R})}$ and $ K=\mathrm{SO}(2) $. Their result is ameliorated by Y. Ben Natan et al. In their paper, the authors studied the analog of the Wiener Tauberian theorem for the Banach algebra $ L^1( \mathrm{SL(2,\mathbb{R})} //\mathrm{SO}(2))$.
In this talk, we will discuss an analog of the Wiener Tauberian theorem for the Lorentz spaces $L^{p,1}(\mathrm {SL}(2, \mathbb{R}))$, $1\leq p<2$.
‘Growth’ is a geometrically defined property of a group that can reveal algebraic aspects of the group. For instance, Gromov showed that a group has polynomial growth if and only if it is virtually nilpotent. In this talk, we will focus on growth of groups that act on a CAT(0) cube complex. Such spaces are combinatorial versions of the more general CAT(0) (negatively curved) spaces. For instance, the fundamental group of a closed hyperbolic 3-manifold acts non-trivially on a CAT(0) cube complex. Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has ‘uniform exponential growth’ or it is virtually abelian. I will present some generalizations of their theorem. This is joint work with Kasia Jankiewicz and Thomas Ng.
We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.
We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.
The notion of Heisenberg uniqueness pair has been introduced by Hedenmalm and Montes-Rodriguez (Ann. of Math. 2011) as a version of the uncertainty principle, that is, a nonzero function and its Fourier transform both cannot be too small simultaneously. Let $\Gamma$ be a smooth curve or finite disjoint union of smooth curves in the plane and $\Lambda$ be any subset of the plane. Let $\mathcal X(\Gamma)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $\Gamma$ and are absolutely continuous with respect to the arc length measure on $\Gamma.$ Let $\mathcal{AC}(\Gamma,\Lambda)=\{\mu\in \mathcal{X}(\Gamma) : \widehat\mu|_{\Lambda}=0\},$ then we say that $\Lambda$ is a Fourier uniqueness set for $\Gamma$ or $(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair, if $\mathcal{AC}(\Gamma,\Lambda)={0}.$
In this talk, we will discuss the following: Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta^\theta$ be the lattice-cross defined by \begin{equation} \Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right) \cup \left(\{0\}\times\beta\mathbb Z\right), \end{equation} where $\beta$ is a positive real and $\theta=1/{p}$, for some $p\in\mathbb N,$ then $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$ Moreover, the space $\mathcal{AC}\left(\Gamma,\Lambda_\beta^\theta\right)$ is infinite-dimensional provided $\beta>p.$
The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex. This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. I will present a general outline of this construction, starting with Khovanov’s functor. This work is joint with Tyler Lawson and Robert Lipshitz.
Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems and reflections groups. The Dunkl operators $T_{j}$, which were introduced by C. F. Dunkl in 1989, are deformations of directional derivatives by difference operators related to the reflection group. The goal of this talk will be to study harmonic analysis in the rational Dunkl setting. The first part will be devoted to some of results obtained in recent joint works with Jacek Dziubanski (2019, 2020).
improved estimates of the heat kernel $h_t(\mathbf{x},\mathbf{y})$ of the Dunkl heat semigroup generated by Dunkl–Laplace operator $\Delta_k=\sum_{j=1}^{N}T_j^2$ expressed in terms of analysis on the spaces of homogeneous type;
theorem regarding the support of Dunkl translations $\tau_{\mathbf{x}}\phi$ of $L^2$ compactly supported function $\phi$ (not necessarily radial).
The results listed above turn out to be useful tools in studying harmonic analysis in Dunkl setting. We will discuss this kind of applications in the second part of the talk. We will focus on a version of the classical Hormander’s multiplier theorem proved in joint work with Dziubanski (2019). If time permits, we will discussed how our tools can be used to for studying singular integrals of convolution type or Littlewood–Paley square functions in the Dunkl setting.
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
Our aim in this talk is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz condition in the space $L_p, 1 < p \leq 2$ on the sphere. We also prove analogues of Jackson’s direct theorem for the moduli of smoothness of all orders constructed on the basis of spherical shift. Finally, we prove equivalence between moduli of smoothness and $K$-functional for the couple $(L^2 (\sigma^{m-1} ), W^r_2 (\sigma^{m-1} ))$.
This is joint work with S. El Ouadih, O. Tyr and F. Saadi.
Anosov representations are representations of word hyperbolic groups into semisimple Lie groups with many good geometric properties. In this talk I will develop a theory of Anosov representation of geometrically finite Fuchsian groups (a special class of relatively hyperbolic groups). I will only discuss the case of representations into the special linear group and avoid general Lie groups. This is joint work with Canary and Zhang.
Rough analysis, as undertaken and popularised by Robert Strichartz and Jun Kigami, deals with the construction of a Laplacian and the study of associated problems on certain fractal sets, embedded in some Euclidean space, thus naturally exploiting the Euclidean topology. In this talk, we generalise this study to abstract, totally disconnected, metric measure unilateral shift spaces. In particular, we discuss the construction of a Laplacian as a renormalised limit of difference operators defined on finite sets that approximate the entire space. We further propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various (appropriately defined) boundary sets. We then define the Neumann derivative of functions on these boundary sets and establish a relationship between the three important concepts in our analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function.
This is a joint work with my doctoral student, Sharvari Neetin Tikekar.
In this talk I will present a quantization approach which directly relates Fujita-Odaka’s delta-invariant to the optimal exponent of certain Moser-Trudinger type inequality on polarized manifolds. As a consequence we obtain new criterions for the existence of twisted Kahler-Einstein metrics or constant scalar curvature Kahler metrics on possibly non-Fano manifolds.
We consider nonlinear Schrödinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation (strong ill-posedness) results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. We shall also discuss similar results for fractional Hartree equation. The talk is based on a joint work with Remi Carles and Saikatul Haque.
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.
One way to prove a theorem in analysis is to initially recognize what can go wrong. In this talk we will discuss a few recent results that follows a detailed version of this approach to establish existence of maximizers of some functionals.
In this talk, I would like to present some recent results regarding the behaviour of functions which are uniformly bounded under the action of a certain class of non-convex non-local functionals related to the degree of a map. In the literature, this class of functionals happens to be a very good substitute of the $L^p$ norm of the gradient of a Sobolev function. As a consequence various improvements of the classical Poincaré’s inequality, Sobolev’s inequality and Rellich-Kondrachov’s compactness criterion were established. This talk will be focused on addressing the gap between a certain exponential integrability and the boundedness for functions which are finite under the action of these class of non-convex functionals.
We consider a class of oscillatory integrals with polynomial phase functions $P$ over global domains $D$ in $\mathbb{R}^2$. As an analogue of Varchenko’s theorem in a global domain, we investigate the two main issues (i) whether the integral converges or not and (ii) how fast it decays. They are described in terms of a generalized notion of Newton polyhedra associated with $(P,D)$. Finally, we discuss its applications to the Strichartz Estimates associated with the general class of dispersive equations.
In this talk, we are concerned with sharp estimate for the spectral projection $P_\mu$ associated with the twisted Laplacian in the Lebesgue spaces. We provide a complete characterization of the sharp $L^p-L^q$ bound for $P_\mu$, which is similar to that for the spectral projection associated with the Laplacian. As an application, we discuss the resolvent estimate for the twisted Laplacian. This talk is based on a joint work with Sanghyuk Lee and Jaehyeon Ryu.
