Mathematics Colloquiums

Erdos et al. (2022) proved a conjecture of Toroczkai that any large enough sequence of prime gaps is graphic- that is, one may find a graph whose vertices have degrees equal to the prime gaps. With Frot, Gou and Wang (2026), we made this result explicit and showed that a sequence of length at least exp(exp(30.5)) is graphic. We use results from graph theory (Erdos--Gallai theorem) and analytic number theory (explicit bounds on distribution of zeros of the Riemann zeta function and explicit zero free region) to derive the result.

In this talk, we discuss several geometric aspects of the Ray transform and the Radon transform, which play a fundamental role in integral geometry and have important applications in areas such as medical imaging, geophysics, and other branches of science. We will then explain how these transforms arise naturally in the study of inverse problems for partial differential equations and demonstrate how they can be used to recover unknown coefficients or structures from indirect measurements.

The special value at s=0 of Dedekind and Artin L-functions is central to the study of Galois module structures in classical algebraic number theory. One can study the special value at u=1 of the Ihara zeta function in graph theory from a similar perspective. In this talk, I will describe a circle of recent results in graph theory that can be viewed as analogues of classical phenomena in algebraic number theory, including results in the spirit of Brumer’s conjecture, the Herbrand–Ribet theorem, and Iwasawa’s asymptotic class number formula. The emphasis will be on the underlying parallels and the emerging dictionary between the two settings.

In this talk we shall discuss averaging operators over lower-dimensional manifolds. We will explore their boundedness and regularity properties. Finally, we will discuss some bilinear analogues.

We discuss bounded factorizations of matrix groups with entries in a wide variety of rings. Results are often intimately related to deeper properties such as the congruence subgroup property, Kazhdan’s property T, rigidity of representations, etc. We also briefly mention growth zeta functions that arise naturally, whose analytic content encodes group theoretic information.

In this talk, we will discuss the existence and uniqueness problem for linear stochastic PDEs involving a bilaplacian operator. Our results on the existence and uniqueness are obtained through an application of a monotonicity inequality. As an application of these results, we also obtain a probabilistic representation of the solution for a linear PDE involving the bilaplacian operator.

In this talk, we discuss operators that are represented by upper triangular 2 × 2 block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators. These operators emerged from the study of Brownian isometries studied by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. We address the issue of subnormality of Brownian-type operators and relate it to a spectral inclusion of a pair of commuting normal operators. This talk is based on a joint work with Z. J. Jablonski, I. B. Jung and J. Stochel.

A free-by-cyclic group is a semi-direct product of a free group with the group of integers. Alternatively, a free-by-cyclic group can be viewed as the mapping torus of a free group automorphism. In this talk, we will see that such a group can in fact fiber in multiple ways. We will then consider the properties of different monodromies that are invariants of the group. This is joint work with Spencer Dowdall, Yassine Guerch, Jean Pierre Mutanguha and Caglar Uyanik.

Let p be a prime number. For a number field K, let Kφ denote the maximal unramified p-extension of K, with Galois group Gφ. We define the deficiency of a finitely generated pro-p group G by Def(G) := r(G) − d(G), where d(G) is the minimal number of generators of G and r(G) is the minimal number of relations. While d(G) can be computed easily, there is no known algorithm to compute r(G) in general. Shafarevich, and independently Koch, showed that 0 ≤ Def(Gφ) ≤ ν, where ν denotes the number of units. Hajir, Maire, and Ramakrishna obtained improved bounds for Def(Gφ) and developed new techniques that may help in computing r(Gφ). In ongoing work, we aim to implement their methods to compute r(Gφ) in order to better understand the structure of Gφ, especially for families of biquadratic number fields and odd primes p, which may come close to producing a counterexample to the Fontaine–Mazur conjecture.

The aim of the talk is to give an overview of A^1 homotopy theory and its potential applications to classical problems of algebraic groups, e.g. the norm principle as formulated by Merkurjev. The talk is based on a recent joint work with Anand Sawant.

