Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field. Objects of broad interest in commutative algebra, representation theory, and algebraic geometry like generic determinantal rings, Plücker coordinate rings of Grassmannians, symmetric determinantal rings, rings defined by Pfaffians of alternating matrices etc. arise as the invariant rings $S^G$ of such group actions.

In characteristic zero, reductive groups are linearly reductive and therefore the embedding of the invariant ring $S^G$ in the ambient polynomial ring $S$ splits. This explains a number of good algebro-geometric properties of the invariant ring in characteristic zero. In positive characteristic, reductive groups are typically no longer linearly reductive. We determine, for the natural actions of the classical groups, precisely when $S^G$ splits from $S$ in positive characteristic.

This is joint work with Melvin Hochster, Jack Jeffries, and Anurag K. Singh.

More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. After introducing the problem and known results in the first part of the talk, I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups, and some $p$-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.

Let $F$ be a field that has a primitive $p$-th root of unity. According to the Bloch–Kato conjecture, now a theorem by Voevodsky and Rost, the norm-residue map \begin{equation} k_*(F)/pk_*(F) \rightarrow H^*(F, \mathbb{F}_p) \end{equation} from the reduced Milnor $K$-theory to the Galois cohomology of $F$ is an isomorphism of $\mathbb{F}_p$-algebras.

This isomorphism gives a presentation of the rather mysterious Galois cohomology ring through generators and relations. In joint work with Jan Minac, Cihan Okay, Andy Schultz, and Charlotte Ure, we have obtained a second cohomology refinement of the Bloch–Kato conjecture. Using this we can characterize the maximal $p$-extension of $F$, as the “decomposing field” for the cohomology of the absolute Galois group.

Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo–Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo–Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.

Let $f:Y \to X$ be a log resolution of singularities which is an isomorphism over the smooth locus of $X$, and the exceptional locus $E$ is a simple normal crossing divisor on $Y$. We prove vanishing (and non-vanishing) results for the higher direct images of differentials on $Y$ with log poles along $E$ in the case when $X$ is a toric variety. Our consideration of these sheaves is motivated by the notion of $k$-rational singularities introduced by Friedman-Laza. This is joint work with Anh Duc Vo and Wanchun Shen.

This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann-Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann-Fock spaces on the complex plane. This problem has also been studied for the Bargmann-Fock spaces on the higher dimensional complex Euclidean spaces. Very few results on interpolating and sampling hypersurfaces in higher dimensions are known. We have proven certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerd'{a}, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of them. In dimension two, we provided sufficient conditions for a smooth algebraic hypersurface to be uniformly flat in terms of its projectivization.

The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.

This thesis comprises two main parts. The details of the two parts are as follows:

The first part of the thesis deals with the monopole-dimer model. The
dimer (resp. monomer-dimer) model deals with weighted enumeration of
perfect matchings (resp. matchings). The monopole-dimer model is a
signed variant of the monomer-dimer model which has determinantal
structure. A more general model called the *loop-vertex model* has
also been defined for an oriented graph and the partition function in
this case can also be written as a determinant. However, this model
depends on the orientation of the graph. The monopole-dimer model
interprets the loop-vertex model independent of the orientation for
planar graphs with Pfaffian orientation. The first part of the thesis
focuses on the extension of the monopole-dimer model for planar graphs
(Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show
that the partition function of this model can be expressed as a
determinant of a generalised signed adjacency matrix. We then show that
the partition function is independent of the orientations of the planar
graphs so long as they are Pfaffian. When these planar graphs are
bipartite, we show that the computation of the partition function
becomes especially simple. We then give an explicit product formula for
the partition function of three-dimensional grid graphs a la Kasteleyn
and Temperley–Fischer, which turns out to be fourth power of a
polynomial when all grid lengths are even. Further, we generalise this
product formula to higher dimensions, again obtaining an explicit
product formula. We also discuss about the asymptotic formulas for the
free energy and monopole densities.

In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional Möbius and Klein grids, and evaluate the partition function for three-dimensional Möbius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and Möbius boundary conditions, generalising a result of Lu and Wu.

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.

The classical theory of Toeplitz operators on Hardy space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the theory for Toeplitz operators on Hardy space over a multiply connected domain $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^\infty + C(\partial D)$ symbols, and the characterization of Fredholm operators with $H^\infty+C$ symbols to those for the multiply connected domain $D$. Also, he came up with the striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.

In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.

We define atomic Hardy space `$H^p_{\mathcal{L}, at}(\mathbb{C}^n), 0<p\leq 1$`

for the
twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy
space defined using the maximal function corresponding to the heat
semigroup $e^{-t\mathcal{L}},t>0$. We also prove sharp `$L^p, 0<p\leq 1$`

estimates for $\left(\mathcal{L}\right)^{-\beta/2}e^{i\sqrt{\mathcal{L}}}$. More
precisely we prove that it is a bounded operator on $H^p_{\mathcal{L},
at}(\mathbb{C}^n)$ when $\beta\geq (2n-1)\left(1/p-1/2\right)$.

Finite Ramsey theory is the study of structure that becomes unavoidable in large finite objects. In this talk, we will provide a brief taste of this rich and beautiful subject. We will start with the following question: In any group of six people, can we always find three who know one another or three who don’t? A far-reaching generalization of this question was first answered in a paper on logic by Frank Ramsey in 1928. Our approach to it will involve graph theory and combinatorics, with a dash of probability. No prerequisites will be needed to understand the talk.

The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case i.e. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.

In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Goerenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.

We can apply Obstruction bundle gluing introduced by Hutchings and Taubes in Embedded contact homology to the setting of Morse theory. The goal is to understand this gluing technique in a simpler set up and analyse Morse theory with assumptions weaker than Smale condition on surfaces. This is work in progress with Yuan Yao.

This Ph.D. thesis consists of two parts. In both the parts we study two new notions of canonical Kahler metrics introduced by Pingali viz. ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (higher cscK) metric’ both of whose definitions are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kähler (cscK) metric respectively. On a compact Kahler manifold a higher extremal Kähler metric is a Kahler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is a Kahler metric whose top Chern form is a real constant multiple of its volume form or equivalently whose top Chern form is harmonic. In both the parts we consider a special family of minimal ruled complex surfaces called as ‘pseudo-Hirzebruch surfaces’ which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and the zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical metrics on them.

In the first part of this Ph.D. thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every Kahler class admits a higher extremal Kahler metric which is not a higher cscK metric. The construction of the required metric boils down to solving an ODE depending on a parameter on a closed and bounded interval with some boundary conditions, but the ODE is not directly integrable and requires a very delicate analysis for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude from this that higher cscK metrics (momentum-constructed or otherwise) do not exist in any Kahler class on this Kahler surface. We will briefly see the analogy of this problem with the related problem of constructing extremal Kähler metrics which are not cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by Tønnesen-Friedman and Apostolov et al..

In the second part of this Ph.D. thesis we will see that if we relax the smoothness condition on our metrics a bit and allow our metrics to develop ‘conical singularities’ along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface then we do get ‘conical higher cscK metrics’ in each Kahler class of the Kahler surface by the momentum construction method. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions. We will show that our constructed metrics are conical Kahler metrics satisfying the strongest condition for conical metrics viz. the ‘polyhomogeneous condition’ of Jeffres-Mazzeo-Rubinstein, and we will interpret the conical higher cscK equation globally on the surface in terms of currents by using Bedford-Taylor theory. We will then employ the top ‘log Bando-Futaki invariant’ to obtain the linear relationship between the cone angles of the conical singularities of the metrics at the zero and infinity divisors of the surface.

We characterize the existence of an Ulrich vector bundle on a variety $X\subset{\bf P}^N$ in terms of the existence of a subvariety satisfying certain conditions. Then we use this fact to prove that $(X,\mathcal{O}_X(a))$ where $X$ is a complete intersection of dimension $n\geq 4$, which if n = 4, is either ${\bf P}^4$ with $a\geq 2$, or very general with $a\geq 1$ and not of type (2), (2, 2), does not carry any Ulrich bundles of rank $r\leq 3$. Work in collaboration with A.F. Lopez.

We discuss representation of integers as a sum of $n$ squares. We explain the quaternionic composition law for sums of four squares and a proof of a theorem of Lagrange on which positive integers can be expressed as a sum of four squares. We outline general connections to the theory of quadratic forms.

You might have carved a piece of potato to create a stamp to print with. How many patterns can we get from a single potato stamp? One approach to answering this question sheds light on the rich connections between objects and their symmetries, and leads us to a more general counting strategy.

A set $\Omega$ is a spectral set for an operator $T$ if the spectrum of $T$ is contained in $\Omega$, and von Neumann’s inequality holds for $T$ with respect to the algebra $R(\Omega)$ of rational functions with poles off of the closure of $\Omega$. It is a complete spectral set if for all $n \in \mathbb{N}$, the same is true for $M_n(\mathbb C) \otimes R(\Omega)$. The rational dilation problem asks, if $\Omega$ is a spectral set for $T$, is it a complete spectral set for $T$? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (e.g., over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is an example of a variety naturally associated to a constrained subalgebra of the disk algebra, namely $\mathbb{C} + z^2 A(\mathbb D)$. This talk discusses why rational dilation fails for a large class of such varieties associated to constrained algebras.

The study of the class L+2 of Hilbert space operators which are the product of two bounded positive operators first arose in physics in the early ’60s. On finite dimensional Hilbert spaces, it is not hard to see that an operator is in this class if and only if it is similar to a positive operator. We extend the exploration of L+2 to separable infinite dimensional Hilbert spaces, where the structure is much richer, connecting (but not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The (generalized) spectral properties of elements of L+2 are also outlined, as well as membership in L+2 among various special classes of operators, including algebraic and compact operators.

Here the proof that a positive (but not necessarily strictly positive) trigonometric polynomial with matrix coefficients can be written as a finite sum of hermitian squares of analytic polynomials is sketched. The difficulties in the case of such polynomials with coefficients which are operators on an infinite dimensional Hilbert space is also briefly discussed.

The Fejer–Riesz thorem states that a positive (i.e., non-negative) trigonometric polynomial of degree $d$ on the unit circle is the hermitian square of an analytic polynomial of the same degree. Rosenblum extended this to polynomials with operator coefficients. The goal of these talks will be to outline a proof of a similar theorem in two variables. Since the techniques used in some proofs of the single variable case play an important role in the two variable proof, this particular talk concentrates primarily on these ideas. An application to strictly positive operator valued multivariable trigonometric polynomials is also considered.

In the colored asymmetric simple exclusion process (ASEP), one places a particle of “color” $-k$ at each integer site $k \in \mathbb{Z}$. Particles attempt to swap places to the left with rate $q \in [0,1)$ and to the right with rate 1; the swap succeeds if the initiating particle has a higher color than the other particle (thus the particles tend to get more ordered over time). We will discuss the space-time scaling limit of this process (as well as a related discrete analog known as the stochastic six-vertex model), captured via a height function given by certain colored particle counts. The limit lies in the Kardar-Parisi-Zhang universality class, and is given by the Airy sheet and directed landscape, which were first constructed in 2018 by Dauvergne-Ortmann-Virág as limits in a very different setting – of fluctuations of a model of a random directed metric. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) with certain Gibbs or spatial Markov properties will play fundamental roles in our discussion. This is based on joint work with Amol Aggarwal and Ivan Corwin.

This talk focuses on the asymptotic analysis of an optimal control problem constrained by the stationary Stokes equations within a periodically perforated domain. The problem involves distributive controls applied to the interior region, where the Stokes operator includes oscillating coefficients for the state equations. We aim to demonstrate the convergence of the solutions of the considered optimal control problem to those of the limit OCP governed by the stationary Stokes equations and to establish the convergence of the associated cost functional.

This talk considers an optimal control problem governed by a semi-linear heat equation within a two-dimensional pillar-type domain $\Omega_{\epsilon}$ .The problem features highly oscillatory periodic coefficients in both the state equation and the cost function $A_\epsilon$ and $B_\epsilon$. Our objective is to analyze the convergence of the optimal solutions (as $\epsilon \to 0$ ) and to identify the limit of the optimal control problem in a fixed domain that effectively captures the impact of the oscillatory coefficients.

In this talk, we shall study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We will focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.

In the initial part of this talk, we shall present a couple of results on local continuous extension of proper holomorphic mappings $F:D \to \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $bD$ and $b\Omega$. These results are motivated by a well-known work by Forstneric–Rosay. However, our results allow us to have much lower regularity, for the patches of $bD, b\Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $b\Omega$ to contain boundary points of infinite type.