This talk primarily concerns the sharp bound on the spectral projection of the Hermite operator in the $L^p$ spaces. In comparison with the spectral projection of the Laplacian, the sharp bound has not been not so well understood. We consider the estimate for the spectral Hermite projection in general $L^p-L^q$ framework and obtain various new sharp estimates in an extended range. Especially, we provide a complete characterization of the local estimate and prove the endpoint $L^2$–$L^{2(d+3)/(d+1)}$ estimate which has been left open since the work of Koch and Tataru. We also discuss application of the projection estimate to related problems, such as the resolvent estimate for the Hermite operator and Carleman estimate for the heat operator.
For a commuting $d$-tuple of operators $\boldsymbol T=(T_1, \ldots , T_d)$ defined on a complex separable Hilbert space $\mathcal{H}$, let $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ be the $d \times d$ block operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ of commutators: $[T_j^*,T_i] := T_j^* T_i - T_i T_j^*$. We define an operator on the Hilbert space $\mathcal{H}$, to be designated the determinant operator, corresponding to the block operator $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$. We show that if the $d$-tuple is cyclic, the determinant operator is positive and the compression of a fixed set of words in $T_j^*$ and $T_i$ – to a nested sequence of finite dimensional subspaces increasing to $\mathcal{H}$ – does not grow very rapidly, then the trace of the determinant of the operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a certain small class of commuting $d$-tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
Let $\Omega$ be an irreducible classical bounded symmetric domain of rank $r$ in $\mathbb{C}^d$. Let $\mathbb{K}$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb{K}$ consisting of linear transformations acts naturally on any $d$-tuple $\mathbf{T}$ of commuting bounded linear operators by the rule: \begin{equation} k \cdot \mathbf{T} = \big( k_1(T_1, \dots, T_d), \dots, k_d(T_1, \dots, T_d) \big), \ k \in \mathbb{K}, \end{equation} where $k_1(\mathbf{z}), \dots, k_d(\mathbf{z})$ are linear polynomials. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\mathbf{T}$ is $\mathbb{K}$-homogeneous. We realize a certain class of $\mathbb{K}$-homogeneous $d$-tuples $\mathbf{T}$ as a $d$-tuple of multiplication by the coordinate functions $z_1, \dots, z_d$ on a reproducing kernel Hilbert space $\mathcal{H}_K$. (The Hilbert space $\mathcal{H}_K$ consisting of holomorphic functions defined on $\Omega$, with $K$ as reproducing kernel.) Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\Omega)$. For an irreducible bounded symmetric domain $\Omega$ of rank 2, an explicit description of the operator $\sum_{i=1}^d T_i^* T_i$ is given. Based on this formula, a conjecture giving the form of this operator in any rank $r \geq 1$ was made. This conjecture was recently verified by H. Upmeier.
A countable family $\{\psi_n: n \in \mathbb{N}\}$ of elements in a Hilbert space $\mathcal{H}$ constitutes a frame if there are constants $0< A\leq B < \infty$ s.t. $\forall f \in \mathcal{H}$ we have: \begin{equation} A\|f\|^2 \leq \sum\limits_n|\langle f,\psi_n \rangle|^2\leq B \|f\|^2, \end{equation} where $\langle\cdot, \cdot\rangle$ denotes an inner-product in $\mathcal{H}$. Frames were introduced by Duffin and Schaeffer in 1952 to deal with problems in nonharmonic Fourier series, and have been used more recently to obtain signal reconstruction for signals embedded in certain noises.
For a given pair of frames $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$, the associated mixed frame operator $S: \mathcal{H} \to \mathcal{H}; f \mapsto \sum_n \langle f, \psi_n \rangle\varphi_n$ is a bounded linear operator. The translation invariance of this operator plays a significant role in investigating reproducing formulas for frame pairs $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$.
In the present talk, we examine necessary and sufficient conditions for $S$ to be invariant under translations on $\mathbb{R}^n$ when $\{\psi_n\}$ and $\{\varphi_n\}$ belong to a special class of structured frame systems in $L^2(\mathbb{R}^n)$.
Given a finite (complex-valued) measure $\mu$ on the circle $\mathbb{R}/\mathbb{Z}$ we write supp$(\widehat{\mu})$ to denote the support of its Fourier coefficients. A subset $P$ of the integers is called a Riesz set if for all measures $\mu$ on $\mathbb{R}/\mathbb{Z}$ for which supp$(\widehat{\mu}) \subset P$, $\mu$ is absolutely continuous with respect to the Lebesgue measure. It is well-known that $P=-\mathbb{N}$, the negative integers, form a Riesz set given a famous result by F. and M. Riesz. Rudin proved that if one appends a lacunary set to the set of negative natural numbers, then it is a Riesz set. This idea was picked up by Meyer who wrote a beautiful paper on the subject, christened such sets (as Riesz sets) and proved, for instance, that appending the squares to the negative integers is still a Riesz set. For this proof a surprising (only initially though) use of the Bohr topology plays an important role.
Can we do this for the cubes? I believe that this is still open and the only progress made uses Fermat’s last theorem. I am not a specialist on the subject and have some rudimentary understanding of harmonic analysis. Most of what I have learnt is by talking to some specialists and reading a few papers/text books. Still, I will attempt to give you a primer on what I know and what I believe nobody does.
We will consider some results for non-local ODE with good conformal properties. These include equations with a fractional Laplacian and a Hardy-type critical potential. It turns out that conformal geometry provides a powerful tool to handle such non-local ODE. We will classify the asymptotic behavior of solutions and give a generalized notion of Wronskian, for instance, together with some applications.
The study of projective representations of a group has a long history starting from the work of Schur. Two essential ingredients to study the group’s projective representations are describing its Schur multiplier and representation group. In this talk, we describe these for the discrete Heisenberg groups. We also include a few general results regarding projective representations of finitely generated discrete nilpotent groups. This talk is based on the joint work with Sumana Hatui and E.K. Narayanan.
The key-point of this talk will be some exploration of function spaces concepts arising from time-frequency analysis respectively Gabor Analysis. Modulation spaces and Wiener amalgams have proved to be indispensable tools in time-frequency analysis, but also for the treatment of pseudo-differential operators or Fourier integral operators.
More precisely, we will recall a short summary of the concepts of Wiener amalgam spaces and modulation spaces, as well as the concept of Banach Gelfand Triples, with the associated kernel theorem (in the spirit of the L. Schwartz kernel theorem). We will indicate in which sense these spaces allow to capture more precisely the mapping properties of operators which may be unbounded in the Hilbert space setting. The subfamily of translation and modulation invariant spaces plays a specific role, with naturally associated regularization operators involving smoothing by convolution and localization by pointwise multiplication.
This thesis studies the mixing times for three random walk models. Specifically these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by 3-cycles of the form $(i, n − 1, n)$ and $(i, n, n − 1)$. We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $(n − 3/2)\log (n)$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group Bn generated by the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i \leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n \log (n)$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i < n$ also has a cutoff at $(n − 1/2)\log (n)$.
The random walk on the complete monomial group: Let $G_1 \subseteq \cdots \subseteq G_n \subseteq \cdots$ be a sequence of finite groups with $|G_1| > 2$. We study the properties of a random walk on the complete monomial group $G_n \wr S_n$ (wreath product of $G_n$ with $S_n$) generated by the elements of the form $(e,\dots, e, g; id)$ and $(e,\dots, e, g^{−1}, e,\dots, e, g; (i, n))$ for $g \in G_n$, $1 \leq i < n$. We call this the warp-transpose top with random shuffle on $G_n \wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n \log (n) + (1/2) n \log(|G_n| − 1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n \log (n)$ if $|G_n| = o( n^\delta )$ for all $\delta > 0$.
We describe a general context, related to metric spaces, in which a weak version of the celebrated Bishop-Gromov inequality is valid and suggest that this could serve of a synthetic version of a lower bound on the Ricci curvature.
