Consider the following questions.

**Question 1:** Does the volume of a union of balls decrease if their centres are brought pairwise closer?

**Question 2:** Does communication over an additive white Gaussian noise channel worsen if the transmitters are brought pairwise closer?

These questions appeal to our basic intuition about geometry and information transmission, which seems to suggest the answer to both of them is yes. The first question is open; the Kneser–Poulsen conjecture asserts that it has an affirmative answer. In this talk, based on well-known analogies between convex geometry and information theory, we will frame (and prove) the natural entropic formulation of the Kneser–Poulsen conjecture. As a corollary, an affirmative answer to the second question is obtained. This talk is based on joint work with Dongbin Li.

In this talk I will give a brief overview of the theory of maximal directional averages and singular integrals in 2 and higher dimensions. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for maximal directional singular integrals in codimension one and general ambient dimension. This reports on joint work with O. Bakas (U. of Patras), F. Di Plinio (Napoli, Federico II) and L. Roncal (BCAM).

Elementary root separation arguments imply that a non-negative polynomial on the real line is a sum of two squares of polynomials. A non-constructive and ingenious enumerative geometry observation of Hilbert shows that in two or more variables, non-negative polynomials are not always sums of squares. This led to Hilbert’s 17-th problem, asking whether such a decomposition is possible in the field of rational functions. The answer is yes, due to Emil Artin. The gap between polynomials which are positive on semi-algebraic sets and corresponding weighted sums of squares was elucidated by Tarski’s elimination of quantifiers principle. The first part of the lecture will contain accessible details and historical notes on these topics, now part of Real Algebra and Real Algebraic Geometry.

In the second part of the lecture, I will show how sums of squares decompositions led F. Riesz to the definitive form of the spectral theorem for self-adjoint transforms of a Hilbert space. Implying for instance novel positivity results of harmonic analysis. Reversing the historical arrow, I will show how operator theory has put some essential marks on purely Real Algebra chapters. We will also touch positivity in non-commutative *-algebras and Lie-algebras.

The third part of the lecture will contain applications of relatively recent positivity certificates to global, non-convex optimization, stability of dynamical systems and construction of wavelet frames.

We discuss an algebraic version of Schoenberg’s celebrated theorem [*Duke Math. J.*, 1942] characterizing entrywise matrix transforms that preserve positive definiteness.
Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for
matrices of a fixed dimension. When the dimension of the matrices is at least 3, we prove that, surprisingly, the positivity preservers are precisely the positive multiples
of the field’s automorphisms. Our work makes crucial use of the well-known character-sum bound due to Weil, and of a result of Carlitz [*Proc. Amer. Math. Soc.*, 1960] that
leads to characterizing the automorphisms of Paley graphs. This is joint with Dominique Guillot and Himanshu Gupta.

After a review of the Markov–Krein transform in the case of a one variable, and the Hilbert space interpretation of it (the phase shift), we will specialize the Markov–Krein transform to 2D. This will bring us to a relaxation of the Heisenberg commutation relation, this time filled by bounded linear transforms. The spectral invariant of this class of so called hyponormal operators is called the principal function. It is a measurable function of compact support, carrying a degree of shade. We will sketch the main specific results pertaining to hyponormal operators.

For the rest of the lecture we will link the resulting inverse spectral problem to image processing, potential theory, Hele–Shaw flows, integrable systems, and the regularity of free boundaries. Current advances with precise open questions will be detailed.

This will be a broad discussion on a few different types of Fourier uncertainty principles for bandlimited functions (i.e. functions that have Fourier transforms compactly supported), and their connections to multiplication operators in certain Hilbert spaces of entire functions. Some of these problems are related to applications in analysis, PDEs and number theory, and I might describe a few of these if time permits. The talk should be accessible to a broad audience.

There are several analogs of Cauchy’s integral transform in the theory of functions of several complex variables. Fantappie’s transform is one of them, possibly the simplest and oldest, with some remarkable properties we will discuss in detail. First, it is immediately connected to Radon transform, much exploited today in inverse problems. The characterization of Fantappie transforms of positive measures resonates with Bernstein’s theorem linking Laplace transforms and completely monotonic functions. One of the spectacular applications of Fantappie’s transform is the multivariate analog of Koethe–Grothendieck duality of spaces of analytic functions. The multiplicative structure of Fantappie’s transform brings us to the classical Markov–Krein correspondence, much investigated these days by probabilists.