We will also discuss another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph–Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds – viewed as complex submanifolds embedded in some ambient complex manifold – with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $b\Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge–Ampere equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called “visibility”.

In the concluding part of this talk, we will explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincare disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali–Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.

In the 1950s, topologists introduced the notion of equivariant cohomology $H_G(E)$ for a topological space $E$ with an action by a compact group $G$. If the action is free, $H_G(E)$ should be $H(E/G)$, and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential forms for a compact Lie group acting on a differential manifold $E$, and proved a result amounting to stating that the cohomology of that complex computes $H_G(E)$. In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of $G$.

Our aim is to reconsider such considerations, by replacing vector spaces by a $k$-linear symmetric monoidal category, requiring that this category contain an odd unit to account for the supersymmetric dimension plus some further properties, and considering modules of a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. (Joint work with Siddhartha Sahi.)

The strength of the French mathematical school goes back to the XVIIth century, with major figures like Descartes, Viète, Fermat or Pascal. But history shows that periods when French mathematics thrived alternated with less fruitful times. The factors are diverse, ranging from the role of singular geniuses to social and political causes: structure of higher education and research institutions, political upheavals, wars…

We will focus on the last 150 years: in the period before 1870, Germany had become the dominant scientific power, major advances were being made in England and Italy, to the great dismay of French scientists. In 1870, France was defeated by Prussia, the Emperor was overthrown, and a very favorable period started, allowing the emergence of a remarkable generation of French mathematicians. World War I had a disastrous effect on science (not only on science, of course), abruptly bringing the momentum to a halt. The reconstruction took some years: it is only in the 1950s that French mathematics flourished one again. By that time, however, mathematics had become much more specialized, and applied mathematics were left behind. It took again many years to reach a more balanced landscape where the pure and applied parts can thrive.

For $n$ a natural number, consider the sequence of $n$ rational numbers $n/1, n/2, n/3, \dots, n/n$. Round each to the nearest integer to obtain sequence of $n$ integers. How many are odd?

In this talk we will see how knowledge of sums of squares and a result of Gauss will help to lead us to a somewhat surprising result. Time permitting, we will discuss similar results.

Given a box packed with identical cubes of cheese, what is the maximum damage one can cause with a single straight cut through the box? This seemingly simple puzzle represents an old but recurrent mathematical theme that slices through numerous fields such as number theory, functional analysis, probability theory, and computational complexity theory. The cross-sections of convex bodies hold many mysteries, some of which continue to puzzle mathematicians today. We will focus on the deceptively simple case of the cube to demonstrate some of these ideas and open questions. No cheese will be harmed in the making of this talk.

For positive integers $n$, $g$ and $d$, the moduli space $M(n,g,d)$ of degree d holomorphic maps to $\mathbb{CP}^n$ from non-singular projective curves of genus g is smooth and irreducible for $d > 2g-2.$ It is contained as an open subset within the compact moduli space $K(n,g,d)$ of “stable maps”, i.e., degree d holomorphic maps to $\mathbb{CP}^n$ from at-worst-nodal projective curves of arithmetic genus $g.$ An unfortunate feature of this very natural compactification is that $M(n,g,d)$ is far from being dense in $K(n,g,d)$. Concretely, this means that many stable maps are not “smoothable”, i.e., they don’t arise as limits of non-singular ones. In my talk, I will explain this phenomenon and a new sufficient condition for smoothability of stable maps, obtained in joint work with Fatemeh Rezaee.

This dissertation consists of three parts, and two important types of complex hessian equations, namely – the J-equation and the deformed Hermitian Yang Mills (dHYM) equation.

In the first part, the main aim is to find out some appropriate “singular” solutions of the equations when they don’t admit smooth solutions (or equivalently, when the so-called “Nakai criteria” fails) - this is the so-called unstable case. An algebro-geometric characterization of the slopes for both the equations is formulated – which we call the “minimal J-slope” in the case of the J-equation and the “maximal dHYM-slope” for the dHYM equation. On compact Kahler surfaces we show that there exists a unique closed $(1,1)$- Kahler current that solves the “weak version” of the equations (i.e. the wedge product is replaced by the “non-pluripolar product”) with the modified slopes. In the higher dimensional case, we conjecture analogous existence and uniqueness results.

In the second part, the convergence behavior of the J-flow is studied on certain generalized projective bundles using Calabi symmetry. For the bundles an “invariant version” of the minimal J-slope is introduced. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the J-equation with the invariant minimal J-slope. This result resolves our conjecture for J-equation on these examples with symmetry.

In the third part, the convergence behavior of a dHYM flow, called the “cotangent flow”, is studied in the unstable case on the blowup of $\mathbb{CP}^2$ or $\mathbb{CP}^3$. Analogous to our results in the second part, it is shown that this flow converges to some unique limit in the unstable case, and the limiting current solves the dHYM equation with the (invariant) maximal dHYM-slope.

How far is it from Bangalore to Chennai? Is there a single correct answer to this question? In this talk we will explore different notions of distance as well as why you might choose one over another depending on the context. This will take us on a brief sight seeing tour through geometry, graph theory, and number theory.

Sub-Riemannian (sR) geometry is the geometry of bracket-generating metric distributions on a manifold. Peculiar phenomena in sR geometry include the exotic Hausdorff dimension describing the growth rate of the volumes of geodesic balls. As well as abnormal geodesics that do not satisfy any variational equation. In this talk I will survey my results, to appear in a forthcoming book, which show how both these phenomena are reflected in the spectral theory of the hypoelliptic Laplacian in sR geometry.

In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.

Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.

One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category `$\mathrm{Bord}_n^{(n -1)}$`

, a variant of the fully extended `$\mathrm{Bord}_{n}$`

. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.

In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.

We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.

We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.

In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.

Topological rigidity is the property that every homotopy equivalence between two closed $n$-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces.

We prove that all non-compact, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any $\pi_1$-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane.

An oriented manifold $M$ is said to be Hopfian if every self-map $f\colon M\to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface $S$ is of finite type if and only if every proper self-map of $S$ of degree one is homotopic to a homeomorphism.

It is an open problem to describe the shape of the reductions of local Galois representations attached to cusp forms at primes away from the level, or more generally, the shape of the reductions of two-dimensional crystalline representations. Partial results go back to Deligne, Fontaine and Edixhoven. One folklore conjecture (attributed to Breuil, Buzzard and Emerton) is that if the weight is even and the slope is fractional, then the reduction is always irreducible.

In this talk we shall state and prove our zig-zag conjecture which deals with large exceptional weights and half-integral slopes. These weights fall squarely outside the scope of the above conjecture. The conjecture states that the reduction in these cases is given by an alternating sequence of irreducible and reducible representations depending on the size of two auxiliary parameters. Special cases of zig-zag have been proved over the years by various authors using Langlands correspondences.

The present general proof uses the reverse of a recent limiting argument due to Chitrao-Ghate-Yasuda in the Colmez-Chenevier rigid analytic blow up space of trianguline representations to reduce the study of the reduction of crystalline representations to results on the reductions of semi-stable representations due to Breuil-Mezard, Guerberoff-Park and most recently Chitrao-Ghate.

Let us denote by *Rubio de Francia square function* the square function formed by frequency projections on a collection of
disjoint intervals of the real line. J. L. Rubio de Francia established in 1985 that this operator is bounded on $L^p$ for
$p\ge 2$ and on $L^p(w)$, for $p>2$, with weights $w$ in the Muckenhoupt class $A_{p/2}$. What happens in the endpoint
$L^1(w)$ for $w \in A_1$ was left open, and Rubio de Francia conjectured the validity of the estimate in this endpoint.

In this talk we will show a new pointwise sparse estimate for the Rubio de Francia square function. Such a bound implies quantitative weighted estimates which, in some cases, improve the available results. We will also confirm that the $L^2(w)$ conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue to the Rubio de Francia square function. In general, the $L^2$ weighted inequality is still an open problem.

Joint work with Francesco Di Plinio, Mikel Flórez-Amatriain, and Ioannis Parissis.

The video of this talk is available on the IISc Math Department channel.

In the area of Affine Algebraic Geometry, there are several problems on polynomial rings which are easy to state but difficult to investigate. Late Shreeram S. Abhyankar was the pioneer in investigating a class of such problems known as Epimorphism Problems or Embedding Problems. In this non-technical survey talk, we shall highlight some of the contributions of Abhyankar, Moh, Suzuki, Sathaye, Russell, Bhatwadekar and other mathematicians.

It was in ancient India that zero received its first acceptance as an integer in its own right. There was an awareness about its perils and yet ancient Indian mathematicians not only embraced zero as an integer but allowed it to participate in all four arithmetic operations, including as a divisor in a division.

But division by zero is strictly forbidden in the present edifice of mathematics. Verses from mathematical giants like Brahmagupta and Bhaskaracharya referring to numbers with “zero in the denominator” shock the modern reader. Certain examples in the Bijaganita of Bhaskaracharya appear as absurd nonsense.

But then there was a time when square roots of negative numbers were considered non-existent and forbidden; even the validity of subtracting a bigger number from a smaller number (i.e., the existence of negative numbers) took a long time to gain universal acceptance. Is it possible that we have simply bound ourselves to a certain safe convention and that there could be other approaches (“mathematical models” in fancy language) in which the ideas of Brahmagupta and Bhaskaracharya, and even the examples of Bhaskaracharya, will appear not only valid but even natural?

Enterprising modern mathematicians have created elaborate legal (or technical) machinery to overcome the limitations imposed by the prohibition against use of zero in the denominator. The most familiar are the methods of calculus with its concept of limit, results like l’Hopital’s rule, and a language which enables one to express intuitive ideas like $\frac{1}{0} = \infty$ through legally permitted euphemisms. Less well-known are the devices of commutative algebra and algebraic geometry like “localisation” which describes a legal structure for directly writing fractions with zero in the denominator without any subterfuge, and the more sophisticated ideas of “valuation theory” which admit multiple levels of infinities and thereby provide higher-dimensional analogues of l’Hopital’s rule.

In this talk we shall present an algebraic model proposed by Prof. Avinash Sathaye for understanding Bhaskaracharya’s
treatment of *khahara*, (numbers with) zero in the denominator. A crucial ingredient of this model is the important
concept of “idempotent” in modern abstract algebra (elements $e$ satisfying the relation $e^2=e$). To historians of
mathematics who have tried to interpret Bhaskaracharya’s *khahara* in the light of calculus, the examples in Bijaganiita
appear as absurdities. But all difficulties disappear in the light of the algebraic treatment based on idempotents. A verse
from the commentary of Kr.s.n.adaivaj˜na indicates that idempotence was indeed envisaged as a natural property of numbers
like zero and its reciprocal, the *khahara*.

Prof. Sathaye’s interpretation of Bhaskaracharya’s *khahara* also gives a new meaning to certain mysterious utterances of
Ramanujan recorded by P.C. Mahalanobis. In the light of valuation theory, Bhaskaracharya’s *khahara* not only deserves our
praise, perhaps they indicate unexplored possibilities!

The Brownian motion is the scaling limit of random walks where the step distribution has finite second moment. Various random objects constructed from the Brownian motion, e.g., the Brownian continuum random tree and the Brownian map, arise naturally in the study of random trees, graphs, and maps. In the first talk, we will give a gentle introduction to these objects. In the second talk, we will discuss some recent advances in establishing certain Brownian objects as the scaling limits of different models of random discrete structures.

Let $S_{g,k}$ be a connected oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S_{g,k})$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher-rank representations.

For a representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, the Hilbert length of a curve $\gamma\in \pi_1(S_{g,k})$ for $n >2$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock-Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$ representation. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.

Let $X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \tau(A)= \sum_{i=1}^{n}\log |\lambda_i(A)|^2, \end{equation} where $\lambda_i(A)$ are the eigenvalues of $A$. We show that the same $j$ dominates $\rho$ with respect to the translation length at the origin as well. Lindström’s Lemma for planar networks and Weyl’s Majorant Theorem are some of the key ingredients of the proof.

In both cases, if $S_{g,k}$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.

Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.

In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.

We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.

We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.

Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.