Given a Galois extension of number fields $K/F$ and two elliptic curves $A$ and $B$ with equivalent residual Galois representation mod $p$, for an odd prime $p$, we will discuss the relation between the $p$-parity conjecture of $A$ twisted by $\sigma$ and that of $B$ twisted by $\sigma$ for an irreducible, self dual, Artin representation $\sigma$ of the Galois group of $K/F$.
This is a joint work with Somnath Jha and Tathagata Mandal.
We show $L^p\to L^q$ estimates for local and global $r$-variation operators associated to the family of spherical means. These can be understood as a strengthening of $L^p$-improving estimates for the spherical maximal function. Our bounds turn out to be sharp up to the endpoints (except for dimension 3) although we also provide positive results in certain endpoints. The results imply associated sparse domination and consequent weighted inequalities.
This is joint work with David Beltran, Richard Oberlin, Andreas Seeger, and Betsy Stovall.
On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution. This is joint work with Takuro Mochizuki (RIMS).
In 2004, Corvaja and Zannier proved an extension of Roth’s theorem on rational approximation of algebraic numbers. With a collaboration of Dr. Veekesh Kumar, we proved a simultaneous version of Corvaja and Zannier’s result. These results are applications of a strong form of the Subspace Theorem. In this talk, we shall discuss the motivation of Corvaja and Zannier’s result and our generalization.
Yau’s solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (arXiv:1612.06816) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. For the most part of the talk, we will explain the main ideas in showing that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give some ideas showing that this is also the number of acyclic orientations with unique sink of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.
This is joint work with P. Tetali (GATech) and D. Hathcock (CMU), and is available at arXiv:2010.11236.
(Note: unusual day.)
In this talk I present a heat semigroup approach to some intertwining formulas which arise in conformal CR geometry. This is recent joint work with G. Tralli.
William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. A natural question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, one can consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmueller space via a natural projection. In this talk, we will report progress on the torus case.
In 1987, A. Bonami and S. Poornima proved that a non-constant function which is homogeneous of degree zero cannot be a Fourier multiplier on homogeneous Sobolev spaces. In this talk, we will discuss the Fourier multipliers on Heisenberg group $\mathbb{H}^n$ and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. We define a notion of homogeneity of degree zero for bounded operators on $L^{2}(\mathbb{R}^n)$ and establish analogous results for Fourier multipliers on Heisenberg group and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. This talk is based on the recent work with Rahul Garg and Sundaram Thangavelu.
The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation.
The “small radius limit” of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K"ahler (cscK) metrics.
In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability.
I will also explain my similar result on the supercritical dHYM equation.
In this talk, we will discuss some of the existing techniques for distinguishing newforms. We will also report on a recent joint work with Kumar Murty and Biplab Paul.
Let $X_1,X_2, X_3$ be Banach spaces of measurable functions in $L^0(\mathbb R)$ and let $m(\xi,\eta)$ be a locally integrable function in $\mathbb R^2$. We say that $m\in \mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ if \begin{equation} B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i \langle\xi+\eta, x\rangle}d\xi d\eta, \end{equation} defined for $f$ and $g$ with compactly supported Fourier transform, extends to a bounded bilinear operator from $X_1 \times X_2$ to $X_3$.
In this talk we present some properties of the class $\mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ for general spaces which are invariant under translation, modulation and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus in the case $m(\xi,\eta)=M(\xi-\eta)$ and find conditions for these classes to contain non zero multipliers in terms of the Boyd indices for the spaces.
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.
We present the denominator identities for the periplectic Lie superalgebras and discuss their relations to representations of $\mathbf{p}(n)$ and $\mathbf{gl}(n)$. Joint work with Crystal Hoyt and Mee Seong Im.
In this expository lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-Sobolev inequalities with weights, the degeneracy. These results improve some well known estimates due to Fabes-Kenig-Serapioni from the 80’s in connection with the local regularity of solutions of degenerate elliptic equations and also some more recent results by Bourgain-Brezis-Minorescu. Our approach is different from the usual ones and is based on methods that come from Harmonic Analysis. This is especially visible when the connection of this theory with the BMO space and its different variants will be shown.
I will discuss some aspects of SYZ mirror symmetry for pairs $(X,D)$ where $X$ is a del Pezzo surface or a rational elliptic surface
and $D$ is an anti-canonical divisor. In particular I will explain the existence of special Lagrangian fibrations, mirror symmetry
for (suitably interpreted) Hodge numbers and, if time permits, I will describe a proof of SYZ mirror symmetry conjecture for del Pezzo surfaces.
This is joint work with Adam Jacob and Yu-Shen Lin.
We introduce a weight-dependent extension of the inversion statistic, a classical Mahonian statistic on permutations. This immediately gives us a new weight-dependent extension of $n!$. By restricting to $312$-avoiding permutations our extension happens to coincide with the weighted Catalan numbers that were considered by Flajolet in his combinatorial study of continued fractions. We show that for a specific choice of weights the weighted Catalan numbers factorize into a closed form, hereby yielding a new $q$-analogue of the Catalan numbers, different from those considered by MacMahon, by Carlitz, or by Andrews. We further refine the weighted Catalan numbers by introducing an additional statistic, namely a weight-dependent extension of Haglund’s bounce statistic, and obtain a new family of bi-weighted Catalan numbers that generalize Garsia and Haiman’s $q,t$-Catalan numbers and appear to satisfy remarkable properties. This is joint work with Shishuo Fu.
In his seminal paper (Acta Math. 1960), H"ormander established the $L^p$-$L^q$ boundedness of Fourier multipliers on $\mathbb{R}^n$ for the range $1<p \leq 2 \leq q<\infty.$ Recently, Ruzhansky and Akylzhanov (JFA, 2020) extended H"ormander’s theorem for general locally compact separable unimodular groups using group von Neumann algebra techniques and as a consequence, they obtained the $L^p$-$L^q$ boundedness of spectral multipliers for general positive unbounded invariant operators on locally compact separable unimodular groups.
In this talk, we will discuss the $L^p$-$L^q$ boundedness of global pseudo-differential operators and Fourier multipliers on smooth manifolds for the range $1<p\leq 2 \leq q<\infty$ using the nonharmonic Fourier analysis developed by Ruzhansky, Tokmagambetov, and Delgado. As an application, we obtain the boundedness of spectral multipliers, embedding theorems, and time asymptotic the heat kernels for the anharmonic oscillator.
This talk is based on my joint works with Duván Cardona (UGent), Marianna Chatzakou (Imperial College London), Michael Ruzhansky (UGent), and Niyaz Tokmagambetov (UGent).
Ramanujan’s Master theorem states that (under certain conditions) if a function $f$ on $\mathbb R$ can be expanded around zero in a power series of the form \begin{equation} f(x)=\sum_{k=0}^\infty (-1)^k a(k) x^k, \end{equation} then \begin{equation} \int_0^\infty f(x) x^{-\lambda-1}\,dx=-\frac{\pi}{\sin\pi\lambda} a(\lambda), \, \text{ for }\lambda\in\mathbb C. \end{equation} This theorem can be thought of as an interpolation theorem, which reconstructs the values of $a(\lambda)$ from its given values at $a(k), k\in \mathbb N\cup {0}$. In particular if $a(k)=0$ for all $k\in \mathbb N\cup {0}$, then $a$ is identically $0$. By selecting particular values for the function $a$, Ramanujan applied this theorem to compute several definite integrals and power series. This explains why it is referred to as the “Master Theorem”.
Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, Bertram, Olafsson-Pasquale proved an analogue of this theorem on Riemannian symmetric spaces of noncompact type.