The purpose of this talk is to give an overview of some applications of the theory of derived deformations of Galois representations. In particular, I will describe two results obtained jointly with Jacques Tilouine. One in Iwasawa theory extending results of Hida on the trivial zeros of algebraic p-adic L-function, and the other one on the generalized Leopoldt conjecture.

The broad research area of my talk lies in holomorphic dynamics. The main focus of the talk will be to explore various rigidity properties of Hénon maps which are of fundamental importance in multivariable holomorphic dynamics. Loosely speaking, by rigidity properties, we mean those properties of Hénon maps which determine the underlying Hénon maps almost uniquely.

A central question in knot theory is to recognise knots from their diagrams. I will begin with a survey of algorithms for knot recognition and then focus on a specific algorithm for hyperbolic knots which uses Pachner moves. A Pachner move is a local combinatorial change to the triangulation of a manifold. Any two geometric ideal triangulations of a cusped complete hyperbolic 3-manifold are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of a lower bound on the dihedral angles of the geometric triangulations. This leads to an effective algorithm to check the equivalence of geometrically triangulated hyperbolic manifolds and therefore of hyperbolic knots. This is joint work with Sriram Raghunath.

The famous conjecture that asserts the non-vanishing of the Ramanujan tau function can be generalized in terms of Poincare series. In joint work with M. Manickam, we develop a new method to study Poincare series, and use it to significantly extend previously known results on their non-vanishing. The talk will be accessible to a wide audience.

We investigate a class of quasilinear elliptic equations that combine both local and nonlocal operators. Our aim is to understand the regularity behavior of their solutions, including local boundedness, Harnack-type inequalities, and other qualitative properties. These results highlight how the interaction between the local and nonlocal components shapes the overall solution behavior. We also discuss how these regularity findings can be applied to singular mixed local–nonlocal problems.

We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form an angle whose measure is a rational multiple of pi. As a corollary, we find all tetrahedra whose six dihedral angles are all rational multiples of pi. While these questions, and their answers, are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by physicists in the 1960s. Joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein.

We know that the field R of real numbers is a completion of the field Q of rational numbers. About a hundred years ago, it was discovered by Hensel that there are other completions of Q. In fact, there is a completion Q_p for each prime p, and there are no others. Collectively, fields such as R, Q_p, and their finite extensions are called local fields of characteristic 0. The local field R has only two finite extensions and C has only one, but every other local field has infinitely many finite extensions. Can we parametrise all finite extensions of other local fields? This question will be the main topic of the colloquium. We will show how to parametrise, for a given local field K, all extensions of prime degree and, more generally, all primitive extensions of K. The notions of primitivity and ramification will be defined.

In this talk, we examine a stochastic nonlinear heat equation in any dimension d ≥ 1 driven by a Gaussian noise in the Stratonovich form along with a constraint on the L^2-norm of the solution. We show the existence of an H^1∩L^p-valued martingale solution. Moreover, we prove that this solution is invariant in a Hilbertian manifold, in particular unit sphere, i.e., if the initial data is in M, then all its corresponding trajectories stay in M. Finally, the pathwise uniqueness of the solution concludes the existence of a strong solution via a Yamada-Watanabe type result. This is a joint work with Ashish Bawalia and Zdzislaw Brzezniak.

One of the intriguing conjectures in differential geometry, posed by S.-T. Yau as Problem 52 in his 1993 list of open problems, asserts that every closed almost complex manifold of real dimension at least 6 should admit a complex structure. While this is known to fail in dimension four, the high-dimensional case remains completely open. In this talk, I will begin by discussing exotic smooth structures on complex projective spaces CP^n—manifolds that are homeomorphic but not diffeomorphic to the standard CP^n for n ≤ 9. I will describe examples such as distinct smooth structures on CP^9, where one admits a metric of nonnegative scalar curvature and another does not. I will then turn to the existence of almost complex structures on these exotic complex projective spaces, aiming to identify promising candidates for testing Yau’s conjecture.