Consider a multipartite graph $G$ with maximum degree at most $n-o(n)$, parts $V_1,\ldots,V_k$ have size $|V_i|=n$, and every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such $G$ has an independent set, referred to as an ‘independent transversal’, which contains exactly one vertex from each part $V_i$. They further conjectured that the vertex set of $G$ can be decomposed into pairwise disjoint independent transversals. We resolve this conjecture approximately by showing that $G$ contains $n-o(n)$ pairwise disjoint independent transversals. As applications, we give approximate answers to questions on packing list colorings and multipartite Hajnal-Szemerédi theorem. We use probabilistic methods, including a ‘two-layer nibble’ argument. This talk is based on joint work with Tuan Tran.

In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.

For a planar domain and an admissible weight function on it, we discuss some aspects of the corresponding weighted Bergman kernel. First, we see a precise relation between the weighted Bergman kernel and the classical Bergman kernel near a smooth boundary point of the domain. Second, the weighted kernel gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the unit disc with respect to the weighted Bergman metrics arising from weights of the form $K(z,z)^{-d}$, where $K$ denotes the classical Bergman kernel and $d$ is a non-negative integer. Specific examples that we discuss in detail include those in which the isometry takes values in polydisk or a cartesian product of a disc and a unit ball, where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.

In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.

A continuous function in an Euclidean domain is harmonic if and only if it satisfies the spherical mean value property for all spheres contained in that domain. But what happens if a continuous function satisfies instead the following ‘restricted mean value property’: for each point in the domain it satisfies the mean value property precisely on one such sphere (centered at the point). Then is the function still going to be harmonic? This is the classical ‘one-circle problem’ posed by Littlewood. We will see some results dealing with sufficient conditions in terms of the boundary behavior of the function for the above problem to have an affirmative answer in the setting of (1) domains in Riemannian manifolds and (2) Hadamard manifolds of pinched negative sectional curvature, extending classical results of Fenton for the Euclidean unit disc. This is based on a joint work with Prof. Kingshook Biswas.

The video of this talk is available on the IISc Math Department channel.

Given two distinct cuspidal automorphic `$L$`

-functions (of `$\mathrm{GL}_n$`

and `$\mathrm{GL}_m$`

over `$\mathbb{Q}$`

) one expects that their quotients will have infinitely poles, but this is surprisingly hard to prove. In this talk, I will discuss my recent work on the case `$m=n-2$`

and the primitivity of the `$L$`

-functions of cuspidal automorphic `$L$`

-functions of `$\mathrm{GL}_3$`

. These methods also work for Artin `$L$`

-functions and, more generally, for the `$L$`

-functions of Galois representations under further hypotheses.

One of the central themes in number theory is the study of special values of `$L$`

-functions, in particular, the investigation of their transcendental nature and algebraic relations among them. A special function governing linear relations among the values `$L(1,\chi)$`

as `$\chi$`

varies over Dirichlet characters modulo `$q$`

, is the digamma function, which is the logarithmic derivative of the gamma function. In this talk, we discuss the arithmetic nature and related properties of values of the digamma function at rational arguments, and emphasize their connection with a seemingly unrelated conjecture of Erdos, which is still open.

We establish Hardy, Hardy-Rellich and Rellich identities and inequalities with sharp constants for Grushin vector fields. We provide explicit remainder terms which substantially improve those known in the literature. This is based on a joint work with Debdip Ganguly and Prasun Roychowdhury.

The video of this talk is available on the IISc Math Department channel.

Families of `$p$`

-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing the correct setup for the study of `$p$`

-adic deformations of automorphic forms. In order to obtain arithmetic applications, such as constructing `$p$`

-adic `$L$`

-functions or proving explicit reciprocity laws for Euler systems, one needs to perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the `$p$`

-stabilization of the automorphic form we are interested in.

While the geometry of an eigenvariety at points of cohomological weight is well understood thanks to classicality results, the study at classical points which are limit of discrete series (such as weight `$1$`

Hilbert modular forms or weight `$(2,2)$`

Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch–Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjecture.

Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint `$p$`

-adic `$L$`

-functions.

In this talk we will illustrate this philosophy based on ideas of Joël Bellaïche.

In this talk, I will report a work in progress in which we show $L^p$ bounds for singular integral operators formed by $(n-1)$-dimensional Hörmander-Mihlin multipliers. In our case, the multipliers act depending on $(n-1)$-dimensional variable subspaces, which depend only on the first $n-1$ variables.

We prove $L^p$ boundedness for these operators for $p>3/2$. Assuming that the frequency support of the function is contained in an annulus, we can show $L^p$ boundedness for $p>1$.

The video of this talk is available on the IISc Math Department channel.

Define $g(k) = \min \{ s :$ every positive integer can be written as a sum of $k$th powers of natural numbers with atmost $s$ summands$\}$. Lagrange proved that $g(2) = 4$. Waring conjectured that $g(3) = 9, g(4) = 19$ and so on.

In fact, in this question, there has been a lot of contribution from Indian mathematicians. The method of attacking this problem is called the circle method and it originates from a seminal paper of Hardy and Ramanujan. The final result owes a lot to the contributions of S.S. Pillai. The analogous question over number fields was settled by C.P. Ramanujam. We shall explain their contributions toward this problem.

For all positive powers of primes `$p \geq 5$`

, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight `$1/2$`

weakly holomorphic modular form or a suitable vector-valued harmonic Maassform. To this end, we work with the logarithmic derivatives of these twisted Borcherds products, and offer various numerical examples of non-trivial linear congruences between them modulo `$p=11$`

. In the case of positive powers of primes `$p = 2, 3$`

, we obtain similar results by multiplying the logarithmic derivative with a Hilbert class polynomial as well as a power of the modular discriminant function. Both results confirm a speculation by Ono. (joint work with Andreas Mono).

For a given function $a(x,\xi)$ on $\mathbb{R}^n \times \mathbb{R}^n$, consider the pseudo-differential operator $a(x,D)$ defined by

\begin{equation} a(x,D) (f)(x) =\int_{\mathbb{R}^n} a(x,\xi) \widehat f(\xi) e^{2\pi i x\cdot \xi} d\xi, \end{equation}

where $\widehat{f}$ denotes the Fourier transform of a function $f$. Let $S^0$ be the set of all smooth functions $a: \mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{C}$ satisfying

\begin{equation} \left| \frac{\partial^\beta_x}{\partial_{\xi} ^\alpha} a (x,\xi)\right| \leq {C_{\alpha,\beta} }\, {( 1+ |\xi| )^{-|\alpha|}} \end{equation}

for all $x,\xi \in \mathbb{R}^n $ and for all multi indices $\alpha$ and $\beta$. It is well known that for $a\in S^0$, the associated
pseudo-differential operator $a(x,D)$ extends to a bounded operator on $L^p(\mathbb{R}^n)$ to itself, for `$1<p<\infty$`

.

In this talk, we will discuss an analogue of this result on radial sections of line bundles over the Poincaré upper half plane. More precisely, we will focus on the group $G=\mathrm{SL}(2,\mathbb{R})$, where we will explore the boundedness properties of the pseudo-differential operator defined on functions of fixed $K=\mathrm{SO}(2)$-type in $G$. Additionally, we will explore the case where the symbol exhibits restricted regularity in the spatial variable.

This talk is based on a joint work with Michael Ruzhansky.

The video of this talk is available on the IISc Math Department channel.

We consider the monomial expansion of the $q$-Whittaker polynomials
given by the fermionic formula and via the *inv* and *quinv*
statistics. We construct bijections between the parametrizing sets of
these three models which preserve the $x$- and $q$-weights, and which
are compatible with natural projection and branching maps. We apply
this to the limit construction of local Weyl modules and obtain a new
character formula for the basic representation of
$\widehat{\mathfrak{sl}_n}$.

How many rational points with denominator of a given size lie within a given distance from a compact hypersurface? In this talk, we shall describe how the geometry of the surface plays a key role in determining this count, and present a heuristic for the same. In a recent breakthrough, J.J. Huang proved that this guess is indeed true for hypersurfaces with non-vanishing Gaussian curvature. What about hypersurfaces with curvature only vanishing up to a finite order, at a single point? We shall offer a new heuristic in this regime which also incorporates the contribution arising from “local flatness”. Further, we will describe how several ideas from Harmonic Analysis can be used to establish the indicated estimates for hypersurfaces of this type immersed by homogeneous functions. Based on joint work with N. Technau.

The video of this talk is available on the IISc Math Department channel.

The matrix $M$ of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) The value of $M$ is equal to zero. (ii) For all principal minors of $M^T$ (transpose of $M$) the value is non-negative. (iii) For any optimal mixed strategy $y$ of the maximizer either $y_i>0$ or $(My)_i>0$ for each coordinate $i$.

For any $1 < q <\infty,$ standard representation formulas and the Calderon–Zygmund estimates imply $u \in W^{2,q}_{\text{loc}}\left(\mathbb{R}^{n}\right)$ if $\Delta u \in L^{q}_{\text{loc}}(\mathbb{R}^{n}).$ Combined with the Sobolev–Morrey embeddings for $q>n,$ we deduce that $\nabla u$ is locally Hölder continuous. However, as soon as we pass from the linear case to the quasilinear operator, we no longer have any representation formula for the solution of the following problem \begin{equation} {-}{\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) = f \end{equation} if $p \neq 2$ and CZ estimates for second derivatives of the solution are not yet known. In fact, the solution can fail to be $C^{2}$ even when $f \equiv 0.$

However, one can still establish Hölder continuity of the gradient whenever

${\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) \in L^{q}_{\text{loc}}$ and $q>n.$ These type of results are often called
“Nonlinear Calderon–Zygmund theory”, as the regularity for the gradient is the same, i.e. “as if” Calderon–Zygmund estimates for second derivatives are
valid! This result relies heavily on a fundamental regularity result, commonly known as the DeGiorgi–Nash–Moser estimate, for $p$-harmonic functions.
However, such regularity results are specific to equations and are in general false for elliptic systems. In another groundbreaking work, Uhlenbeck extended
gradient Hölder continuity estimates for solutions to special type of systems, which includes the homogeneous $p$-Laplacian systems.

In this lecture, I would sketch the main ideas involved to establish nonlinear Calderon–Zygmund theory for scalar equations and elliptic systems with Uhlenbeck
structures. In the second half, I would discuss how to extend these estimates to the following $p$-Laplacian type system for vector-valued differential forms
\begin{equation}
d^{\ast}\left(\left\lvert d u \right\rvert^{p-2}d u\right) = f.

\end{equation}
This includes systems which are, strictly speaking, even non-elliptic.

This Talk is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of developments, although breakthroughs are expected in the not-too-distant future.

Nonlinear constitutive relationships which, at a given point in the space, present a behaviour for large arguments that is described by monomials of order $p$ and $q$ is considered in this presentation.

The original contribution this work makes is that the nonlinear problem can be approximated by a weighted $p$-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p=2$, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials.

The main result of this work is that for “large” Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted $p$-Laplace problem.

Solvability of the Dirichlet problem with data in $L^p$ for some finite $p$ for elliptic operators, such as the Laplacian, amounts to showing that the associated elliptic/harmonic measure satisfies a Reverse Hölder inequality. Under strong connectivity assumptions, it has been proved that such a solvability is equivalent to the fact that that all bounded null-solutions of the operator in question satisfy Carleson measure estimates. In this talk, we will give a historical overview of this theory and present some recent results in collaboration with M. Cao and P. Hidalgo where, without any connectivity, we characterize certain weak Carleson measure estimates for bounded null-solutions in terms of a Corona decomposition for the elliptic measure. This extends the previous theory to non-connected settings where, as a consequence of our method, we establish Fefferman-Kenig-Pipher perturbation results.

The video of this talk is available on the IISc Math Department channel.

Since the work of Kubota in the late 1960s, it has been known that certain Gauss sum twisted (multiple) Dirichlet series are closely connected to a theory of automorphic functions on metaplectic covering groups. The representation theory of such covering groups was then initiated by Kazhdan and Patterson in the 1980s, who emphasized the role of a certain non-uniqueness of Whitattaker functionals.

Motivated on the one hand by the recent theory of Weyl group multiple Dirichlet series, and on the other by the so-called “quantum” geometric Langlands correspondence, we explain how to connect the representation theory of metaplectic covers of $p$-adic groups to an object of rather disparate origin, namely a quantum group at a root of unity. This gives us a new point of view on the non-uniqueness of Whittaker functionals and leads, among other things, to a Casselman–Shalika type formula expressed in terms of (Gauss sum) twists of “$q$”-Littlewood–Richardson coefficients, objects of some combinatorial interest.