In the first part of the talk, we shall discuss an analogue of this theorem for radial sections of line bundles over Poincare upper half plane. This is a joint work with Swagato K Ray.
In the second half, we shall discuss an analogue of this theorem for Sturm-Liouville operators. This is a joint work with Jotsaroop Kaur.
We prove some new $L^p$ estimates for the maximal function associated to bilinear Bochner Riesz means in all dimensions $n\geq 1$. This is a joint work with Saurabh Shrivastava.
The pseudo-hyperbolic space $H^{2,n}$ is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of $H^{2,n}$, we construct a unique complete maximal surface bounded by this circle. This construction relies on Gromov’s theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.
The affine Demazure modules are the Demazure modules that occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We call them $\mathfrak{g}$-stable if they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a $\mathfrak{g}$-stable affine Demazure module is isomorphic to the fusion (tensor) product of smaller $\mathfrak{g}$-stable affine Demazure modules, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact that was used in Chari et al. (op cit.), to prove the decomposition of $\mathfrak{g}$-stable affine Demazure modules. Our proof for this key fact is uniform, avoids the case-by-case analysis, and works for all finite-dimensional simple Lie algebras.
In this talk, we discuss Carleman estimates for Laplacian, which implies strong unique continuation for $-\Delta u+Vu$ with potential $V \in L^{\infty}.$ We briefly discuss unique continuation in certain critical situations such as when the potential is in $L^{n/2}_{loc}$ assuming Fourier restriction theorems. Then $L^{n/2}_{loc}$ case is a well-known result of Kenig-Jerison. Our proof of unique continuation is based on the Carleman estimate where it is a consequence of the spectral gap of Laplace Beltrami on the sphere.
We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. The category $R(G)$ of smooth representations of $G(F)$ has a decomposition into a product of indecomposable subcategories called Bernstein blocks and to each block is associated a non-negative real number called Moy-Prasad depth. We will begin with recalling all this basic theory. Then we will focus the discussion on ‘regular’ blocks. These are ‘most’ Bernstein blocks when the residue characteristic of $F$ is suitably large. We will then talk about an approach of studying blocks in $R(G)$ by studying a suitably related depth-zero block of certain other groups. In that context, I will explain some results from a joint work with Jeffrey Adler. One of them being that the Bernstein center (i.e., the center of a Bernstein block) of a regular block is isomorphic to the Bernstein center of a depth-zero regular block of some explicitly describable another group. I will give some applications of such results.
In this talk, we will try to characterize eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers via Roe-Strichartz type theorems in rank one symmetric spaces of noncompact type. This work has its origin in a simple result of Roe, which says that if all the derivatives and antiderivatives of a given function on the real line are uniformly bounded, then the function is a linear combination of sin(x) and cos(x). We will talk about ramification of this result in context of characterizing eigenfunctions of the Laplace-Beltrami operator. The talk will be based on a joint work with Prof. Rudra P. Sarkar.
The spherical averages often make their appearance in partial differential equations. For instance, the solution of the wave equation \begin{equation} u_{tt}=\Delta u,\ \ u(x,0)=0,\ \ u_{t}(x,0)=f(x),\ \ in\ \ \mathbb{R}^{3}\ \ is \end{equation} \begin{equation} u(x,t)=\frac{t}{4\pi}\int_{\mathbb{S}^{2}}f(x-ty)d\sigma(y), \end{equation}
where $d\sigma$ is the rotation invariant, normalized surface measure on the sphere $\mathbb{S}^{2}$. In [Proc. Natl. Acad. Sci. USA (1976)], Stein proved the following result:
Theorem. Let $n \geq 3$. Then \begin{equation}\Vert \sup_{t>0} \int_{\mathbb{S}^{n-1}}f(x-ty)d\sigma(y) \Vert_{L^{p}(\mathbb{R}^{n})} \leq C_{p}\Vert f\Vert_{L^{p}(\mathbb{R}^{n})} \end{equation} if, and only if $\frac{n}{n-1}<p\leq\infty$.
The above result was extended to dimension $n=2$, by Bourgain in [J. d’Anal. Math. (1986)]. Later, in [J. d’Anal. Math. (2019)] Lacey proved sparse domination for both lacunary and full spherical maximal functions.
In this talk, I shall talk about the bilinear spherical maximal functions of product type, which is defined in the spirit of bilinear Hardy–Littlewood maximal function. The lacunary and full bilinear spherical maximal functions are defined by \begin{equation} \mathcal{M}_{lac}(f_1,f_2)(x):= \sup_{j\in\mathbb{Z}} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-2^{j}y_i)d\sigma(y_i), \end{equation} \begin{equation} \mathcal{M}_{full}(f_1,f_2)(x):= \sup_{r>0} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-ry_i)d\sigma(y_i), \end{equation} where $f_{1},f_{2}\in\mathcal{S}(\mathbb{R}^{n})$, the Schwartz class. We have investigated the sparse domination and weighted boundedness of both the operators $\mathcal{M}_{lac}$ and $\mathcal{M}_{full}$ with respect to the bilinear Muckenhoupt weights $A_{\vec{p},\vec{r}}$. (Joint with Saurabh Shrivastava and Luz Roncal.)
A traditional way of assessing the size of a subset X of the integers is to use some version of density. An alternative approach, independently rediscovered by many authors, is to look at the closure of X in the profinite completion of the integers. This for example gives a quick, intuitive solution to questions like: what is the probability that an integer is square-free? Moreover, in many cases, one finds that the density of X can be recovered as the Haar measure of the closure of X. I will discuss some things that one can learn from this approach in the more general setting of rings of integers in global fields. This is joint work with Luca Demangos.
The first part of this talk deals with identifying and proving the scaling limit of a uniform tree with given child sequence. A non-negative sequence of integers $\mathbf{c}=(c_1, c_2, …, c_l)$ with sum $l-1$ is called a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1, v_2,…, v_l$ of the nodes, $v_i$ has exactly $c_i$ many children. Consider for each $n$, a child sequence $\mathbf{c}^n$ with sum $n-1$, and let $\mathbf{t}_n$ be the plane tree with $n$ nodes, which is uniformly distributed over the set of all plane trees having $\mathbf{c}^n$ as their child sequence. Broutin and Marckert (2012) prove that under certain assumptions on $\mathbf{c}^n$, the scaling limit of $\mathbf{t}_n$, suitably normalized, is the Brownian Continuum Random Tree (BCRT). We consider a more general setting, where a finite number of vertices of $\mathbf{t}_n$ are allowed to have large degrees. We prove that the scaling limit of $\mathbf{t}_n$ in this regime is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff sense.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerny, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_{\star}$, such that if a random walk is run up to time $un$ with $u<u_{\star}$, $n$ being the total number of nodes in the graph, a giant component of size $\text{O}(n)$ of the subgraph spanned by the vacant nodes i.e. the nodes that are not visited by the random walk, is seen. Whereas if the random walk is run up to time $un$ with $u>u_{\star}$, the size of the largest component of the subgraph spanned by the vacant nodes becomes $\text{o}(n)$. With the help of simulations, we try to investigate whether there is such a phenomenon for supercritical configuration models with heavy-tailed degrees.
An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff on $\mathbb{R}^d$ using iterates of the Laplacian. In 1934, Ingham used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of an integrable function on $\mathbb{R}$. In this talk, we discuss analogues of the theorems of Chernoff and Ingham for Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.
I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.
I will give a gentle introduction to total positivity and the theory of Polya frequency (PF) functions. This includes their spectral properties, basic examples including via convolution, and a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how PF functions connect to the Laguerre–Polya class and hence Polya–Schur multipliers, and mention 21st century incarnations of the latter.