An application to mathematical economics will be sketched.

The focus of this talk will be on computing the point counts
of algebraic varieties, i.e., number of solutions of a system of
polynomial equations over finite fields. The zeta function encodes the
point counts over an infinite tower of finite field extensions and enjoys
the property of being a rational function over `$\mathbb{Q}$`

. Further, the zeta
function can be recovered from certain invariants of the variety in
question, using an appropriate cohomology theory. I will review the state of the art on efficient algorithms to compute the
zeta function of varieties, including the dimension one case of curves
(covering the works of Schoof, and Pila) and report on our generalisations
for the first cohomology (joint work with Diptajit Roy and Nitin
Saxena) and ongoing work on the second cohomology, which addresses a question
of Edixhoven.

In this talk I will discuss Strichartz estimates on the Heisenberg group for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available.

Our approach, inspired by the Fourier transform restriction method initiated by Tomas and Stein, is based on Fourier restriction theorems, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

There are some remarkable bijections that connect the spectrum of classical random matrix ensembles (such as Gaussian and Laguerre unitary ensembles) to certain planar random growth models. I shall review some of these connections and discuss a few results on either side that uses the interplay.

Based on joint works with Jnaneshwar Baslingker, Sudeshna Bhattacharjee and Manjunath Krishnapur.

Contrary to Fourier, Laplace, Cauchy, or Poisson transforms, the power moments of a positive measure, fast decaying at infinity on the real line, do not determine the original measure. The non-uniqueness phenomenon was analyzed in depth by Stieltjes, via continued fraction expansions of the formal generating series of moments. I will sketch the main ideas of Stieltjes celebrated memoir, to continue with an account of a not less foundational contribution put by Carleman in book format. Then we will touch on Marcel Riesz’s pioneering work on extensions of positive linear functionals, to return from another perspective to the Christoffel-Darboux kernel. All in 1D.

Some unfinished parallel studies in $n$D, marred by pitfalls and open problems will be discussed.

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

Teichmuller space admits several ray structures, such as the Teichmuller geodesic ray, Thurston stretch ray, harmonic map (dual) ray, grafting ray, etc. In the first part of this talk, we will depict harmonic map ray structures on Teichmuller space as a geometric transition between Teichmuller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces, the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to a Teichmuller geodesic. In the second part, we will discuss applications to the Thurston metric. This is a joint work with Michael Wolf.

I will focus on two interacting particle systems with inhomogeneous rates on the finite ring with site-dependent rates. The first is the Totally Asymmetric Zero Range Process (TAZRP) and the second is the PushTASEP, where TASEP stands for the Totally Asymmetric Simple Exclusion Process. In both cases, I will present results which say that the distribution of the path of the process restricted to certain intervals is unchanged under permutations of the rates. Our proofs build on Weber’s theorem for exponential queues. I will aim to keep the talk self-contained.

This is joint work with O. Mandelshtam and J. Martin (arXiv:2209.09859, Math. Z., to appear) and with J. Martin (arXiv:2310.09740).

“Explicit” number theory is the name given to the study of what are called zero-free regions of the Riemann zeta function and other L-functions. An explicit determination of such regions often reveals deep arithmetic properties of the underlying object attached to the concerned L-function. More generally, it could refer to the use of “explicit”, often technical methods to understand an arithmetic object. On the other hand, “probabilistic” number theory attempts to investigate arithmetic properties of an object by treating the object as one in a family of many, and exploring these families of varying sizes through the viewpoint of probability. In this talk, we will explore both perspectives and compare the wealth of information each perspective presents to us. In particular, we will address the study of Fourier coefficients of certain modular forms (called the Hecke eigenforms) through both the above viewpoints.

Algebraic Statistics is a relatively new field of research where tools from Algebraic Geometry, Combinatorics and Commutative Algebra are used to solve statistical problems. A key area of research in this field is the Gaussian graphical models, where the dependence structure between jointly normal random variables is determined by a graph. In this talk, I will explain the algebraic perspectives on Gaussian graphical models and present some of my key results on understanding the defining equations of these models. In the end, I will talk about the problem of structural identifiability and causal discovery and how algebraic techniques can be implemented to tackle them.