Joint work with Valentin Buciumas.

The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case ie. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.

In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Gorenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.

In this talk, we will explore the $L^{p}$-boundedness of both bilinear and multilinear maximal averages defined on non-degenerate hypersurfaces. Additionally, we will delve into the $L^2(\mathbb{R}^d)\times L^2(\mathbb{R}^d)\times\cdots\times L^2(\mathbb{R}^d) \to L^{2/m}(\mathbb{R}^d)$ estimates for $m$-linear maximal averages, focusing on hypersurfaces with $1\leq \kappa < md-1$ non-zero principal curvatures.

The video of this talk is available on the IISc Math Department channel.

Baladi and Vallée shows the limit Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers with bounded denominators based on ergodic methods. We give an analogue of the result for complex continued fractions over imaginary quadratic number fields and discuss applications in value distribution of `$L$`

-functions of `$\mathrm{GL}_2$`

(joint with Dohyeong Kim and Seonhee Lim).

This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann–Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann–Fock spaces on the complex plane. This problem has also been studied for the Bargmann–Fock spaces on the higher dimensional complex Euclidean spaces. Very few results about the interpolating and sampling hypersurfaces in higher dimensions are known. We have proved certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerda, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied the uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of it. In dimension two, we provided sufficient conditions for a smooth hypersurface to be uniformly flat in terms of its projectivization.

The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.

After a gentle introduction to the spectral approach to geometry, we extend the framework in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by operator algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc.

We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.

There are many ways to associate a graph (combinatorial structure) to a
commutative ring $R$ with unity. One of the ways is to associate a
*zero-divisor graph* $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$
are all elements of $R$ and two vertices $x, y \in R$ are adjacent in
$\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a
combinatorial structure $\Gamma(R)$ and show that the representatives of
some algebraic invariants are eigenvalues of the Laplacian of
$\Gamma(R)$. Moreover, we discuss association of another combinatorial
structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two
positive integers. The Young’s partition lattice $L(m,n)$ is defined to
be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq
\cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset
as Young diagrams ordered by inclusion. We conclude this talk with a
discussion on Stanley’s conjecture regarding symmetric saturated chain
decompositions (SSCD) of $L(m,n)$.

On a so-called Stein manifold the $\overline{\partial}$-problem can be solved in each degree $(p,q)$ where $q\geq 1$, or in other words the Dolbeault cohomology vanishes in these degrees. Sufficient conditions on complex manifolds which ensure that the Dobeault cohomology in degree $(p,q)$ is finite dimensional or vanishes have been studied since Andreotti-Grauert, who introduced the notions of $q$-convex/$q$-complete manifolds, which generalize Steinness. For manifolds with boundary, Hormander and Folland-Kohn introduced the condition now called $Z(q)$ which ensures finite-dimensionality of the cohomology in degree $q$ as well as $\frac{1}{2}$ estimates for the $\overline{\partial}$-Neumann operator. These conditions ($q$-convexity/completeness and $Z(q)$) are biholomorphically invariant characteristics of the underlying complex manifold.

In the context of Hermitian manifolds, a different type of sufficient condition implies that the $L^2$-cohomology in degree $(p,q)$-vanishes. Here one assumes that the sum of any $q$-eigenvalues is positive, and this also leads to the vanishing of the $L^2$-cohomology via the Bochner-Kohn-Morrey formula. These conditions are not biholomorphically invariant (they depend on the choice of the metric).

In this report on ongoing joint work with Andy Raich and Phil Harrington, we discuss the relationship between the two types of the condition. We give new sufficient conditions for the vanishing of the $L^2$-cohomology in degree $(p,q)$ in a domain in a complex manifold and discuss to what extent the conditions are necessary.

Consider critical Bernoulli bond percolation on $\mathbb{Z}^2$. We show that the two arm exponent is strictly larger than twice the one arm exponent. This answers a question of Schramm and Steif (2010), and shows that their proof of the existence of exceptional times on the triangular lattice also applies to the square lattice. We use an interpolation formula via noise to obtain asymptotic correlation of crossings and apply this at each scale to obtain the strict inequality of arm exponents. This talk is based on joint work with Vincent Tassion.

This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.

Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.

We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.

The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.

For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).

This is joint work in progress with Dan Cristofaro-Gardiner. We explore the topological dynamics of Reeb flows beyond periodic orbits and find the following rather general phenomenon. For any Reeb flow for a torsion contact structure on a closed 3-manifold, any point is arbitrarily close to a proper compact invariant subset of the flow. Such a statement is false if the invariant subset is required to be a periodic orbit. Stronger results can also be proved that parallel theorems of Le Calvez-Yoccoz, Franks, and Salazar for homeomorphisms of the 2-sphere. In fact, we can also extend their results to Hamiltonian diffeomorphisms of closed surfaces of any genus.

Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.

In this talk, I will survey some of the main results, techniques, and questions in this area.

In this talk, I will describe a complete geodesic metric $d_p$ on the finite energy space $\mathcal{E}^p(X,\theta)$ for $p\geq 1$ where $\theta$ represents a big cohomology class. This work generalizes the complete geodesic metrics in the Kahler setting to the big setting. When p=1, the metric $d_1$ in the Kahler setting has found various applications in the understanding of Kahler-Einstein and Constant Scalar Curvature Kahler metrics. In this talk, I’ll describe how to construct the metric and explain some properties that could have useful applications in the future.

The Macdonald polynomials are a homogeneous basis for the algebra of symmetric polynomials, which generalize many important families of special functions, such as Schur polynomials, Hall-Littlewood polynomials, and Jack polynomials.

The interpolation polynomials, introduced by F. Knop and the speaker, are an inhomogeneous extension of Macdonald polynomials, which are characterized by very simple vanishing properties.

The binomial coefficients are special values of interpolation polynomials, which play a central role in the higher rank $q$-binomial theorem of A. Okounkov.

We will give an elementary self-contained introduction to all three objects, and discuss some recent results, open problems, and applications.

Let $\mathbb B_d$ be the open unit ball in $\mathbb C^d$ and $\boldsymbol T$ be a commuting $d$-tuple of bounded linear operators
on a complex separable Hilbert space $\mathcal H$. Let $\mathcal U(d)$ be the linear group of unitary transformations acting on
$\mathbb C^d$ by the rule: $\boldsymbol z \mapsto u\cdot \boldsymbol z$, $\boldsymbol z \in \mathbb C^d$, where $u\cdot \boldsymbol z$
is the usual matrix product. We say that $\boldsymbol T$ is $\mathcal U(d)$-homogeneous if $u \cdot \boldsymbol T$ is unitarily
equivalent to $\boldsymbol T$ for all $u\in \mathcal U(d)$.
In this talk, we describe $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol M$ of multiplication by the coordinate functions acting
on a reproducing kernel Hilbert space `$\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq {\rm Hol}(\mathbb B_d, \mathbb C^n)$,`

where
$n$ is the dimension of the joint kernel of $\boldsymbol T^*$. The case $n=1$ is well understood, here, we focus on the case $n=d.$ We
describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative definite kernels quasi invariant under the
action of the group $\mathcal U(d).$ As a result, we obtain criterion for boundedness, irreducibility and mutual unitary equivalence
among these operators.

This is a joint work with Soumitra Ghara, Gadadhar Misra and Paramita Pramanick.

We consider probability measures on $GL(n,\mathbb{R})$ that are invariant under the left action of the orthogonal group $O(n,\mathbb{R})$ and satisfy a mild integrability condition. For any such measure we consider the following two quantities: (a) the mean of the log of the absolute value of the eigenvalues of the matrices and (b) the Lyapunov exponents of random products of matrices independently drawn with respect to the measure. Our main result is a lower bound for (a) in terms of (b).

This lower bound was conjectured by Burns-Pugh-Shub-Wilkinson (2001), and special cases were proved by Dedieu-Shub (2002), Avila-Bochi (2003) and Rivin (2005). We give a proof in complete generality by using some results from the theory of spherical functions and Jack polynomials.

This is joint work with Diego Armentano, Gautam Chinta, and Michael Shub. (arXiv:2206.01091), (Ergodic theory and Dynamical systems, to appear).

For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is a powerful tool to study the geometry of $X$ because of results such as the degeneration of the Hodge-de Rham spectral sequence (when $X$ is proper). For singular varieties, it follows from the work of Deligne and Du Bois that there is a substitute called the Du Bois complex which satisfies many of the nice properties enjoyed by the de Rham complex in the smooth case. In this talk, we will discuss some classical singularities associated with this complex, namely Du Bois and rational singularities, and some recently introduced refinements, namely $k$-Du Bois and $k$-rational singularities. This is based on joint work with Wanchun Shen and Anh Duc Vo.

This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.

The focus of the first part is when the domain is the complex plane. In this case, given a finite twisted ideal polygon, there exists a harmonic map heat flow $u_t$ such that the image of $u_t$ is asymptotic to that polygon for all $t\in[0,\infty)$. Moreover, we prove that given any twisted ideal polygon in $\mathbb{H}^3$ with \textit{rotational symmetry}, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.

In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $\text{PSL}_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely \textit{marked and bordered surfaces} (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (\text{PSL}_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$. In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.

The basic ideas of Calculus started with Archimedes, and reached a highly developed form in the 17th Century with Newton and Leibniz often being credited as its inventors. What was not so well-known until only a few decades ago is that between the 14th and 17th Century there was an unbroken lineage of profound mathematicians working in Kerala who had independently discovered many of the great themes of Calculus. This talk is an introduction to the lives and works of some of the prominent members of the Kerala School of Mathematics. Most of the talk will be accessible to a general audience. Only towards the end of the talk some elementary mathematics will be assumed to explain a few of their contributions.

A matrix factorisation of a polynomial $f$ is an equation $AB = f \cdot {\rm I}_n$ where $A,B$ are $n \times n$ matrices with polynomial entries and ${\rm I}_n$ is the identity matrix. This question has been of interest for more than a century and has been studied by mathematicians like L.E. Dickson. I will discuss its relation with questions arising in algebraic geometry about the structure of subvarieties in projective hypersurfaces.

Let $G$ be a finite simple graph (with no loops and no multiple edges), and let $I_G(x)$ be the multi-variate independence polynomial of $G$. In 2021, Radchenko and Villegas proved the following interesting characterization of chordal graphs, namely $G$ is chordal if and only if the power series $I_G(x)^{-1}$ is Horn hypergeometric. In this talk, I will give a simpler proof of this fact by computing $I_G(x)^{-1}$ explicitly using multi-coloring chromatic polynomials. This is a joint work with Dipnit Biswas and Irfan Habib.

Quantum toroidal algebras are the next class of quantum affinizations after quantum affine algebras, and can be thought of as “double affine quantum groups”. However, surprisingly little is known thus far about their structure and representation theory in general.

In this talk we’ll start with a brief recap on quantum groups and the representation theory of quantum affine algebras. We shall then introduce and motivate quantum toroidal algebras, before presenting some of the known results. In particular, we shall sketch our proof of a braid group action, and generalise the so-called Miki automorphism to the simply laced case.

Time permitting, we shall discuss future directions and applications including constructing representations of quantum toroidal algebras combinatorially, written in terms of Young columns and Young walls.

In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.

For a planar domain $D \subset \mathbb C$ and an admissible weight function $\mu$ on it, we discuss some aspects of the corresponding weighted Bergman kernel $K_{D, \mu}$. First, we see a precise relation between $K_{D, \mu}$ and the classical Bergman kernel $K_D$ near a smooth boundary point of $D$. Second, the weighted kernel $K_{D, \mu}$ gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \geq 0$. Specific examples that we discuss in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.

In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.

A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Some well-known examples are free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds. I will show in the talk that semidirect products of hyperbolic groups with $\mathbb{Z}$ which are again hyperbolic are cubulated, and give some consequences.

Two prominent examples of our setup are

- mapping tori of fundamental groups of closed hyperbolic surfaces over pseudo-Anosov automorphisms, and
- mapping tori of free groups over atoroidal automorphisms.

Both these classes of groups are known to be cubulated by outstanding works. Our proof uses these two noteworthy results as building blocks and places them in a unified framework. Based on joint work with François Dahmani and Jean Pierre Mutanguha.

I will give a gentle introduction to the combinatorial Rogers–Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots.