Cohomology theories are one of the most important algebraic invariants of topological spaces and this has inspired the definition of several different cohomology theories in algebraic geometry. In this talk, we focus on algebraic K-theory, which is one such classical cohomological invariant of algebraic varieties. After motivating and introducing this notion, we discuss several fundamental properties of algebraic K-theory of varieties with algebraic group actions. Well-known examples of varieties with group actions include toric varieties and flag varieties.
Let $G$ be the group $SL_2$ over a finite extension $F$ of $\mathbb{Q}_p$, $p$ odd. I will discuss certain distributions on $G(F)$, belonging to what is called its Bernstein center (I will explain what this and many other terms in this abstract mean), supported in a certain explicit subset of $G(F)$ arising from the work of A. Moy and G. Prasad. The assertion is that these distributions form a subring of the Bernstein center, and that convolution with these distributions has very agreeable properties with respect to orbital integrals. These are ‘depth $r$ versions’ of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.
We prove well-posedness of McKean–Vlasov stochastic differential equations (McKean–Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. We propose (moment) stable numerical schemes for this class of McKean–Vlasov SDEs, namely tamed Euler and tamed Milstein schemes. Further, their rates of convergence in strong sense are shown to be 1/2 and 1 respectively. We employ the notion of measure derivative introduced by P.-L. Lions in his lectures delivered at the College de France. The strong convergence of the tamed Milstein scheme is established under mild regularity assumptions on the coefficients. To demonstrate our theoretical findings, we perform several numerical simulations on popular models such as mean-field versions of stochastic 3/2 volatility models and stochastic double well dynamics with multiplicative noise.
The talk is based on my recent joint works with Neelima (Delhi University), Christoph Reisinger (Oxford University) and Wolfgang Stockinger (Oxford University).
In classical Iwasawa theory, one studies a relationship called the Iwasawa main conjecture, between an analytic object (the p-adic L-function) and an algebraic object (the Selmer group). This relationship involves codimension one cycles of an Iwasawa algebra. The topic of higher codimension Iwasawa theory seeks to generalize this relationship. We will describe a result in this topic using codimension two cycles, involving an elliptic curve with supersingular reduction. This is joint work with Antonio Lei.
This thesis studies the mixing times for three random walk models. Specifically, these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $\left(n-\frac{3}{2}\right)\log n$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n\log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i< n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
The random walk on the complete monomial group: Let $G_1\subseteq\cdots\subseteq G_n \subseteq\cdots $ be a sequence of finite groups with $|G_1|>2$. We study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(e,\dots,e,g;$id$)$ and $(e,\dots,e,g^{-1},e,\dots,e,g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n+\frac{1}{2}n\log (|G_n|-1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n\log n$ if $|G_n|=o(n^{\delta})$ for all $\delta>0$.
Let $\boldsymbol T=(T_1, \ldots , T_d)$ be $d$ -tuple of commuting operators on a Hilbert space $\mathcal{H}$. Assume that $\boldsymbol T$ is hyponormal, that is, $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ] :=\big (\big ( \big [ T_j^*,T_i] \big )\big )$ acting on the $d$-fold direct sum of the Hilbert space $\mathcal{H}$ is non-negative definite. The commutator $[T_j^*,T_i]$, $1\leq i,j \leq d$, of a finitely cyclic and hyponormal $d$-tuple is not necessarily compact and therefore the question of finding trace inequalities for such a $d$-tuple does not arise. A generalization of the Berger-Shaw theorem for commuting tuple $\boldsymbol T$ of hyponormal operators was obtained by Douglas and Yan decades ago. We discuss several examples of this generalization in an attempt to understand if the crucial hypothesis{\rm in their theorem requiring the Krull dimension of the Hilbert module over the polynomial ring defined by the map $p\to p(\boldsymbol T)$, $p\in \mathbb C[\boldsymbol z]$, is optimal or not. Indeed, we find examples $\boldsymbol T$ to show that there a large class operators for which $\text{trace}[T_j^*,T_i]$, $1\leq j,i \leq d$, is finite but the $d$-tuple is not finitely polynomially cyclic, which is one of the hypothesis of the Douglas-Yan theorem. We also introduce the weaker notion of “projectively hyponormal operators” and show that the Douglas-Yan theorem remains valid even under this weaker hypothesis. However, one might look for a function of $ \big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ which may be in trace class. For this, we define an operator valued determinant of a $d\times d$-block operator $\boldsymbol B := \big (\big ( B_{i j} \big ) \big )$ by the formula
\begin{equation} \text{dEt}\big (\boldsymbol{B}\big ):=\sum_{\sigma, \tau \in \mathfrak S_d} \text{sgn}(\sigma)B_{\tau(1),\sigma(\tau(1))}B_{\tau(2),\sigma(\tau(2))},\ldots, B_{\tau(d),\sigma(\tau(d))}. \end{equation}
It is then natural to investigate the properties of the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big ] \big ),$
in this case, $B_{i j} = [T_j^*,T_i]$.
Indeed, we show that the operator dEt equals the generalized commutator
$\text{GC} \big (\boldsymbol T^*, \boldsymbol T \big )$ introduced earlier by
Helton and Howe. Among other things, we find a trace inequality for the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big] \big ),$
after imposing certain growth and cyclicity condition on the operator $\boldsymbol T$, namely,
\begin{equation} \text{trace} \big( {\rm dEt} \big( [[ \boldsymbol{T}^*, \boldsymbol{T} ]] \big) \big) \leq m \vartheta d! \prod_{i=1}^{d} |T_i|^2 \end{equation}
for some $\vartheta \geq 1.$ We give explicit examples illustrating the abstract inequality.
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.
The theory of projective representations of groups, extensively studied by Schur, involves understanding homomorphisms from a group into the projective linear groups. By definition, every ordinary representation of a group is also projective but the converse need not be true. Therefore understanding the projective representations of a group is a deeper problem and many a times also more difficult in nature. To deal with this, an important role is played by a group called the Schur multiplier.
In this talk, we shall describe the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their $t$-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.
This is a joint work with Pooja Singla.
Let (M,g) be a Riemannian manifold and ‘c’ be some homology class of M. The systole of c is the minimum of the k-volume over all possible representatives of c. We will use combine recent works of Gromov and Zhu to show an upper bound for the systole of [S^2x{}] under the assumption that [S^2x{}] contains two representatives which are far enough from each other.
The systematic study of determinantal processes began with the work of Macchi (1975), and since then it has appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths), and physics (fermions, repulsion arising in quantum physics). The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a separable Hilbert space. Let $H$ and K be two finite-dimensional subspaces of a Hilbert space, and $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons (2003) showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalues of Wishart matrix ensembles $W(N, N)$ and $W(N-1, N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M, N)$ has the same distribution as the directed last-passage time $G(M, N)$ on $\mathbb{Z}^2$ with the i.i.d. exponential weight. This was recently used by Basu and Ganguly to obtain stochastic domination between $G(N, N)$ and $G(N-1, N+1)$. Similar connections are also known between the largest eigenvalue of the Meixner ensemble and the directed last-passage time on $\mathbb{Z}^2$ with the i.i.d. geometric weight. We prove another stochastic domination result, which combined with Lyons’ result, gives the stochastic domination between the eigenvalue processes of Meixner ensembles $M(N, N)$ and $M(N-1, N+1)$.
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lectures 1–4 (Jan 14, 16, 29, Feb 5):
Lecture 5. Multivariable Scattering Theory (Tuesday, March 10)
The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with weak notions of principal G-bundles and connections on them. The bundle transition functions for such bundles are not continuous and thus there is no obvious notion of a topological isomorphism class.