Given a positive measure in Euclidean space, rapidly decaying at infinity, a point $a$, and a filtration of the polynomial ring by the degree, the optimal bounds for point evaluations at $a$ in Lebesgue space $L^2$ are provided by a reproducing kernel known as the Christoffel–Darboux kernel. For a century and a half, this object continues to intrigue by surprising new turns.

We will touch the asymptotics of orthogonal polynomials in the complex plane, with emphasis on Szego’s Limit Theorem. Then turn to spectral analysis on the line, culminating in Weyl’s circle phenomenon.

Recent applications to dynamical systems (via Koopman’s operator formalism) and the statistics of geometric data will be presented together with some numerical experiments.

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

I will discuss a theory of function spaces defined in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. This includes inhomogeneous and homogeneous Sobolev, Besov and Triebel–Lizorkin spaces on Lie groups and Grushin settings.

In this talk, we will study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In conclusion, we will obtain the Sato-Tate distribution for the trace of certain families of Elliptic curves in arithmetic progressions. As a special case we will recover a result of Birch proving Sato-Tate distribution for certain families of elliptic curves. Moreover, we will see that these results follow from asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. Finally, if time permits, we will discuss the bias conjecture in the finite field setting. These are joint works with Kathrin Bringmann, Ben Kane, and Zichen Yang.

On a Riemannian manifold, consider the Laplace-Beltrami operator $-\Delta$, and the associated extension problem

\begin{equation} \Delta v+\frac{(1-2\sigma)}{t}\frac{\partial v}{\partial t}+\frac{\partial^2 v}{\partial t^2}=0, \quad 0 <\sigma < 1,\quad t>0, \end{equation}

introduced by Caffarelli and Silvestre on Euclidean space to recover the fractional Laplacian $(-\Delta)^{\sigma}$, as $t$ approaches zero.
On hyperbolic spaces this gives rise to a family of convolution operators, including the Poisson operator $e^{-t\sqrt{-\Delta}}$, `$t>0$`

; moreover,
the kernels of these operators are subordinated to the heat kernel.

Motivated by Euclidean results for the Poisson semigroup, but also by results on the heat semigroup on Riemannian manifolds and the influence of underlying geometry, we study the long-time asymptotic behavior of solutions to the extension problem for $L^1$ initial data. If the initial datum is radial, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence can break down in the non-radial case. The results extend to all noncompact symmetric spaces of arbitrary rank.

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

The problem of maximum packing density of the $n$-dimensional real space with spheres is a classic one. Exact values of the density are known only in a few cases ($n=1,2,3,8,24$), and there have been several recent improvements of the bounds for other small dimensions. In a parallel development, researchers have studied the maximum size of packings of the $n$-dimensional Hamming space, known as error-correcting codes. While existence bounds in both cases are found by random choice, the best known impossibility results are obtained by an application of a general method commonly known as Delsarte’s linear programming. The best known upper bound on the maximum size of a code with a given minimum distance for large $n$ was obtained in 1977, and it has proved surprisingly resistant to various improvement attempts, including the semidefinite programming extension of LP.

In the first part of the talk we will introduce the general problem and give an overview of the known results on upper bounds on codes and related problems such as equiangular lines and families of finite sets with restricted intersections. In the second part, we will delve into the details of the proofs for the case of codes and highlight some obstacles for further improvements.

This thesis consists of three parts. Two important complex Hessian equations are studied on certain compact Kahler manifolds from different perspectives. The first one is the J-equation introduced independently by S.K. Donaldson and X.X. Chen from different point of view. The second one is the deformed Hermitian Yang Mills (dHYM) equation which has connection to the mirror symmetry in string theory.

There is a notion of (global) slopes for both equations. It is known that they admit smooth solutions with the global slopes if and only if certain Nakai-Moishezon (NM) type criterion holds. In the first part, our aim is to find some appropriate singular solutions of the equations when the NM-type criterion fails–this is the so-called unstable case. An algebro-geometric characterization of the slopes is formulated – which we call the minimal J-slope for the J-equation and the maximal dHYM-slope for the dHYM equation. There is a natural weak (i.e. singular) version of the equations replacing the standard wedge product with a more generalized product, called the non-pluripolar product. We settle the existence and uniqueness problem for the singular J and dHYM equation on compact Kahler surfaces. More precisely, for the J-equation we show that there exists a unique closed $(1,1)$-Kahler current solving the singular J-equation on a compact K"ahler surface with the minimal J-slope. Analogous result is established for the singular dHYM equation on compact Kahler surfaces with the maximal dHYM-slope. Furthermore, we conjecture analogous existence and uniqueness result for higher dimensions.