In his seminal paper in 2001, Henri Darmon proposed a systematic construction of `$p$`

-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of local (`$p$`

-adic) cohomology classes/cycles in the `$p$`

-adic Galois representation attached to a cuspidal cohomological automorphic representation of `$\mathrm{PGL}_2$`

over any number field, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato conjecture. This work generalises previous constructions of Rotger-Seveso for elliptic cusp forms and earlier joint work with Williams for Bianchi cusp forms. Time permitting, I will also talk about the plectic analogues of these objects.

Branching rules are a systematic way of understanding the multiplicity of irreducible representations in restrictions of representations of Lie groups. In the case of $GL_n$ and orthogonal groups, the branching rules are multiplicity free, but the same is not the case for symplectic groups. The explicit combinatorial description of the multiplicities was given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed that this multiplicity corresponds to the dimension of the multiplicity space, which was a representation of $SL_2$ $(=Sp(2))$. In this talk, we give an alternate proof of the same without invoking any partition function machinery. The only assumption for this talk would be the Weyl character formula.

In a joint work in progress with K. Prasanna, we
study period relations for the base change to `$\mathrm{GL}_4$`

of a
cohomological cuspidal representation on `$\mathrm{GSp}_4$`

.
An unexpected period occurs in the period relations.

In 80’s Goldman introduced a Lie bracket structure on the free homotopy classes of oriented closed curves on an oriented surface known as the Goldman Lie bracket. In this talk, I will give a brief overview of Goldman Lie algebra and discuss two criteria for a homotopy class of a curve to be simple in terms of the Goldman Lie bracket.

In this talk, I will discuss about the structure of ideals in enveloping algebras of affine Kac–Moody algebras and explain a proof of the result which states that if $U(L)$ is the enveloping algebra of the affine Lie algebra $L$ and “$c$” is the central element of $L$, then any proper quotient of $U(L)/(c)$ by two sided ideals has finite Gelfand–Kirillov dimension. I will also talk about the applications of the result including the fact that $U(L)/(c-\lambda)$ for non zero $\lambda$ is simple. This talk is based on joint work with Susan J. Sierra.

Let $S$ be an oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S)$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher rank representations.

For a representation $\rho : \pi_1(S) \rightarrow PSL_n(\mathbb{C})$ where $n >2$, the Hilbert length of a curve $\gamma\in \pi_1(S)$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock–Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$-representations. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.

Let $ X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \ell_{X_n}(A)= \sum_{i=1}^{n}\log (\sigma_i(A))^2, \end{equation} where $\sigma_i(A)$ are the singular values of $A$. We show that the same $j$ dominates $\rho$ in the translation length spectrum as well. Lindström’s Lemma for planar networks is one of the key ingredients of the proof.

In both cases, if $S$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.

Associated to two given sequences of eigenvalues is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as the number of eigenvalues tends to infinity.

Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdös–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. This is joint work with Scott Sheffield and Terence Tao.

Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form

- $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or
- $(g)^{(w)}:=(\dots,\mathbf{e},g,\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w \gt 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$,

such that $\{(u,v)_G,(g)^{(w)} : x_{u,v} \gt 0,\; y_w\alpha_g \gt 0,\;1\leq u \lt v \leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u \lt v\leq n$ and $g^{(w)},g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous’ spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.

A convex function $\Phi$ from $[0,\infty]$ to $[0,\infty]$ with properties
\begin{equation}
\Phi (0)=0,\qquad \lim_{t\to \infty}\Phi (t)=\Phi (\infty )=\infty ,
\end{equation}
is called a *Young function*. For any Young function $\Phi$, the Orlicz space $L^\Phi$ is a Banach space,
and consists of all measurable functions $f$ such that $\Phi (t\cdot |f|)\in L^1$ for some $t>0$. By choosing
$\Phi$ in suitable ways we gain the definition of any (Banach) Lebesgue space $L^p$, as well as sums of such
spaces like $L^p+L^q$, $p,q\in [1,\infty ]$. In particular, the family of Orlicz spaces contain any Lebesgue space.

The Orlicz modulation space $M^{\Phi}$ is obtained by imposing $L^\Phi$ norm conditions of the short-time Fourier transforms of the involved functions and distributions. In the same way we may discuss Orlicz modulation spaces $M^{\Phi ,\Psi}$ of mixed normed types. Again, by choosing the Young functions $\Phi$ and $\Psi$ in suitable ways, $M^{\Phi ,\Psi}$ becomes the classical Feichtinger’s modulation space $M^{p,q}$.

In the talk we explain some basic properties and give some examples on interesting Orlicz spaces and Orlicz modulation spaces. We also explain some classical results on pseudo-differential operators acting on Lebesgue or modulation spaces, and give examples on how such results can be extended to the framework of Orlicz spaces and Orlicz modulation spaces.

As another example we discuss suitable Orlicz modulation spaces and the entropy functional $f\mapsto E_\phi (f)$ with $\phi$ as the coherent state, considered by E. H. Lieb when discussing kinetic energy in quantum systems. Here we find an Orlicz modulation space $M^\Phi$ which satisfies \begin{equation} M^{p_1}\subsetneq M^\Phi \subsetneq M^{p_2},\qquad p_1<\frac 12,\ p_2\ge \frac 12 \end{equation} for which $E_\phi$ is continuous on $M^{p_1}$ and $M^\Phi$, but discontinuous on $M^{p_2}$. We hope that this should shed some light on how to find suitable Banach spaces when dealing with non-linear functionals.

The talk is based on joint works with A. Gumber, E. Nabizadeh Morsalfard, N. Rana, S. Öztop and R. Üster.

Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and fractional Fourier
transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces,
and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on
modulation spaces. Especially we extend some results in our recent works and those of Bhimani, Cordero,
Gröchenig, Manna, Thangavelu, and others. We also show that general forms of fractional harmonic oscillator
propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially
we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are
continuous on any Pilipovic space and corresponding distribution space, which are *not* Gelfand–Shilov spaces.
(The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.)

Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.

In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.

We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.

We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.

Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.

Einstein-Bogomolnyi metrics, which physically models the Cosmic Strings, solve the Einstein’s Fields Equation coupled with an Abelian gauge field and a Higgs field. In this talk, I will present a general existence and uniqueness theorem for Einstein-Bogomolnyi metrics on Riemann sphere. I will also discuss the behaviors of the metrics as the volume approaches the lower bound and infinity respectively, and the moduli space problem. Part of this talk is based on the joint work with Luis-Alvarez, Garcia-Fernandez, Garcia-Prada and Pingali.

In the first part of the talk, we will discuss the main statement of local class field theory that describes the abelian extensions of a non-archimedean local field $F$ in terms of the arithmetic of the field $F$. Then we will discuss the statement of the local Langlands conjectures, a vast generalization of local class field theory, that gives a (conjectural) parametrization of the irreducible complex representations of $G(F)$, where $G$ is a connected, reductive group over $F$, in terms of certain Galois representations. We will then discuss a philosophy of Deligne and Kazhdan that loosely says that to obtain such a parametrization for representations of $G(F’)$, with $F’$ of characteristic $p$, it suffices to obtain such a parametrization for representations of $G(F)$ for all local fields $F$ of characteristic $0$. In the second half of the talk, we will mention some instances where the Deligne-Kazhdan philosophy has been applied successfully to obtain a Langlands parametrization of irreducible representations of $G(F’)$ in characteristic $p$ and focus on some recent work on variants/generalizations of the work of Kazhdan.

The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss a structure of a pair of commuting $C_0$-semigroups of isometries and obtain a BCL type result.

The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
```
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
```

for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a *factorization* of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.

Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.

Non-normal matrices are ubiquitous in various branches of science, such as fluid dynamics, mathematical physics, partial differential equations, and many more. Non-normality causes notorious sensitivity of the eigenvalues, and the eigenvalue analysis often turns out to be misleading. These motivate the study of pseudospectrum, and the spectral properties of random perturbation of non-normal matrices. In the first part of the talk, we will introduce these issues and their resolutions through some fun experiments and simulations. In the latter half, we will move to describe spectral properties of random perturbations of non-normal Toeplitz matrices, where over the last few years a coherent theory has emerged.

We discuss about flat structures on surfaces of finite type $S_{g,n}$, possibly with punctures. For a given representation $\chi\colon \pi_1(S_{g,n})\to \textnormal{Aff}(\mathbb C)$, we wonder if there exists a flat structure having the given representations as the holonomy representation. For closed surfaces $(n=0)$, holonomy representations has been determined by works of Haupt for representations in $\mathbb C$ and subsequently by Ghazouani for a generic representation in $\textnormal{Aff}(\mathbb C)$. It turns out that for surfaces of hyperbolic type, i.e. $2-2g-n<0$, the resulting structures must have special points, called branched points, around which the geometry fails to be modelled on $\mathbb C$. In the present seminar we discuss the case of punctured surfaces and provide conditions under which a representation $\chi$ is a holonomy representation of some flat structure. In this case, being surfaces no longer closed, it is even possible to find flat structures with no branched points. This is a joint work with Subhojoy Gupta and partially with Shabarish Chenakkod.

Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we can define the notions of Nakano positivity, Griffiths positivity and ampleness. All these notions of positivity are equivalent for line bundles. In general, Nakano positivity implies Griffiths positivity and Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. In this talk, I will briefly discuss about the solution of these systems on the vortex bundle using method of continuity.

In this talk we will first see the definitions of ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (hcscK) metric’, both of which are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kahler (cscK) metric respectively. Informally speaking, on a compact K ̈ahler manifold a higher extremal Kahler metric is a Kahler metric whose corresponding top Chern form and volume form differ by a smooth real-valued function whose gradient is a holomorphic vector field, and an hcscK metric is a Kahler metric whose top Chern form and volume form differ by a real constant or equivalently whose top Chern form is harmonic. We will then prove that on a special type of minimal ruled complex surface, which is an example of a ‘pseudo-Hirzebruch surface’, every Kahler class admits a higher extremal Kahler metric which is constructed by using the well-known momentum construction method involving the Calabi ansatz procedure. We will then check that this specific higher extremal Kahler metric yielded by the momentum construction method cannot be an hcscK metric. By doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude that hcscK metrics do not exist in any Kahler class on this Kahler surface. We will then see briefly what changes in the calculations in the momentum construction method when we take a general pseudo- Hirzebruch surface which is basically the projectivization of a certain kind of rank two holomorphic vector bundle over a compact Riemann surface of genus greater than or equal to two. It can be seen that the results about the existence of higher extremal Kahler metrics and the non-existence of hcscK metrics obtained in the special case of our minimal ruled surface can be generalized to all pseudo-Hirzebruch surfaces. If time permits we will see the motivation for studying this problem and its analogy with the related and previously well- studied problem of constructing extremal Kahler metrics on a pseudo-Hirzebruch surface.

Of fundamental importance in number theory is the question of non-vanishing of central L-values of L-functions. One approach, explained in the talk, is to make use of the Relative trace formula (which will be introduced from scratch); a basic example of interest involves twists of L-functions of classical modular forms. If time permits, we will explain the recent work with Michel and Yang on $U(2)$-twists of $U(3)$ L-functions.

We study the spectral gap phenomena for weighted $L^p$-Hardy inequalities on $C^{1,\gamma}$-domain with a compact boundary, where $\gamma\in (0,1]$. We show that the weighted Hardy constant is attained by some appropriate minimizer if and only if the spectral gap (the difference between the weighted Hardy constant and the weighted Hardy constant at infinity ) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. In this talk, we will try to understand how the ideas in criticality theory help us to extend the spectral gap phenomena from $C^2$-domains to $C^{1,\gamma}$-domains. This talk is based on the joint work with Yehuda Pinchover, Baptiste Devyver.

Any compact Riemann surface is shown to have a canonical projective structure (which is different from the canonical one given by the uniformization theorem). Some properties of this projective structure are established. (Joint work with Elisabetta Colombo, Paola Frediani and Gian Pietro Pirola.)

Recent advances in the nonconforming FEM approximation of elliptic PDE eigenvalue problems include the guaranteed lower eigenvalue bounds (GLB) and its adaptive finite element computation. The first part of the talk explains the derivation of GLB for the simplest second-order (and fourth-order) eigenvalue problems with relevant applications, e.g., for the localization of the critical load in the buckling analysis of the Kirchhoff plates. The second part mentions an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace (and bi-Laplace) operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. Numerical experiments in the third part of the presentation shows benchmarks in which the naive adaptive mesh-refining and the post processed GLB do not lead to efficient GLB. The fourth part outlines a new extra-stabilised scheme based on extended Crouzeix-Raviart (resp. Morley) finite elements that directly computes approximations as GLB and that allows optimal convergence rates at the same time.