In this talk, we shall discuss a few natural classes of weak bundles with connections which can be approximated in the appropriate norm topology by smooth connections on smooth bundles. We also show how we can associate a topological isomorphism class to such bundle-connection pairs, which is invariant under weak gauge changes. In stark contrast to classical notions, this topological isomorphism class is not independent of the connection.
In the 1980’s Tate stated the Brumer–Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Gross–Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. In 2008 and 2018 Dasgupta and Dasgupta–Spieß, conjectured formulas for this unit. During this talk I shall explain Tate’s conjecture and then the ideas for the constructions of these formulas. I will finish by explaining the results I have obtained from comparing these formulas.
It is commonly expected that $e$, $\log 2$, $\sqrt{2}$, $\pi$, among other “classical” numbers, behave, in many respects, like almost all real numbers. For instance, they are expected to be normal to base 10, that is, one believes that their decimal expansion contains every finite block of digits from ${0, \ldots , 9}$. We are very far away from establishing such a strong assertion. However, there has been some small recent progress in that direction. After surveying classical results and problems on normal numbers, we will adopt a point of view from combinatorics on words and show that the decimal expansions of $e$, of any irrational algebraic number, and of $\log (1 + \frac{1}{a})$, for a sufficiently large integer $a$, cannot be ‘too simple’, in a suitable sense.
We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class having polynomial Euler product and satisfying Selberg’s orthonormality condition. We show that on every vertical line $s=\sigma+it$ in the complex plane with $\sigma \in(1/2,1)$, these $L$-functions simultaneously take “large” values inside a small neighborhood.
This is joint work with Kamalakshya Mahatab and Lukasz Pankowski.
We start with reviewing Dwork’s seminal work on a certain $p$-adic hypergeometric function, which has an application to the unit-root $L$-function of the Legendre family of elliptic curves in characteristic $p>2$. Then I would like to overview what can be said about unit-root $L$-function of the family of abelian varieties over a curve, and discuss its potential applications.
An endomorphism $\phi: G\to G$ of a group yields an action of $G$ on itself, known as the $\phi$-twisted conjugacy action, where $(g,x)\mapsto gx\phi(g^{-1})$. The group $G$ is said to have the property $R_\infty$ if, for any automorphism $\phi$ of $G$, the orbit space of the $\phi$-twisted conjugacy action is infinite. This notion, and the related notion of Reidemeister number, originated from Nielsen fixed point theory.
It is an interesting problem to decide, given an infinite group $G$ whether or not $G$ has property $R_\infty$. We will consider the problem in the case when $G=GL(n,R), SL(n,R), n\ge 2$, when $R$ is either a polynomial ring or a Laurent polynomial ring over a finite field $\mathbb{F}_q$.
The talk is based on recent joint work with Oorna Mitra.
Let $G$ be an algebraic group defined over a finite field $\mathbb{F}_q$ and let $m$ be a positive integer. Shintani descent is a relationship between the character theories of the two finite groups $G(\mathbb{F}_q)$ and $G(\mathbb{F}_{q^m})$ of $\mathbb{F}_q$ and $\mathbb{F}_{q^m}$-valued points of $G$ respectively. This was first studied by Shintani for $G=GL_n$. Later, Shoji studied Shintani descent for connected reductive groups and related it to Lusztig’s theory of character sheaves. In this talk, I will speak on the cases where $G$ is a unipotent or solvable algebraic group. I will also explain the relationship with the theory of character sheaves.
We discuss a set of purely sequential strategies to estimate an unknown negative binomial mean $\mu$ under different forms of loss functions. We develop point estimation techniques where the thatch parameter $\tau$ may be known or unknown. Both asymptotic first-order efficiency and risk efficiency properties will be elaborated. The results will be supported by an extensive set of data analysis carried out via computer simulations for a wide variety of sample sizes. We observe that all of our purely sequential estimation strategies perform remarkably well under different situations. We also illustrate the implementation of these methodologies using real datasets from ecology, namely, weed count data and data on migrating woodlarks. (This is a Skype talk.)
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)
Lecture 5. Multivariable Scattering Theory
The quintic threefold (the zero set of a homogeneous degree 5 polynomial on CP^4) is one of the most famous examples of a Calabi Yau manifold. It is one of the most studied in the field of Enumerative Geometry. For example, how many lines are there on a Quintic threefold? In this talk we will explain some approaches to count curves on the Quintic threefold. In particular, we will try to explain the following idea: If Y is a submanifold of X, and we understand the Enumerative Geometry of X, how can we answer questions about the Enumerative Geometry of Y? We will try to explain the idea used by Andreas Gathman to compute all the genus zero Gromov-Witten invariants of the Quintic Threefold.
The talk will be self contained and will not assume any prior knowledge of Enumerative Geometry or Gromov-Witten Invariants.
For a finite abelian group $G$ and $A \subset [1, \exp(G) - 1]$, the $A$-weighted Davenport Constant $D_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$-weighted zero-sum subsequence. The original motivation for studying Davenport Constant was the problem of non-unique factorization in number fields. The precise value of this invariant for the cyclic group for certain sets $A$ is known but the general case is still unknown. Typically an extremal problem deals with the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. In a recent work with Prof. Niranjan Balachandran, we introduced an Extremal Problem for a finite abelian group related to Weighted Davenport Constant. In this talk I will talk about the behaviour of it for different groups, specially for cyclic group.
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)
Lecture 5. Multivariable Scattering Theory
In 1976, E.M. Stein proved $L^p$ bounds for spherical maximal function on Euclidean space. The lacunary case was dealt on later by C.P. Calderon in 1979. In a recent paper, M. Lacey has proved sparse bound for these functions and $L^p$ bounds will follow immediately as a result.
In this talk, we will look at various maximal functions corresponding to spherical averages and find sparse bounds for those functions. We will also observe some weighted and unweighted estimates that will follow as a consequences.
First, we will show sparse bound for lacunary spherical maximal function on Heisenberg group . Next we move on to full spherical maximal function. Then we study lacunary maximal function corresponding to the spherical average on product of Heisenberg groups. Finally, we will revisit generalized spherical averages on Euclidean space and prove sparse bounds for the related maximal functions.
Solitons are solutions of a special class of nonlinear partial differential equations (soliton equations, the best example is the KdV equation). They are waves but behave like particles. The term “soliton” combining the beginning of the word “solitary” with ending “on” means a concept of a fundamental particle like “proton” or “electron”.
The events: (1) sighting, by chance, of a great wave of translation, “solitary wave”, in 1834 by Scott–Russell, (2) derivation of KdV equation by Korteweg de Vries in 1895, (3) observation of a very special type of wave interactions in numerical experiments by Kruskal and Zabusky in 1965, (4) development of the inverse scattering method for solving initial value problems by Gardener, Greene, Kruskal and Miura in 1967, (5) formulation of a general theory in 1968 by P.D. Lax and (6) contributions to deep theories starting from the work by R. Hirota (1971-74) and David Mumford (1978-79), which also gave simple methods of solutions of soliton equations, led to the development of one of most important areas of mathematics in the 20th century.
This also led to a valuable application of solitons to physics, engineering and technology. There are two aspects of soliton theory arising out of the KdV Equation:
The subject is too big but I shall try to give some glimpses (1) of the history, (2) of the inverse scattering method, and (3) show that an algorithm based on algebraic-geometric approach is much easier to derive soliton solutions.
This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about a problem on certain holomorphic differential operators on the space of Jacobi forms.