Periods are numbers comparing different rational or integral modules which are isomorphic over C. Modular forms, and more generally automorphic forms, give rise to two types of integral structures and hence give rise to periods. On the other hand, there is a notion of Langlands transfer F of a form f from a group to another group. The question arises to compare, first rationally and then integrally, the periods of f and F. We will explain some results in this direction.

After a brief overview of hypersurface geometry, I will describe a classification for complete hypersurfaces with constant isotropic curvature in Riemannian space forms. This is a joint work with Niteesh Kumar.

Let M be an open manifold of dimension at least 3 that admits a complete metric of positive scalar curvature. For a function v with bounded growth of derivative, whether M admits a metric of positive scalar curvature with volume growth of the same growth type as v is unknown. We answer this question positively in the case of manifolds which are infinite connected sums of closed manifolds that admit metrics of positive scalar curvature. To define a metric of positive scalar curvature with a certain volume growth type on M, we use the Gromov-Lawson construction of metrics with positive scalar curvature on connected sums and Grimaldi-Pansu's construction of metrics of bounded geometry of certain volume growth type on open manifolds. We generalise this result to manifolds which are infinite connected sums of similar closed manifolds along lower-dimensional spheres. This is a joint work with Anushree Das.

In this talk, we will discuss inverse problems related to linear and nonlinear partial differential equations of parabolic and hyperbolic type. Our focus will be on unique determination of coefficients of these PDEs, using the information gathered from full or partial boundary measurements of solution.

This talk will be on introductory topics in differential geometry.

Modular curves have traditionally played a very important role, especially in the study of elliptic curves over integers, and eventually to the solution of Fermat's last theorem. Modular curves are the first examples of certain interesting algebraic curves, named after Shimura, and called Shimura curves. Their higher dimensional generalisation is at the center of the Langlands program.

Reversibility is a classical concept in group theory and dynamical systems, referring to the property of an element being conjugate to its inverse. This notion arises naturally in various mathematical and physical contexts, from time-reversibility of dynamical flows to symmetries in algebraic structures. In this talk, I will begin with a gentle and accessible introduction to the idea of reversibility in groups and survey recent works. I will then focus on the case of Hecke groups, which are a family of Fuchsian groups that generalize the modular group and have rich connections to hyperbolic geometry, number theory, and automorphic forms. Despite their relatively simple definitions, Hecke groups exhibit intricate algebraic and geometric behavior. I will discuss recent results concerning reversible elements in these groups, including classification questions and asymptotic properties. Some of this work is part of ongoing efforts, jointly with Debattam Das, to understand reversibility in Hecke groups.

We describe how Euler added up all positive integers into a mysterious fraction when he was 28 years old, and I try to legitimize his method p-adically. This is a story of number theory from the 17th century on. We only need some knowledge of polynomials and fractions of polynomials and very basics of differentiation. If time allows, I enter into some results related to Ramanujan I found when I was 28 years old. For the results exposed here, a detailed proof can be found in my book: Elementary Theory of L-functions and Eisenstein Series, LMSST vol. 26, 1993, Cambridge U. Press.

We will begin by contrasting semisimple groups over local fields against those over complex numbers. Then we will explain root systems and concave functions. Let $\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ over an algebraically closed residue field $k$. Set $K_{_n} := \text{Fract}~\mathcal{O}_{_n}$. Let $G$ be an almost-simple, simply-connected affine algebraic group over $k$ with a maximal torus $T$. Given a $n$-tuple of concave functions on the root system of $G$, as in Bruhat-Tits we define bounded subgroups of $G(K_{_n}) $. Our main theorem shows that they are all schematic. This generalizes Bruhat-Ttis theory to higher dimensions in the most geometric setup and provides a new approach to classical Bruhat-Tits theory.

In this talk, we will discuss some necessary and sufficient conditions under which a closed curve on a bouquet of n circles can be lifted to a class of finite-sheeted normal coverings. We will also discuss some applications of these results to finite index normal subgroups of free groups. This is a joint work with Deblina Das.