In the second part, we study the convergence behaviour of the J-flow, which is the parabolic version of the J-equation, on certain generalized projective bundles using the Calabi Symmetry in the J-unstable case. An invariant version of the minimal J-slope is introduced for these bundles. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the singular J-equation with the invariant minimal J-slope. This result resolves the invariant version of our conjecture for the J-equation on these examples with symmetry.

In the third part, we study the convergence behaviour of a flow, called the cotangent flow, for the dHYM equation in the dHYM-unstable case on the blowup of $\mathbb{C}\mathbb{P}^2$ or $\mathbb{C}\mathbb{P}^3$. Analogous to our results in the second part, we show that the flow converges to some unique limit, and the limiting current solves the singular dHYM equation with the (invariant) maximal dHYM-slope.

The study of invariant dispersive PDE on noncompact symmetric spaces, such as the wave equation or the Schrödinger equation, requires to analyze oscillating integrals arising from the inverse spherical Fourier transform. While this can be achieved by classical though nontrivial tools in rank one, a major problem in higher rank lies in the fact that the Plancherel density is not a differentiable symbol in general, and thus integration by parts produces no additional global decay at infinity. In this talk, we will explain a way to overcome this problem by introducing a smooth barycentric decomposition of Weyl chambers, which leads eventually to the same dispersive and Strichartz estimates as in rank one. This work started 15 years ago as a joint project with S. Meda, V. Pierfelice, M. Vallarino and was finally achieved in collaboration with H.-W. Zhang.

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

In this talk, I will discuss the semilinear hypoelliptic damped wave equation with power-type nonlinearity associated with a Rockland operator on graded Lie groups. Specifically, we will concentrate on the case when the initial data belongs to Sobolev spaces of negative order. We show the global-in-time existence of small data Sobolev solutions of lower regularity for the supercritical range and a finite-time blow-up of weak solutions for the subcritical range. For the particular settings of the Heisenberg group and Euclidean space, we will prove that the critical exponent belongs to the blow-up case. Furthermore, to precisely characterize the blow-up time, we derive sharp upper and lower bound estimates for the lifespan in the subcritical cases.

This talk is based on my joint research with Aparajita Dasgupta (IIT Delhi), Shyam Swarup Mondal (ISI Kolkata), Michael Ruzhansky (Ghent University), and Berikbol Torebek (Ghent University).

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

In this talk, we consider *p* and *h-p* least-squares spectral element methods for elliptic boundary layer problems in one dimension. We derive stability
estimates and design a numerical scheme based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter
robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter for the *p* and *hp*-version. Numerical results
are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results.

A couple of years ago Ian Agol proved a long-standing conjecture showing that a relation called “Ribbon concordance” on knots is a partial order. The proof was a six page paper involving a blend of topology, combinatorial group theory and real algebraic geometry, with ribbon concordance implying a relation between representation varieties. In this talk I describe Agol’s proof and some background results.

We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichm"uller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.

Let $X$ be a Banach space. Let $C$ be a subset of $X$. Let $x^*$ be a functional on $X$. Then $S(C, x^*, \alpha) := \{ x \in C : x^*(x) > \sup x^*(C) - \alpha \}$, $\alpha > 0$, is called the open slice of $C$ determined by $x^*$ and $\alpha$. $X$ has Radon Nikodym Property if and only all closed bounded convex sets admit slices of arbitrarily small diameter i.e. these sets are dentable. The geometry of Banach space is an area of research which characterizes the topological and measure theoretic concepts in Banach spaces in terms of geometric structure of the space. The related concepts were initiated developed and extensively studied in the context of Radon Nikodym Property and Krein Milman Property by Ghoussoub, Godefroy, Maurey, and Scachermayer [Memoirs AMS 1987]. In this work, we look at Banach spaces where the unit ball admits slices of arbitrarily small diameter. We look at some related properties as well. We prove that all these properties are stable under $l_p$ sum for $1 \leq p \leq ∞$, sum and Lebesgue Bochner spaces. We show that these are three space properties under certain conditions on the quotient space. We also study these properties in ideals of Banach spaces. This is based on two papers jointly written with my graduate student, Susmita Seal in [J. Math. Anal. Appl. 2022] and [J. Convex Anal. 2023]. The only prerequisite for this talk is the statement of the Hahn Banach Theorem.