In the first part we will briefly discuss how the statistics of signs of newforms determine them (work of Matomaki-Soundararajan-Kowalski) and then introduce certain ‘Linnik-type’ problems (the original problem was concerning the size of the smallest prime in an arithmetic progression in terms of the modulus) which ask for the size of the first negative eigenvalue (in terms of the analytic conductor) of various types modular forms, which has seen a lot of recent interest. Also specifically we will discuss the problem in the context of Yoshida lifts (a certain subspace of the Siegel modular forms), where in the thesis, we have improved upon the previously known result on this topic significantly. We will prove that the smallest $n$ with $\lambda(n)<0$ satisfy $n < Q_{F}^{1/2-2\theta+\epsilon}$, where $Q_{F}$ is the analytic conductor of a Yoshida lift $F$ and $0<\theta <1/4$ is some constant. The crucial point is establishing a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level.
We will focus on a similar question concerning the first negative Fourier coefficient of a Hilbert newform. If ${C(\mathfrak{m})}_{\mathfrak{m}}$ denotes the Fourier coefficients of a Hilbert newform $f$, then we show that the smallest among the norms of ideals $\mathfrak{m}$ such that $ C(\mathfrak{m})<0$, is bounded by $Q_{f}^{9/20+\epsilon}$ when the weight vector of $f$ is even and $Q_{f}^{1/2+\epsilon}$ otherwise. This improves the previously known result on this problem significantly. Here we would show how to use certain ‘good’ Hecke relations among the eigenvalues and some standard tools from analytic number theory to achieve our goal.
Finally we would talk about the statistical distribution of the signs of the Fourier coefficients of a Hilbert newform and essentially prove that asymptotically, half of them are positive and half negative. This was a breakthrough result of Matomaki-Radziwill for elliptic modular forms, and our results are inspired by those. The proof hinges on establishing some of their machinery of averages multiplicative functions to the number field setting.
In the second part of the talk we will introduce Jacobi forms and certain differential operators indexed by $\{D_{v}\}_{0}^{2m}$ that maps the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the direct sum of the differential operators $D_{v}$ for $v={1,2,…,2m}$ maps $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. Inspired by certain conjectures of Hashimoto on theta series, S. Bocherer raised the question whether any of the differential operators be removed from that map while preserving the injectivity. In the case of even weights S. Das and B. Ramakrishnan show that it is possible to remove the last operator. In the talk we will discuss the case of the odd weights and prove a similar result. The crucial step (and the main difference from the even weight case) in the proof is to establish that a certain tuple of congruent theta series is a vector valued modular form and finding the automorphy of the Wronskian of this tuple of theta series.
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
The date and time for the third and fourth lectures will be announced in due course.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras
Lecture 5. Multivariable Scattering Theory
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogeneous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y\mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid \cdot \mid$ is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal{P}_{\lambda}$ is endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
\begin{equation} \left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right) \end{equation}
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$ -dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of isolated vertices converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
The date and time for the third and fourth lectures will be announced in due course.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras
Lecture 5. Multivariable Scattering Theory
Many models of one dimensional random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For a few such models, the limiting interface profile, after scaling by characteristic KPZ scaling exponents of one-third and two-third, is known to be the Airy_2 process shifted by a parabola. This limiting process is expected to be “locally Brownian”, and a recent result gives a quantified bound on probabilities of events under the Airy_2 process on a unit order interval in terms of probabilities of the same events under Brownian motion (of rate two). This comparison also holds in the prelimit for the particular model of Brownian last passage percolation. In this talk, we will introduce KPZ universality and discuss this result and a number of consequences, using last passage percolation as an expository framework.
Joint work with Jacob Calvert and Alan Hammond.
We will discuss the celebrated Kneser–Tits conjecture for algebraic groups and report on some recent results. We will keep the technicalities to the minimum.
We will discuss some work on the Ricci flow on manifolds with symmetries. In particular, cohomogeneity one manifolds, i.e. a Riemannian manifold M with an isometric action by a Lie group G such that the orbit space M/G is one-dimensional. We will also explain how this relates to diagonalizing the Ricci tensor on Lie groups and homogeneous spaces.
The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.
Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, we found that a certain averaging of the height function at the rough smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show after suitable centering and rescaling that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough. This is joint work with Kurt Johansson and Vincent Beffara.
The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances. He will conclude with a discussion of recent joint work with mathematicians Michael Griffin of Brigham Young University, Larry Rolen of Georgia Tech, and Don Zagier of the Max Planck Institute, which sheds new light on this famous problem.
The Gyárfás–Sumner conjecture states the following: Let $a,b$ be positive integers. Then there exists a function $f$, such that if $G$ is a graph of clique number at most $a$ and chromatic number at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as induced subgraphs. This conjecture is still open, though for several special cases it is known to be true. We study the oriented version of this conjecture: Does there exist a function $g$, such that if the chromatic number of an oriented graph $G$ (satisfying certain properties) is at least $g(s)$ then $G$ contains all oriented trees on at most $s$ vertices as its induced subgraphs. In general this statement is not true, not even for triangle free graphs. Therefore, we consider the next natural special class – namely the 4-cycle free graphs – and prove the above statement for that class. We show that $g(s) \leq 4s^2$ in this case.
We also consider the rainbow (colorful) variant of this conjecture. As a special case of our theorem, we significantly improve an earlier result of Gyárfás and Sarkozy regarding the existence of induced rainbow paths in $C_4$ free graphs of high chromatic number. I will also discuss the recent results of Seymour, Scott (and Chudnovsky) on this topic.
Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for $p=2$. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{r})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are NOT isomorphic for $r \geq 2e+2$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{r}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{r}))]$ are NOT isomorphic for $r\geq4$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We will show both original and known results on Harmonic Analysis for functions defined on the infinite-dimensional torus, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results will include:
Several open problems and other questions will be considered. Some of the results presented are joint work with Emilio Fernandez (Universidad de La Rioja, Spain).
The study of Leavitt path algebras has two primary sources, the work of W.G. Leavitt in the early 1960’s on the module type of a ring, and the work by Kumjian, Pask, and Raeburn in the 1990’s on Cuntz-Krieger graph $C^*$-algebras. Given a directed graph $\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an $F$-algebra essentially built from the directed paths in the graph $\Gamma$. Reasonable necessary and sufficient graph-theoretic conditions for two directed graphs to have isomorphic Leavitt path algebras do not seem to be known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with a directed graph $\Gamma$, that we call the Leavitt inverse semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then $L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$ to have isomorphic Leavitt inverse semigroups.
This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.
We will present some recent studies on ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are matrices whose entries are polynomials of degree at most 1. We will discuss, how a good Groebner basis for these ideals help us compute primary decompositions and gather various other homological informations.
I will discuss a few examples of concepts that have interesting extensions if loops are allowed (but not required). I will include interval graphs, strongly chordal graphs, and other concepts.
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles (non-abelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions. We will discuss how these results can be applied to obtain an alternate proof of a result of Pauly-Ramanan. If time permits, we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.
Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.
The Lehmer conjecture and Serre’s lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.
G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.
We follow G.-C. Rota’s advice. This leads to several new results and improvement of known results. This includes Kostant’s non-vanishing results attached to simple complex Lie algebras, a new non-vanishing zone of the Nekrasov-Okounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of reciprocals of the cubic root of Klein’s absolute $j$- invariant, and hence the $j$-invariant itself. Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
2. Construction and Fourier Analysis of Automorphic Forms
3. Automorphic Forms and Operator Theory
4. Automorphic Forms in Scattering Theory
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
The sphere packing problem asks for the densest packing by congruent non-overlapping spheres in n dimensions. It is a famously hard problem, though easy to state, and with many connections to different parts of mathematics and physics. In particular, every dimension seems to have its own idiosyncracies, and until recently no proven answers were known beyond dimension 3, with the 3-dimensional solution being a tour de force of computer-aided mathematics.