The Weil height induces a natural partial ordering on algebraic numbers of bounded degree, offering a powerful tool for their study. It is well known that an algebraic number alpha has logarithmic Weil height h(alpha)=0 if and only if alpha is a root of unity. When alpha is not a root of unity, obtaining lower bounds on h(alpha) is a deep and extensively studied problem. In particular, for totally real or totally p-adic numbers alpha, explicit lower bounds for h(alpha) are known due to the works of Schinzel and Bombieri-Zannier respectively. In this talk, we will discuss lower bounds on h(alpha) for asymptotically positive infinite extensions. This is the first instance where such a result is obtained for non-Galois extensions. We will also discuss lower bounds on the canonical height of points on a fixed elliptic curve over these infinite extensions. This is joint work with Sushant Kala.

Characterizing embedded circles has been a longstanding topic of interest in low-dimensional topology, with deep connections to various areas of mathematics. In this talk, we will provide a concise overview of widely used invariants employed to address this problem, highlighting their limitations and how these limitations have driven the search for more refined invariants—one such example being Heegaard Floer homology. The presentation will be largely self-contained.

The Hungarian-American Nobel Prize winning physicist Eugene Wigner (1902–1995) was right on target when he observed the surprising and often inexplicable fact that mathematical concepts, developed to describe one set of observations, can be remarkably effective in describing another, seemingly unrelated phenomena. In this talk, we shall explore how this continues to be true as different new areas emerge in different sciences.

We are interested here in questions related to the maximal regularity of solutions of elliptic problems with Dirichlet boundary condition. For the last 40 years, many works have been concerned with questions when the domain is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work. We give here new proofs and some complements for the case of the Laplacian, the Bilaplacian and the operator div (A∇), when A is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the Dirichlet-to-Neumann operator for Laplacian and Bilaplacian. Using the duality method, we can then revisit the work of Lions-Magenes, concerning the so-called very weak solutions, when the data are less regular.

We first discuss Conjectures/Problems (in cyclotomic theory) Iwasawa described in one of his final unpublished manuscripts. Then if time allows, we indicate which of his problems has generalizations in more general settings of adjoint Selmer groups via the theory of modular forms.In the general case, the Iwasawa algebra is replaced by a universal deformation ring (which is a $p$-adic Hecke algebra by a theorem of Taylor--Wiles). We encounter new interesting features related to Iwasawa's question in the general case.

The Riemann hypothesis (RH) is perhaps considered as one of the most challenging unsolved problems in the history of mathematics. Over the years, many mathematicians found different equivalent criteria for the Riemann hypothesis. In 1916, Hardy and Littlewood gave an equivalent criterion for RH while correcting one identity of Ramanujan. In this talk, we shall discuss a generalization of the equivalent criteria given by Hardy and Littlewood. This is joint work with Archit Agarwal and Meghali Garg.

Given an element of a free group on d generators, called a word, and a group G, we can define a map by evaluation. Such maps are called word maps. In the last 3 decades, several great results called Waring-like results for finite simple groups have been obtained. In general, the main question in the subject has been to understand the image w(G). In this talk, we try to understand when the image w(G) is symmetric under inversion for finite groups of Lie type.

This talk will largely touch upon Harish-Chandra's life around his student days at the University of Allahabad.

The question of whether a time-preserving geodesic conjugacy determines a closed, negatively curved Riemannian manifold up to an isometry is one of the central problems in Riemannian geometry. While an answer to the question in this generality has yet remained elusive, this talk will briefly give an overview and discusses certain known results.

This talk will explore the diverse applications of pseudodifferential calculus, starting with a comprehensive overview of various generalizations of pseudodifferential operators. We will highlight their analytic, algebraic, and geometric aspects, showcasing the rich variety of ways these operators can be extended. Finally, we will delve into two specific applications: the well-posedness of hyperbolic operators and their relevance in analytic number theory.