The presentation is on joint work with Dr. Sophie Puttkammer.

Let `$N$`

be a prime number `$>3$`

. Mazur has defined, from the theory of modular forms, a unit `$u$`

in `$\mathbb{Z}/N$`

. This unit turned out to be, up to a `$6$`

-th root of unity,`$\prod_{k=1}^{(N-1)/2}k^k$`

.
In this talk we will describe how the unit is connected to various objects in number theory. For instance:
–The unit `$u$`

can be understood as a derivative of the zeta function at `$-1$`

, (despite living in a finite field).
– Lecouturier has shown that this unit is the discriminant of the Hasse polynomial: `$\sum_{i=0}^{(N-1)/2}a_i X^i$`

modulo `$N$`

, where `$a_i$`

is the square of the `$i$`

-th binomial coefficient in degree `$N$`

.
– Calegari and Emerton have related `$u$`

to the class group of the quadratic field `$\mathbb{Q}(\sqrt{-N})$`

.
For every prime number `$p$`

dividing `$N-1$`

, It is important to determine when `$u$`

is a `$p$`

-th power in `$(\mathbb{Z}/N)^*$`

.
If time allows, I will describe the connections to modular forms and Galois representations, and the general theory that Lecouturier has developed from this unit. For instance,when `$u$`

is not a `$p$`

-th power, a certain Hecke algebra acting on modular forms is of rank `$1$`

over the ring of `$p$`

-adic integers `$\mathbb{Z}_p$`

(the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.

Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak h$. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak h$. These include fixed point subalgebras of $\mathfrak h$ under subgroups of diagram automorphisms of $\mathfrak g$. This is joint work with K.N. Raghavan, R. Venkatesh and S. Viswanath.

Application of the Hahn-Banach Theorem to the space of bounded sequences with a specific sub linear functional $p$ defined on it gives rise to linear functionals which are dominated by $p$ and are extensions of limits of convergent sequences. These are called Banach Limits and were studied by Banach (1932), and their uniqueness is called almost convergence and was characterised by Lonentz (1948).

In the present lecture we will discuss about the absolute analogue of almost convergence which generalizes lp spaces.

The two concepts of variational inequality and complementarily problems are essentially the same concepts which are studied by two different groups of mathematicians: applied mathematics on one hand and operations researchers on the other hand. The proof existence of variational inequality problem uses Hahn-Banach Theorem or Fixed Point Theorem.

In this lecture we will discuss about the existence of solutions of the complementarily problem, under the most general conditions on the operator and the cone.

In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume.
Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a harmonic function, where $n\geq 2$. One way to quantify the
growth of $u$ in the ball $B(0,1) \subset \mathbb{R}^n$ is via the *doubling index* $N$, defined by
\begin{equation}
\sup_{B(0,1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|.
\end{equation}
I will present a result, obtained jointly with A. Logunov and A. Sartori, where we prove an
almost sharp result, namely:
\begin{equation}
\mathcal{H}^{n-1}({u=0} \cap B(0,2)) \gtrsim_{n,\varepsilon} N^{1-\varepsilon},
\end{equation}
where $\mathcal{H}^{n-1}$ denotes the $(n-1)$ dimensional Hausdorff measure.

The Kubota-Leopoldt $p$-adic $L$-function is an important concept in number theory. It takes special values in terms of generalized Bernoulli numbers, and helps solve Kummer congruences. It is also used in Iwasawa theory. Formalization of $p$-adic $L$-functions has been done for the first time in a theorem prover called Lean 3. In this talk, we shall briefly introduce the concept of formalization of mathematics, the theory behind $p$-adic $L$-functions, and its formalization.

Let `$p$`

be an odd prime, `$f$`

be a `$p$`

-ordinary newform of weight `$k$`

and `$h$`

be a normalized cuspidal `$p$`

-ordinary Hecke eigenform of weight `$\ell < k$`

. Let `$p$`

be an Eisenstein prime for `$h$`

i.e. the residual Galois representation of `$h$`

at `$p$`

is reducible. In this talk, we show that the `$p$`

-adic `$L$`

-function and the characteristic ideal of the `$p^{\infty}$`

-Selmer group of the Rankin-Selberg convolution of `$f$`

, `$h$`

generate the same ideal modulo `$p$`

in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for `$f \otimes h$`

holds modulo `$p$`

. This is a joint work with Somnath Jha and Sudhanshu Shekhar.

Given a closed, oriented surface with genus greater that 2, we study quasi-Fuchsian hyperbolic 3-manifolds homeomorphic to this surface times the interval. Different properties of these manifolds have been carefully studied in previous important works on 3 manifold geometry and topology and some interesting questions about them still remain to be answered. In this talk, we will focus on a new geometric invariant associated to them which we call the measured foliations at infinity. These are horizontal measured foliations of a holomorphic quadratic differential ( the Schwarzian derivative ) associated canonically with each of the two connected component of the boundary at infinity of a quasi-Fuchsian manifold. We ask whether given any pair of measured foliations (F,G) on a surface, is there a quasi-Fuchsian manifold with F and G as it measured foliations at infinity. The answer is affirmative under certain assumptions; first, (F,G) satisfy the property of being an “arational filling pair” and second, the quasi-Fuchsian manifold should be very close to being “Fuchsian” . The goal of this talk would be introducing the concepts and outlining the proof idea.

The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.

By the resolution of the Poincare conjecture in 3D, we know that the only closed three-manifold with the trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation $\pi_1(M)\rightarrow SU(2)$ is the trivial one. Does this imply that $\pi_1(M)$ is trivial? The class of manifolds $M$ for which this question is interesting (and open) are integer homology spheres. We prove a result in this direction: the half-Dehn surgery on any non-trivial fibered knot $K$ in $S^3$ admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the strategy. This is based on work in progress, some jointly with Zhenkun Li and Fan Ye.

Associated to every reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, obtained by killing all “sufficiently long” braid words, as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal.

Motivated by the classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture, and continuing beyond the previous work of Khare, we attempt to obtain a classification of the finite-dimensional nil-Hecke algebras for all reflection groups $W$. These include the usual nil-Coxeter algebras for $W$ of finite type, their “fully commutative” analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). We further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\overline{12}$-avoiding signed permutations), and classify the Frobenius nil-Hecke algebras in the aforementioned cases. (Joint with Apoorva Khare.)

This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.

The focus of the first part is when the domain is the complex plane. In this case, given a finite cyclic configuration
of points $P \subset \partial\mathbb{H}^3=\mathbb{CP}^1$, we construct a harmonic map from $\mathbb{C}$ to
$\mathbb{H}^3$ that is asymptotic to a twisted ideal polygon with ideal vertices contained in $P$. Moreover, we prove
that given any ideal twisted polygon in $\mathbb{H}^3$ with *rotational symmetry*, there exists a harmonic map
from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and
Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.

In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an
irreducible representation $\rho$ of its fundamental group into $PSL_2(\mathbb{C})$, a seminal theorem of Donaldson
asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$.
In this thesis, we consider domain surfaces that are non-compact, namely *marked and bordered surfaces* (introduced
in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of
marked points that are either punctures or marked points on boundary components. Our main result in this part is: given
an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed
representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (PSL_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is
the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from
$\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$.

In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.

Motivated by optimization considerations and the (matrix theory) inequalities of Ky Fan and von Neumann,
we introduce a Fan-Theobald-von Neumann system as a triple $(V,W,\lambda)$, where $V$ and $W$ are real
inner product spaces and $\lambda:V\rightarrow W$ is a (nonlinear) map satisfying the following condition:
For all $c,u\in V$,

```
$$\max \{\langle c,x\rangle: x\in [u] \}=\langle \lambda(c),\lambda(u)\rangle,$$
where $[u]:= \{x:\lambda(x)=\lambda(u)\}$.
```

This simple formulation happens to be equivalent to the Fenchel conjugate formula of the form $(\phi\circ \lambda)^*=\phi^*\circ \lambda$ and a subdifferential formula in some settings and becomes useful in addressing linear/distance optimization problems over “spectral sets” which are of the form $\lambda^{-1}(Q)$, where $Q$ is a subset of $W$. Three standard examples of FTvN systems are: $(\mathbb{R}^n,\mathbb{R}^n,\lambda)$ with $\lambda(x):=x^\downarrow$ (the decreasing rearrangement of the vector $x\in \mathbb{R}^n$); $({\cal H}^n,\mathbb{R}^n,\lambda)$, where ${\cal H}^n$ is the space of $n$ by $n$ complex Hermitian matrices with $\lambda$ denoting the eigenvalue map; and $(M_n,\mathbb{R}^n, \lambda)$, where $M_n$ is the space of $n$ by $n$ complex matrices with $\lambda$ denoting the singular value map. Other examples come from Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). In the general framework of Fan-Theobald-von Neumann systems, we introduce and elaborate on the concepts of commutativity, automorphisms, majorization, etc. We will also talk about “transfer principles” where properties (such as convexity) of $Q$ are carried over to $\lambda^{-1}(Q)$, leading to a generalization of a celebrated convexity theorem of Chandler Davis.

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.

I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino.

This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.

Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.

We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.

The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.

For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).

This thesis consists of two parts. In the first part, we introduce coupled K¨ahler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Amp`ere vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the K¨ahlerness of the symplectic form which arises in the moment map interpretation of Calabi-Yang-Mills equations.

A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.

In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.

In the second part of the talk, we shall discuss the uniqueness of the
solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb
G$. The uniqueness set is the largest set in $\mathbb G$ where all the
solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable
Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction
of an algebraic variety, which coincides with the uniqueness set in
$\mathbb G$. The algebraic variety is called the *uniqueness
variety*. We shall see if an $N$-point solvable Nevanlinna–Pick problem
is such that it has no solutions of supremum norm less than one and that
each of the $(N-1)$-point subproblems has a solution of supremum norm
less than one, then the uniqueness variety corresponding to the $N$-point
problem contains a distinguished variety containing all the initial
nodes, this is called the *Sandwich Theorem*. Finally, we shall see
the converse of the Sandwich Theorem.

The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss the structures of the pairs of commuting $C_0$-semigroups of isometries in generality as well as under certain additional assumptions like double commutativity or dual double commutativity.

The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
```
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
```

for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a *factorization* of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.

Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.

This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:

We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.

Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.

We then talk about multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.

Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.

The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over $\mathbb{Q}$. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group.

Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree $d$?” To obtain more refined asymptotics, one may fix the Galois structure of the number fields in question. There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods to prove many new instances of these conjectures, including some joint results with Altug, Bhargava, Ho, Shankar, and Wilson.

I will discuss how the inverse problem of recovering conductances in an electrical network from its response matrix can be solved using an automorphism of the positive Grassmannian called the twist.

The origins of disordered actomyosin network contraction such as in the cellular cortex remain an active topic of research. We derive an agent-based mathematical model for the evolution of two-dimensional networks. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use simulations of disordered networks and find that both protein friction and actin filament bending are sufficient for contraction.

Asymptotic analysis of a special case of this model implies that bending induces a geometric asymmetry that enables motors to move faster close to filament plus-ends, inhibiting expansion.

We also explore a minimal model for pattern formation through biased turnover of actin filaments. The resulting discrete-time interacting particle system can be interpreted as voter model with continuous opinion space. We fully characterise the asymptotic shape of solutions which are characterised by transient clusters.

In this talk, we first introduce the basic structure of tumors and consequently present some fundamental modelling aspects of tumor growth based on ODE / PDE models. We then introduce the biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. We introduce what is called in-vivo and in-vitro tumors considering an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are saturated by a physiological extracellular fluid. The mass and momentum equations for both the phases are coupled due to the interaction term. Well-posedness results will be discussed in brief. The criterion for necrosis will be shown in terms of the nutrient transport.

Prime numbers have been studied by Humankind for centuries and have applications in Internet Cryptography. We will outline this connection and also talk about how prime numbers give rise to different number systems.

A “buckyball” or “fullerene” is a trivalent graph embedded in the sphere, all of whose rings have length 5 or 6. The term originates from the most famous buckyball, “Buckminsterfullerene,” a molecule composed of 60 carbon atoms. In this talk, I will explain why there are exactly 1203397779055806181762759 buckyballs with 10000 carbon atoms.