Then in 2016, a breakthrough was achieved by Viazovska, solving the sphere packing problem in 8 dimensions. This was followed shortly by joint work of Cohn-Kumar-Miller-Radchenko-Viazovska solving the sphere packing problem in 24 dimensions. The solutions involve linear programming bounds and modular forms. I will attempt to describe the main ideas of the proof.
We describe the leading terms in the asymptotic behavior of the eigenvalues and the eigenfunctions to an elliptic Dirichlet spectral problem in a thin finite cylindrical domain with a periodically oscillating boundary by means of homogenization. Under suitable scaling and structure assumptions, the eigenfunctions show oscillatory behavior, and asymptotically localize with a profile solving a diffusion equation with quadratic potential on the real line. Methods for analysis of spectral asymptotics for heterogeneous media will be briefly discussed.
We will discuss certain rationality results for the critical values of the degree-$2n$ $L$-functions attached to $GL_1 \times O(n,n)$ over a totally real number field for an even positive integer $n$. We will also discuss some relations for Deligne periods of motives. This is part of a joint work with A. Raghuram.
Let W be a Weyl group and V be the complexification of its natural reflection representation. Let H be the discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) – H) is the Artin group described by the Dynkin diagram of W.
We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) – H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group.
The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes–Wall lattice for which some similar results hold.
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$.
We expect similar applications. We also discuss few results on the square-free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, \mathbb{Z})$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau’))$ ($(\tau,z,z,\tau’)$ in Siegel’s upper half-plane of degree 2) to $H \times H$ is its restriction to $z=0$, which we denote by $F|_{z=0}$. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$-norms of such pullbacks to central values of $L$-functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$ $L$-functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$ with the norm of its pullback provides a measure of concentration of $F_g$ along $z=0$. We recall certain conjectures pertaining to the size of the’mass’. We use the amplification method to improve the currently known bound for $N(F_g)$.
We will define an invariant for annular links using the combinatorial link Floer complex that gives genus bounds for annular cobordisms. The celebrated slice-Bennequin inequality relates slice genus of a knot with its contact geometric invariants. We investigate similar relations in our context. In particular, we will define an invariant of transverse knots that refines the transverse invariant $\theta$ in knot Floer homology.
In this talk, we discuss the “local smoothing” phenomenon for Fourier integral operators with amplitude function belongs to the “symbol class”. We give an overview of the regularity results which have been proven to date. We use harmonic analysis of Hermite functions in the study of Fourier integral operators. Finally, we give an application of the local smoothing estimate to the wave equation and maximal operators. This is a joint work with Prof. P. K. Ratnakumar.
If $T$ is a cnu (completely non-unitary) contraction on a Hilbert space, then its Nagy-Foias characteristic function is an operator valued analytic function on the unit disc $\mathbb{D}$ which is a complete invariant for the unitary equivalence class of $T$. $T$ is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all elements $\varphi$ of the group $M$ of biholomorphic maps on $\mathbb{D}$. A stronger notion is of an associator. $T$ is an associator if there is a projective unitary representation $\sigma$ of $M$ such that $\varphi(T) = \sigma(\varphi)^* T \sigma(\varphi)$ for all $\varphi$ in $M$. In this talk we shall discuss the following result from a recent work of the speaker with G. Misra and S. Hazra: A cnu contraction is an associator if and only if its characteristic function $\theta$ has the factorization $\theta(z) = \pi_* (\varphi_z) C \pi(\varphi_z), z \in \mathbb{D}$ for two projective unitary representations $\pi, \pi_∗$ of $M$.
This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms. We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries.
In the first part of this talk I shall recall what the Hot spots conjecture is. Putting it in mathematical terms, I shall provide a brief history of the conjecture. If time permits I shall explain a proof of the conjecture for Euclidean triangles.
I will begin by reviewing the relationship between Hitchin’s Integrable System and 4d N=2 Supersymmetric Quantum Field Theories. I will then discuss two classes of deformations of the Hitchin system which correspond, in the physical context, to relevant and marginal deformations of a conformal theory. The study of relevant deformations turns out to be related to the theory of sheets in a complex Lie algebra and their classification leads to a surprising duality between sheets in a Lie algebra and Slodowy slices in the Langlands dual Lie algebra (work done with J. Distler) . If there is time, I will discuss marginal deformations which are related to studying the Hitchin system as a family over the moduli space of curves including over nodal curves (ongoing project with J. Distler and R. Donagi) .
Modular forms are certain functions defined on the upper half plane that transform nicely under $z\to -1/z$ as well as $z\to z+1$. By a modular relation (or a modular-type transformation) for a certain function $F$, we mean that which is governed by only the first map, i.e., $z\to -1/z$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha \beta = 1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha) = F(iw, \beta)$ or other relations of the form $F(z, \alpha) = F(z, \beta)$ etc. The famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eisenstein series on ${\rm SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha) = F(z, iw, \beta)$ recently obtained by Kesarwani, Moll and the speaker. This implies that one can superimpose the theta structure on the Ramanujan-Guinand formula.
The current work arose from answering a similar question - can we superimpose the theta structure on a recent modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ which generalizes an important result of Ramanujan? In the course of answering this question in the affirmative, we were led to a surprising new generalization of $\zeta(z, a)$. This new zeta function, $\zeta_w(z, a)$, satisfies interesting properties, albeit they are much more difficult to derive than the corresponding ones for $\zeta(z, a)$. In this talk, I will briefly discuss the theory of the Riemann zeta function $\zeta(z)$, the Hurwitz zeta function $\zeta(z, a)$ and then describe the theory of $\zeta_w(z, a)$ that we have developed. In order to obtain the generalized modular relation (with the theta structure) satisfied by $\zeta_w(z, a)$, one not only needs this theory but also has to develop the theory of reciprocal functions in a certain kernel involving Bessel functions as well as the theory of a generalized modified Bessel function. (Based on joint work with Rahul Kumar.)
A beautiful $q$-series identity found in the unorganized portion of Ramanujan’s second and third notebooks was recently generalized by Maji and I. This identity gives, as a special case, a three-parameter identity which is a rich source of partition-theoretic information allowing us to prove, for example, Andrews’ famous identity on the smallest parts function $\mathrm{spt}(n)$, a recent identity of Garvan, and identities on divisor generating functions, to name a few. Guo and Zeng recently derived a finite analogue of Uchimura’s identity on the generating function for the divisor function $d(n)$. This motivated us to look for a finite analogue of my generalization of Ramanujan’s aforementioned identity with Maji. Upon obtaining such a finite version, our quest to look for a finite version of Andrews’ $\mathrm{spt}$-identity necessitated finding finite analogues of rank, crank and their moments. We could obtain finite versions of rank and crank for vector partitions. We were also able to obtain a finite analogue of a partition identity recently conjectured by George Beck and proven by Shane Chern. I will discuss these and some related results. This is joint work with Pramod Eyyunni, Bibekananda Maji and Garima Sood.
Given a closed orientable surface $S$, a $(G,X)-$structure on $S$ is the datum of a maximal atlas whose charts take values on $X$ and transition functions are restrictions of elements in $G$. Any such structure induces a holonomy representation $\rho:\pi_1(\widetilde{S})\to X$ which encodes geometric data of the structure. Conversely, can we recover a geometric structure from a given representation? Is such a structure unique? In this talk we answer these questions by providing old and new results.