The Adams spectral sequence is an important tool in stable homotopy theory, originally introduced by Adams to compute the stable homotopy groups of the sphere (at a prime p), using the Eilenberg-MacLane spectrum. The E_2-term of the Adams spectral sequence may be identified with certain derived functors, which also holds for other Bousfield-Kan types spectral sequences. In this talk, I'll explain how the higher terms of such spectral sequences are determined by truncations of functors, defined in terms of certain (spectrally) enriched functors called mapping algebras.

Microlocal analysis techniques are mainly used to study wave front sets, oscillatory integrals, Fourier integral operators for the analysis of linear as well as nonlinear PDEs. In general, singularities of any distribution are measured by the wavefront set. The classical tool for the analysis of the wavefront set is the Fourier transform. In this talk we use continuous shearlet transform to characterize different types of wavefront set. We also discussed the affine radon transform and its relation between continuous shearlet transform. Finally we discussed some intertwining properties of shearlet transform with Radon transform and its application to characterize wavefront sets.

I will talk about strong unique continuation properties for some scaling critical variable coefficient parabolic differential inequalities. This is based on a couple of joint works with Pritam Ganguly, Nicola Garofalo, Abhishek Ghosh and Ramesh Manna.

I will talk about a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. I will then show that an application of such an estimate for real-analytic leading coefficients leads to a parabolic generalization of the well known Donnelly-Fefferman nodal set estimate. I will also present applications to new Landis type results in the parabolic setting. This is based on joint work with Vedansh Arya and Nicola Garofalo.

In this talk we will present recent results with S.Tappe (2024) on invariant submanifolds for a class of SPDEs whose solutions have the `translation invariance property'. We will show that the analytic property of `translation invariance' of the solutions transforms into the geometric property of existence of invariant submanifolds for the corresponding SPDE generated by the orbits of the translation group acting on tempered distributions.

The theory of Stochastic PDEs has become an important avenue of research in understanding many complex natural phenomena. The Monotonicity Inequality for the operators in Stochastic PDE, introduced by Krylov & Rozovskii (1981), is used crucially in obtaining the uniqueness of strong solutions. Gawarecki, Mandrekar and Rajeev (2008) showed that this inequality can also be used to obtain the existence of strong solutions for Linear Stochastic PDEs in the dual of multi-Hilbertian spaces and the same authors (2009) established the inequality for certain constant coefficient differential operators on the space of tempered distributions. In this talk, we shall explore some recent results on the Monotonicity inequality, based on multiple joint works with Rajeev Bhaskaran and Arvind Kumar Nath.

Let $\mathbbb{X}\,=\,[X_1\righrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H}\,\subset \,T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $\mathcal{H}$ is integrable, we define a version of de Rham cohomology for the pair $(\mathbb{X},\, \mathcal{H})$, and we study connections on principal $G$-bundles over $(\mathbb{X},\, \mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal $G$-bundles over a pair $(\mathbb{X},\, \mathcal{H})$. This is a joint work with I. Biswas, P. Koushik and F Neumann.

We will motivate through population growth models such as logistic growth models, population models with restricted resources, continuous growth models, delay models, Ecological models such as Predator-Prey models, Lotka–Volterra systems, and finally Epidemiology Models like modelling of an epidemic.

In this talk, the analysis and homogenization of a poroelastic model for the hydro-mechanical response of fiber-reinforced hydrogels will be shown. The medium is considered to be a highly heterogeneous two-component media composed of a connected fiber-scaffold with periodically distributed inclusions of the hydrogel. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior (in the sense of two-scale convergence) of the solutions with respect to a scale parameter, characterizing the heterogeneity of the medium. We obtain an effective model where the micro variations of the pore pressure give rise to a micro stress correction at the macro scale.