Nonlocal conservation laws are gaining interest due to their wide range of applications in modeling real world phenomena such as crowd dynamics and traffic flow. In this talk, the well-posedness of the initial value problems for certain class of nonlocal conservation laws, scalar as well as system, will be discussed and monotone finite volume approximations for such PDEs will be proposed. Strong compactness of the proposed numerical schemes will be presented and their convergence to the entropy solution will be proven. Some numerical results illustrating the established theory will also be presented.

If we had two extra thumbs, how would we check if “2024” is divisible by eleven? Or by “11”? We will see a simple test in any base $B$, i.e. usable by species having any number of fingers (whether shaped like hot-dogs or not); and for any divisor $d$. That is, the test works for everything ($d$), everywhere ($B$), all at once.

We will then move to recurring decimals. Note that 1/3 = 0.3333… and 1/3x3 = 0.1111… have the same number of digits - one - in their recurring parts. (Is 3 the only prime with this property in base 10?) More generally, we will see how many digits $1/d$ has in its recurring “decimal” expansion, for us or for any species as above.

Finally, for a species with a given number of fingers (= digits!), are there infinitely many primes $p$ for which the recurring part of $1/p$ has $p-1$ digits? (E.g. for us, 1/7 has the decimal recurring string (142857).) And what does this have to do with Gauss, Fermat, and one of the Bernoullis? Or with Artin and a decimal number starting with 0.3739558136… ? I will end by mentioning why this infinitude of primes holds for at least one species among humans (10), emus (6), ichthyostega (14), and computers (2) - but, we don’t know which one!

We classify similarity classes of tetrahedra whose dihedral angles are all rational multiples of $\pi$ (when measured in radians), answering a question of Conway-Jones from 1976. In the process, we also classify collections of vectors in $\mathbb{R}^3$ whose pairwise angles are rational. The proof uses a mixture of theoretical arguments, exact computations in computer algebra, and floating-point numerical computations. (Joint with Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein.)

We will define one of the most famous functions in all of mathematics, the *Riemann zeta function*,
whose properties are the subject of one of the *Millenium Problems*. We will also look at some of
its analogues for other objects.

We introduce an important family of polynomials, the *cyclotomic polynomials*,
whose roots are the roots of unity of a fixed order. We explore the structure
of these polynomials and the number fields that they generate, including a
brief look at Gauss sums.

This talk will be a lucid introduction to the formal mathematics behind Euclidean Constructions, which we all learn in our middle school curriculum. The rules, regulations and restrictions of this type of construction will be discussed in detail. An alternative will also be suggested. We shall also find out how a completely geometric question can be answered using purely algebraic techniques giving rise to an elegant theory introduced in the nineteenth century by a famous French mathematician named Évariste Galois.

We shall discuss Reeb’s Theorem and basic differential topology of Morse functions. This was used by Milnor to prove the existence of exotic spheres in 7 dimensions. We shall propose a generalization of Reeb’s Theorem and discuss a proof of it. This is joint work with Sachchidanand Prasad.

The problem of algorithmically computing the volumes of convex bodies is a well studied problem in combinatorics and theoretical computer science. The best known results are perhaps those concerning the use of Markov Chain Monte Carlo techniques for approximately computing the volumes of general convex bodies. There are also results of a different kind: Deterministic (approximate) computation of the volumes of (certain)polytopes. In this direction, Alexander Barvinok and John Hartigan gave an algorithm based upon the Maximum Entropy heuristic from Statistical Physics that provides good approximations for certain classes of polytopes, that includes the transportation polytopes.

The Maximum Entropy heuristic, originally introduced by Jaynes in 1957 says the following: Suppose one is faced with an unknown probability distribution over a product space. Further suppose we know the expectations of a certain number of random variables with respect to this measure. Then the Maximum Entropy heuristic says that it ‘is natural’ to work with that probability distribution that has max entropy subject to the given linear constraints. Barvinok and Hartigan’s work uses this idea and combines it with some fundamental results about the computability of entropies of these max entropy distributions.

In this talk, I will show how to adapt this approach to Spectrahedra, which are a naturally occurring class of convex sets, defined as slices of the cone of Positive Semidefinite matrices. The case of spectrahedra shows up several surprises. As a byproduct of this work it will follow that central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. I will then give several examples to illustrate the utility of this method.

This is joint work with Jonathan Leake (U Waterloo) and Mahmut Levent Dogan (T U Berlin).

This talk comprises two parts. In the first part, we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few “labeled points”, our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian often perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data. We introduce a class of graph-based higher order fractional Sobolev spaces (H^s) and establish their consistency in the large data limit, along with applications to the SSL problem. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of a weighted Laplace-Beltrami operator in the continuum.

Obtaining optimal convergence rates for such spectra has so-far been an open question in stochastic homogenization. In the rest of the talk, we answer this question by obtaining optimal, state-of-the-art results for the case of a Poisson point cloud on a bounded domain in Euclidean space with Dirichlet or Neumann boundary conditions.

The first half is joint work with Dejan Slepcev (CMU), and the second half is joint work with Scott Armstrong (Courant).

This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems. The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and L1 data in a circular oscillating domain. As the source term we are considering is in L1, we will examine the renormalized solutions. The second problem we will investigate is an optimal control problem governed by a second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the oscillations occur in $m$ directions, with $1< m < n$. We will explore the asymptotic behavior of this problem by homogenizing the corresponding optimality systems.

The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.

The (tame) class field theory for a smooth variety `$X$`

is the
study of describing the abelianized (tame) {'e}tale fundamental group of
`$X$`

in terms of some groups which are defined using algebraic cycles of `$X$`

.
In this talk, we study the tame class field theory for smooth varieties
over local fields. We will begin with defining few notions and recalling
various results from the past to overview the historical background of the
subject. We will then study abelianized tame fundamental group denoted as
`$\pi^{ab,t}_{1}(X)$`