In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ
scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at
low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some
fields and non-Brownian behavior for others. In other words, the external field can *induce* the phase transition in the directed polymer model.

We study the ruin probability of a gambler in a scheme where the bet size is doubled after every win in a round and halved after every loss; we show some paradoxical results such as that the ruin probability is one if and only if the probability of winning in any round is at least half. We also verify some conjectures about the behaviour of the ruin probability as a function of the initial fortune and the probability of winning.

What happens to an $L_p$ function when one truncates its Fourier transform to a domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s celebrated theorem for the ball (1971) imposes that, to preserve $L_p$-integrability, the boundary of such domain must be flat. What if we truncate on a curved space like a Lie group? And if we truncate the entries of a given matrix? What happens with the singular numbers of it or with its Schatten $p$-norm? We fully characterize the local geometry of such $L_p$-preserving truncations for these (apparently unrelated) problems, in terms of a surprisingly lax notion of boundary flatness. The matrix ones are all diffeomorphic variations of a fundamental example: the triangular projection. The Lie group ones are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms that we call affine and projective. This vastly generalizes Fefferman’s theorem to nontrigonometric and noncommutative scenarios. It confirms the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers – even in the lack of a Fourier transform connection – and complete, for Lie groups, a longstanding search of Fourier $L_p$-idempotents. Joint work with M. de la Salle and E. Tablate.

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

In this thesis, we study some important classification problems related to affine Kac–Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems (which are roots of affine Kac–Moody Lie algebras). Secondly, we understand the correspondence between these symmetric real-closed subsets and the regular subalgebras generated by them, using the aforementioned classification. Motivation for this work comes from the celebrated work of Dynkin (1952), where he classified the semi-simple subalgebras of a given finite-dimensional semisimple Lie algebra 𝔤. He introduced the notion of regular subalgebras in order to achieve this classification. It is not hard to see that the regular subalgebras of 𝔤 correspond to symmetric closed subsets of roots of 𝔤, so the problem of classifying regular subalgebras comes down to the combinatorial problem of classifying these subsets. The analogous problem of studying regular subalgebras of affine Kac–Moody Lie algebras was initiated by Anna Felikson et al. in 2008 and continued by Roy–Venkatesh in 2019, where they addressed the part of the combinatorial problem, namely provided the classification of maximal real closed subroot systems of affine root systems.

In the finite case, it is well known that symmetric closed subsets are in fact closed subroot systems. It is not true in general for affine root systems. So, it is natural to ask the following questions:

- When a given symmetric real-closed subset of affine root system is a closed subroot system?
- Is it possible to classify all symmetric real-closed subsets of affine root systems?

We give affirmative answers to these questions in this thesis. In the untwisted setting, we prove that any symmetric real-closed subset is indeed a closed subroot system. Twisted types need more careful analysis since the finite part of a symmetric real-closed subset has two possibilities in these types, namely closed and semi-closed. For semi-closed cases, the behavior of symmetric real-closed subsets varies for each type. We prove that there are three types of irreducible symmetric real-closed subsets for reduced real affine root systems, one of which did not appear in Roy–Venkatesh’s work. We conclude our classification for the twisted case by determining explicitly when a symmetric closed subset is a closed subroot system, including the case when the ambient Lie algebra is the non-reduced affine Lie algebra $A_{2n}^{(2)}$.

In the second part of the thesis, we explore the correspondence between symmetric real-closed subsets and regular subalgebras generated by them. Roy-Venkatesh proved that the map between closed subroot systems and the regular subalgebras generated by them is injective. We observe that it is not true in general when we extend this map to symmetric real-closed subsets. Let ψ be a real-closed subroot system. We describe the types of symmetric closed subsets that can appear in the fiber of the subalgebra generated by ψ. Moreover, we determine when these fibers are finite. In certain cases, we are able to describe very explicitly the defining parameters of the symmetric closed subsets appearing in the fiber.