, with the help of reciprocity map ```
$\rho^{t}_{X} :
C^{t}(X) \rightarrow \pi^{ab,t}_{1}(X)$
```

and will describe the kernel and
topological cokernel of this map. This talk is based on a joint work with
Prof. Amalendu Krishna and Dr. Rahul Gupta.

Minimal Lagrangian tori in CP^{2} are the expected local model for particular point singularities of Calabi-Yau 3-folds and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods we construct new examples of minimal Lagrangian surfaces of genus 1/2(k-1)(k-2) for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit.

This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following.

(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate.

(3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.

Nanomedicine is an offshoot of nanotechnology that involves many disciplines, including the manipulation and manufacturing of materials, imaging, diagnosis, monitoring, and treatment. An efficient iterative reconstruction algorithm,together with Total Variation (TV), and a good mathematical model, can be used to enhance the spatial resolution and predictive capabilities. In this webinar, I will start with our current results using integrated approach for predicting efficient biomarkers for Acute respiratory distress syndrome (ARDS) and then move to PDE based (Total variation flow) approach for Image denoising which can have promising applications in denoising medical images from different modalities. In principle, I will be discussing the below-mentioned topics and their important concepts in dealing with the main markers of cardiovascular diseases, specifically Pulmonary Hypertension.

**1. 4D FlowMRI Data Assimilation: Integrated approach reveals new biomarkers for Experimental ARDS conditions.**
The purpose of this study is to characterize flow patterns and several other hemodynamic parameters (WSS, OSI, Helicity) using
computational fluid dynamics model by combining imaging data from 4D-Flow MRI with hemodynamic pressure and flow waveforms from
control and hypertensive subjects (related to acute respiratory distress syndrome).
**This work mainly concerns how to facilitate bench-bedside approach using integrated approach by combining CFD and AI.**

**2. An adaptive $C^0$ interior penalty discontinuous galerkin approximation of second order total variation problems.**
Singular nonlinear fourth order boundary value problems have significant applications in image processing and material science.
We consider an adaptive $C^0$ Interior Penalty Discontinuous Galerkin (C0IPDG) method for the numerical solution of singular
nonlinear fourth order boundary value problems arising from the minimization of functionals involving the second order total
variation. The mesh adaptivity will be based on an aposteriori error estimator that can be derived by duality arguments. The
fourth order elliptic equation reads as follows:
```
\begin{align}
u + \lambda \nabla \cdot \nabla \cdot \frac{D^2 u}{|D^2 w|} = & \ 0 \quad \mbox{in} \ Q := \Omega, \\
u = & \ 0 \quad \mbox{on} \ \Gamma,\\
n_{\Gamma} \cdot\frac{D^2 u} {n_{\Gamma}} = & \ 0 \quad \mbox{on} \ {\Gamma}.
\end{align}
```

ChatGPT and other advances in Artificial Intelligence have become popular sensations. In parallel with this has been an enormous advance in the digitization of mathematics through Interactive Theorem Provers and their libraries. Artificial Intelligence has started entering mathematics through these and other routes.

This session will have some presentations/demos about present use of Computer Proofs, Artificial Intelligence together and separately in Mathematics and related fields (including software), both in research and in teaching. After that everyone is welcome to discuss their work, ideas, wish-lists etc related to these themes.

I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.

For decades, mathematicians have been using computers to calculate. More recently there has been some interest in trying to get them to reason. What is the difference? An example of a calculation: compute the first one million prime numbers. An example of reasoning: prove that there are infinitely many prime numbers. Tools like ChatGPT can prove things like this, because they have seen many proofs of it on the internet. But can computers help researchers to come up with new mathematics? Hoping that a computer will automatically prove the Riemann Hypothesis is still science fiction. But new tools and methods are becoming available. I will give an overview of the state of the art.

(This is a Plenary talk in the EECS Research Students’ Symposium)

A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.

In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.

In the second part of the talk, we shall discuss the uniqueness of the
solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb
G$. The uniqueness set is the largest set in $\mathbb G$ where all the
solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable
Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction
of an algebraic variety, which coincides with the uniqueness set in
$\mathbb G$. The algebraic variety is called the *uniqueness
variety*. We shall see if an $N$-point solvable Nevanlinna–Pick problem
is such that it has no solutions of supremum norm less than one and that
each of the $(N-1)$-point subproblems has a solution of supremum norm
less than one, then the uniqueness variety corresponding to the $N$-point
problem contains a distinguished variety containing all the initial
nodes, this is called the *Sandwich Theorem*. Finally, we shall see
the converse of the Sandwich Theorem.

Let `$K$`

be an imaginary quadratic field of class number `$1$`

such that both `$p$`

and `$q$`

split in `$K$`

. We show that under appropriate hypotheses, the `$p$`

-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic `$\mathbb{Z}_q$`

-extension of `$K$`

. This is joint work with Antonio Lei.

The Virasoro algebra, which can be realized as a central extension of (complex) polynomial vector fields on the unit circle, plays a key role in the representation theory of affine Lie algebras, as it acts on almost every highest weight module for the affine Lie algebra. This remarkable phenomenon eventually led to constructing the affine-Virasoro algebra, which is a semi-direct product of the affine Lie algebra and the Virasoro algebra with a common extension. The representation theory of the affine-Virasoro algebra has been studied extensively and is an extremely well-developed classical object.

In this talk, we shall consider a natural higher-dimensional analogue of the affine-Virasoro algebra, popularly known as the full toroidal Lie algebra in the literature and henceforth classify the irreducible Harish-Chandra modules over this Lie algebra. As a by-product, we also obtain the classification of all possible irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, thereby proving Eswara Rao’s conjecture (conjectured in 2004). These directly generalize the well-known result of O. Mathieu for the classical Virasoro algebra and also the recent work of Billig–Futorny for the higher rank Witt algebra.

Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition, for instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure.

Consider a finite group $G$ and a prime number $p$ dividing the order of $G$. A $p$-regular element of $G$ is an element whose order is coprime to $p$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. The quasi $p$-Steinberg character is a generalization of the well-known $p$-Steinberg character. A group, which does not have a non-linear quasi $p$-Steinberg character, can not be a finite group of Lie type of characteristic $p$. Therefore, it is natural to ask for the classification of all non-linear quasi $p$-Steinberg characters of any finite group $G$. In this joint work with Digjoy Paul and Pooja Singla, we classify quasi $p$-Steinberg characters of all finite complex reflection groups.

We report on new ideas of Ki-Seng Tan and myself towards the construction of a `$p$`

-adic `$L$`

-function associated to an automorphic overconvergent `$F$`

-isocrystal over a curve over a finite field. This function should be of interest in the Iwasawa theory for such coefficients.

Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Mike Wolf.

Let `$F$`

be a totally real field. Let `$\pi$`

be a cuspidal cohomological automorphic representation for `$\mathrm{GL}_2/F$`

. Let `$L(s, \mathrm{Ad}^0, \pi)$`

denote the adjoint `$L$`

-function associated to `$\pi$`

. The special values of this `$L$`

-function and its relation to congruence primes have been studied by Hida, Ghate and Dimitrov. Let `$p$`

be an integer prime. In this talk, I will discuss the construction of a `$p$`

-adic adjoint `$L$`

-function in neighbourhoods of very decent points of the Hilbert eigenvariety. As a consequence, we relate the ramification locus of this eigenvariety to the zero set of the `$p$`

-adic `$L$`

-functions. This was first established by Kim when `$F=\mathbb{Q}$`

. We follow Bellaiche’s description of Kim’s method, generalizing it to arbitrary totally real number fields. This is joint work with John Bergdall and Matteo Longo.

From the longest increasing subsequence in a random permutation to the shortest distance in a randomly weighted two dimensional Euclidean lattice, a large class of planar random growth models are believed to exhibit shared large scale features of the so-called Kardar-Parisi-Zhang (KPZ) universality class. Over the last 25 years, intense mathematical activity has led to a lot of progress in the understanding of these models, and connections to several other topics such as algebraic combinatorics, random matrices and partial differential equations have been unearthed. I shall try to give an elementary introduction to this area, describe some of what is known as well as many questions that remain open.

This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:

We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.

Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.

We then talk about the multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.

Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.

The Siegel-Veech transform is a basic tool in homogeneous as well as Teichmuller dynamics. I will introduce the transform and explain how it can be used in counting problems.

It is well known that solvability of the complex Monge- Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s celebrated resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others (including some at IISc). A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. Results of this nature are ubiquitous in Kaehler geometry - existence of weak Kaehler-Einstein metrics on normal varieties and Hermitian-Einstein metrics on reflexive sheaves to name a couple. Much closer to the present theme, is the work of Boucksom-Eyssidieux-Guedj-Zeriahi on solving the complex Monge-Ampere equation in big classes. In the talk, I will first speak about some joint and ongoing work with Ramesh Mete and Jian Song, that offers a reasonably complete resolution in complex dimension two, at least for the J-equation and the dHYM equations. Next, I will discuss some conjectures on what one can expect in higher dimensions.

Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.

This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.

Second, we discuss a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.

Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.

Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.

This thesis consists of two parts. In the first part, we introduce coupled Kähler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Ampère vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the Kählerness of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations.

This will be an introductory talk on some matters relating to Fatou-Bieberbach domains and uniformizing stable manifolds.

In the late 1950s, an important problem in number theory was to extend the notion of `$L$`

-functions attached to cuspforms on the upper-half
plane to automorphic forms on reductive groups. Langlands’s work on non-abelian Harmonic analysis, namely the problem of the spectral decomposition of automorphic forms, led him to a general notion of `$L$`

-functions
attached to cuspforms. We give an introduction to the spectral decomposition of automorphic forms and discuss some contemporary problems.

Over an unramified extension `$F/\mathbb{Q}_p$`

, by the works of Fontaine, Wach, Colmez and Berger, it is well-known that a crystalline representation of the absolute Galois group of `$F$`

is of finite height. Moreover, in this case, to a crystalline representation one can functorially attach a lattice inside the associated etale `$(\varphi, \Gamma)$`

-module called the Wach module. Berger showed that the aforementioned functor induces an equivalence between the category of crystalline representations and Wach modules. Furthermore, this categorical equivalence admits an integral refinement. In this talk, our goal is to generalize the notion of Wach modules to relative `$p$`

-adic Hodge theory. For a “small” unramified base (in the sense of Faltings) and its etale fundamental group, we will generalize the result of Berger to an equivalence between crystalline representations and relative Wach modules as well as establish its integral refinement.

Cauchy’s determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the rank-one matrix $(u_i v_j)$. This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials $p(t)$ with a negative coefficient that entrywise preserve positivity. A key novelty here has been the application of Schur polynomials, which essentially arise from the expansion of $\det(p[{\bf u}{\bf v}^T])$, to positivity.

In the first half of the talk, I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion of $\det(f[{\bf u}{\bf v}^T])$ for all power series $f$. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.) In the second half, joint with Siddhartha Sahi, I will explain how to extend the above determinantal identities to (a) any subgroup $G$ of signed permutations; (b) any character of $G$, or even complex class function; (c) any commutative ground ring $R$; and (d) any power series over $R$.

Andreatta, Iovita, and Pilloni have proven the existence of an adic eigencurve, which includes characteristic `$p$`

points at the boundary. In joint work with Ruochuan Liu, using the theory of Crystalline periods, we show that the Galois representations associated to these points satisfy an appropriate trianguline property.

In 1976 Bernstein, Gelfand, and Gelfand introduced Category $\mathcal{O}$ for a semi-simple Lie algebra $\mathfrak{g}$. This is roughly the smallest sub-category of $\mathfrak{g}$-mod containing the Verma modules and such that the simple modules have projective covers. After work of Beilinson–Bernstein and Beilinson–Ginzburg–Soergel it became clear that the the good homological properties of this category were due to the fact that it can be identified with a category of perverse sheaves on the flag variety $G/B$.

In this talk I will show how this story fits into the physics of 3d mirror symmetry. This leads to conjectural 2-categorifications of category $\mathcal{O}$ that can be computed explicitly for $\mathfrak{g} = \mathfrak{sl}_2$.

The geometry, and the (exposed) faces, of $X$ a “Root polytope” or “Weyl polytope” over a complex simple Lie algebra $\mathfrak{g}$, have been studied for many decades for various applications, including by Satake, Borel–Tits, Casselman, and Vinberg among others. This talk focuses on two recent combinatorial analogues to these classical faces, in the discrete setting of weight-sets $X$.

Chari et al [*Adv. Math.* 2009, *J. Pure Appl. Algebra* 2012]
introduced and studied two combinatorial subsets of $X$ a root system or
the weight-set wt $V$ of an integrable simple highest weight
$\mathfrak{g}$-module $V$, for studying Kirillov–Reshetikhin
modules over the specialization at $q=1$ of quantum affine algebras
$U_q(\hat{\mathfrak{g}})$ and for constructing Koszul algebras. Later,
Khare [*J. Algebra* 2016] studied these subsets under the names
“weak-$\mathbb{A}$-faces” (for subgroups $\mathbb{A}\subseteq
(\mathbb{R},+)$) and “$212$-closed subsets”.
For two subsets $Y\subseteq X$ in a vector space, $Y$ is said to be
$212$-closed in $X$, if $y_1+y_2=x_2+x_2$ for $y_i\in Y$ and $x_i\in X$
implies $x_1,x_2\in Y$.

In finite type, Chari et al classified these discrete faces for $X$ root
systems and wt $V$ for all integrable $V$, and Khare for all
(non-integrable) simple $V$. In the talk, we extend and completely solve
this problem for *all* highest weight modules $V$ over *any*
Kac–Moody Lie algebra $\mathfrak{g}$.
We classify, and show the equality of, the weak faces and
$212$-closed subsets in the three prominent settings of $X$:
(a) wt $V$ $\forall V$,
(b) the hull of wt $V$ $\forall V$,
(c) wt $\mathfrak{g}$ (consisting of roots and 0).
Moreover, in the case of (a) (resp. of (b)), such subsets are precisely
the weights falling on the exposed faces (resp. the exposed faces) of the
hulls of wt $V$.

While statistical decision theory led me to game theory, certain war duel models, and the close connection between the Perron–Frobenius theorem and game theory led me to the works of M.G. Krein on special classes of cones, and spectral properties of positive operators. The influence of Professors V.S. Varadarajan, K.R Parthasarathy and S.R.S Varadhan in early 60’s at ISI is too profound to many of us as young graduate students in 1962-66 period. The talk will highlight besides the theorems, the teacher-student interactions of those days.

The study of diluted spin glasses may help solve some problems in computer science and physics. In this talk, I shall introduce the diluted Shcherbina–Tirozzi (ST) model with a quadratic Hamiltonian, for which we computed the free energy at all temperatures and external field strengths. In particular, we showed that the free energy can be expressed in terms of the weak limits of the quenched spin variances and identified these weak limits as the unique fixed points of a recursive distributional operator. The talk is based on a joint work with Wei-Kuo Chen and Arnab Sen.

This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal
control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems.
The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and $L^1$
data in a circular oscillating domain. As the source term we are considering is in $L^1$, we will examine the
renormalized solutions. The second problem we will investigate is an optimal control problem governed by a
second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the
oscillations occur in $m$ directions, with `$1<m<n$`

. We will explore the asymptotic behavior of this problem by
homogenizing the corresponding optimality systems.

In the first half of the talk I will recall two classical theorems - Dirichlet’s class number formula and Stickelberger’s theorem. Stark and Brumer gave conjectural generalisations of these statements. We will see formulations of some of these conjectures. In the second half of the talk we will restrict to a special case of the Brumer-Stark conjecture. Here p-adic techniques can be used to resolve the conjecture. We will see a sketch of this proof. This is joint work with Samit Dasgupta.

Let $\mathfrak g$ be a Borcherds–Kac–Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak n_+ \oplus \mathfrak h \oplus \mathfrak n_{-}$, and each root space $\mathfrak g_{\alpha}$ is either contained in $\mathfrak n_+$ or $\mathfrak n_{-}$. In particular, each $\mathfrak g_{\alpha}$ involves only the relations (2) and (3). In this talk, we will discuss the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially using heaps of pieces and construct two different bases for these root spaces of $\mathfrak g$.

The Thom conjecture, proven by Kronheimer and Mrowka in 1994, states that complex curves in $\mathbb{C}{\rm P}^2$ are genus minimizers in their homology class. We will show that an analogous statement does not hold for complex hypersurfaces in $\mathbb{C}{\rm P}^3$. This is joint work with Ruberman and Strle.

The intersection theory of the Grassmannian, known as Schubert calculus, is an important development in geometry, representation theory and combinatorics. The Quot scheme is a natural generalization of the Grassmannian. In particular, it provides a compactification of the space of morphisms from a smooth projective curve C to the Grassmannian. The intersection theory of the Quot scheme can be used to recover Vafa-Intriligator formulas, which calculate explicit expressions for the count of maps to the Grassmannian subject to incidence conditions with Schubert subvarieties.

The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with symplectic (or symmetric) bilinear form. I will present explicit formulas for certain intersection numbers of the symplectic and the orthogonal analogue of Quot schemes. Furthermore, I will compare these intersection numbers with the Gromov–Ruan–Witten invariants of the corresponding Grassmannians.

Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the `$\ell$`

-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the `$\ell$`

-primary part of principally polarized Abelian `$3$`

-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over `$\mathbb{Q}$`

defined over at least `$10$`

variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).

We shall discuss Legendre Pairs, an interesting combinatorial object related to the Hadamard conjecture. We shall demonstrate the exceptional versatility of Legendre Pairs, as they admit several different formulations via concepts from disparate areas of Mathematics and Computer Science. We shall mention old and new results and conjectures within the past 20+ years, as well as potential future avenues for investigation.

The video of this talk is available on the IISc Math Department channel.

$\mathrm{Per}_n $ is an affine algebraic curve, defined over $\mathbb Q$, parametrizing (up to change of coordinates) degree-2 self-morphisms of $\mathbb P^1$ with an $n$-periodic ramification point. The $n$-th Gleason polynomial $G_n$ is a polynomial in one variable with $\mathbb Z$-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of $\mathbb C$ with an $n$-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is $\mathrm{Per}_n$ connected? (2) Is $G_n$ irreducible over $\mathbb Q$?

We show that if $G_n$ is irreducible over $\mathbb Q$, then $\mathrm{Per}_n$ is irreducible over $\mathbb C$, and is therefore connected. In order to do this, we find a $\mathbb Q$-rational smooth point of a projective completion of $\mathrm{Per}_n$. This $\mathbb Q$-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits an interpretation in terms of tropical geometry.

(This talk will be pitched at a broad audience.)

Given a bipartite graph $G$ (subject to a constraint), the “cross-ratio degree” of G is a non-negative integer invariant of $G$, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings. Finally, time permitting, I may discuss the tropical side of the story.