I will briefly recall Milnor cycle modules over a field as defined by Rost (1996) and their significance and properties. Recently, ‘modules’ over Milnor–Witt K-theory or alternatively Milnor–Witt cycle modules over fields have been formalized by N. Feld (2020).

I will talk about recent joint work with Chetan Balwe and Amit Hogadi, where we considered the Milnor–Witt cycle modules over excellent DVR and studied a subclass of these that satisfy certain lifting conditions on residue maps associated with horizontal valuations. As an important example, Milnor–Witt K-theory of fields belongs to this subclass. Moreover, this condition is sufficient to deduce the local acyclicity property and $A^1$-homotopy invariance of the associated Gersten complex.

According to a well-known result in geometric topology, we have $(S^2)^n/Sym(n) = \mathbb{CP}^n$, where $Sym(n)$ acts on $(S^2)^n$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^n$ for each $n > 1$. In more detail, we take the standard two triangle crystallisation $S^2_3$ of the 2-sphere $S^2$, in its $n$-fold Cartesian product. We then simplicially subdivide, and prove that naively taking the $Sym(n)$ quotient yields a simplicial cell decomposition of $\mathbb{CP}^n$. Taking the first derived subdivision of this cell complex produces a triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n > 3$. This is a joint work with Jonathan Spreer, University of Sydney.

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.

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.

In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.

Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.

Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.

In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)

\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}

contains a unique maximal element.

In this talk, we give a new combinatorial proof of classical edge universality of Wigner matrices without assuming the entries of the matrices are symmetrically distributed around 0. We complete this proof by giving a new encoding of the Wigner words and a counting strategy which works for traces of very high powers of the matrix. In this talk, we shall introduce the encoding, describe the class of words which capture the randomness and finally give some insight about the proof for general non vanishing odd moments.

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.

Applied benchmark tests for the famous ‘subgraph isomorphism problem’ empirically discovered interesting phase transitions in random graphs. This motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities $p_1,p_2$. In particular, (i) we prove a sharp threshold result for the appearance of $G_{n,p_1}$ as an induced subgraph of $G_{N,p_2}$, (ii) we show two-point concentration of the size of the maximum common induced subgraph of $G_{N,p_1}$ and $G_{N,p_2}$, and (iii) we show that the number of induced copies of $G_{n,p_1}$ in $G_{N,p_2}$ has an unusual ‘squashed lognormal’ limiting distribution.

These results resolve several open problems of Chatterjee and Diaconis, and confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.

Based on joint work with my PhD students Erlang Surya and Emily Zhu; see arXiv:2305.04850.

Discrepancy theory is a well-studied area in mathematics, concerned with the question of partitioning geometry and combinatorial objects in balanced subsets. Starting from seminal works of Spencer, Banaszczyk, Gluskin amongst others, deep connections between the area and other areas in mathematics such as convex geometry and probability were established. But, for most of its history, the arguments establishing the existence of good partitions were non-constructive, even believed to be fundamentally non-algorithmic.

But, the past decade has seen a flurry of work in algorithmic discrepancy theory, leading to efficient algorithms for several of the most famous settings in discrepancy theory. Perhaps surprisingly, these algorithmic techniques have further strengthened the connection between discrepancy and other areas such as convex geometry and probability.

In this talk, we will survey recent results and techniques in algorithmic discrepancy with the aim to convey connections to various areas. Time permitting, we will end with natural conjectures that would lead to progress on long standing conjectures in discrepancy theory.

No prior exposure to algorithms, computer science or discrepancy will be assumed.

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.

A recurrent theme encountered in many models of random geometry is that of two trees glued to one another with a space-filling curve snaking in between them. In this talk, we first recall a few examples of this, namely, Brownian geometry, Liouville quantum gravity, and the Brownian web. Subsequently, we discuss the construction of a pair of interlaced trees and the corresponding Peano curve in the directed landscape, the conjectural universal scaling limit of models in the Kardar-Parisi-Zhang universality class. Finally, we look at the question of determining the precise Holder and variation regularity of this space-filling curve and discuss some of the ideas involved in the proof. Based on the works arxiv:2304.03269 (joint with Riddhipratim Basu) and arxiv:2301.07704.

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.

In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.

Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.

Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.

In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)

\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}

contains a unique maximal element.

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.