We will describe the problem of mod p reduction of p-adic local Galois representations. For two dimensional crystalline representations of the local Galois group Gal(Q¯ p|Qp), the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003. We will give a sketch of the background and history of the problem and discuss few interesting patterns in the behavior of the reduction map. Most of the above results are applicable only for odd primes or sufficiently large primes. If time permits, we will look into the curious case of p = 2, which is an ongoing work with A. Venugopal. We will compare our results with some existing results for odd primes, for example with Buzzard-Gee’s results for slope 1/2 and in general with Zigzag conjecture of Ghate.

Heisenberg’s uncertainty principle has several manifestations in Fourier analysis, including Hardy’s theorem. This talk discusses an operator-theoretic version connected with the boundedness of certain convolution-type operators on the Fock space, where Bargmann, Schrödinger, Fock, and Hermite all play a role.

We discuss issues surrounding the spectra of complete Riemannian manifolds, and sometimes orbifolds, of non-positive curvature. Topics include absolute and singularly continuous spectra, small eigenvalues, and eigenfunction decay. This describes previous and ongoing joint work with Ballmann and Polymerakis.

Atmospheric winds and gravitational tides force the world's oceans, exciting fast-evolving dispersive waves and slowly evolving vorticity-rich eddies. This talk gives an overview of turbulent oceanic flow structures and mathematical models, including deterministic, stochastic, and machine-learning models, for resolving turbulent energy transfers across spatio-temporal scales.

Modular forms are holomorphic functions on the upper-half complex plane with many symmetries. These objects have applications to number theory. Hida’s idea combined modular forms with Iwasawa-theoretic variation, leading to p-adic families of modular forms, also known as Lambda-adic forms. These lectures introduce these forms and lead up to the construction of their Galois representations.

We characterise weighted Lebesgue spaces on the torus for which the solution of the heat equation converges pointwise to the initial data as time tends to zero. A similar question on the cylinder is also discussed. This is joint work with Rupak Dalai.

The talk uses probabilistic ideas to predict how many rational points are located near a compact manifold. After a basic introduction, it discusses the role of oscillatory integrals and curvature in proving such predictions, based on joint work with Rajula Srivastava.

How many rational points with denominator of a given size lie near a compact non-degenerate manifold? This talk discusses recent progress using harmonic analysis, oscillatory integrals, and homogeneous dynamics, with applications to Hausdorff dimension and measure refinements for well-approximable points.

The theory of bilinear multipliers is an active area of research in Euclidean harmonic analysis. There have been many interesting developments in this area in the recent past. In this talk, we will discuss some of the recent results in the context of the bilinear Bochner-Riesz multiplier.

The talk concerns local L2 bounds of Eisenstein series on general reductive groups. It discusses Maass–Selberg relations, the Finis–Lapid–Müller fine spectral expansion of the Arthur–Selberg trace formula, and applications to Sarnak’s Optimal Lifting Conjecture after Assing–Blomer.

The classical Schur-Weyl duality brings together the representation theories of the general linear group GL_n(C) and the symmetric group S_k via their mutually commuting actions on the k-fold tensor space of the defining representation of GL_n(C). Over the years, this type of duality (also referred to as the Schur-Weyl duality) emerging from mutually commuting actions of various other algebraic structures have been discovered. For example, by restricting the action of GL_n(C) to the orthogonal group in the classical Schur-Weyl duality, R. Brauer exhibited a duality between the orthogonal group and a diagram algebra, now known as the Brauer algebra. Further, V. F. R. Jones and P. Martin, independently, restricted the action of GL_n(C) to the symmetric group S_n and exhibited a duality between S_n and another diagram algebra, namely the partition algebra. In this talk, we will overview various known Schur-Weyl dualities and also give new such dualities. We will discuss how these dualities and the representation theory of partition algebras also contribute to understanding various problems pertinent to the representation theory of symmetric groups and to algebraic combinatorics. The talk contains joint works with D. Paul, S. Narayanan, and V. Mazorchuk.

Let H∞ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by B(H∞) the Banach space of all bounded linear operators from H∞ into itself. We prove the denseness of norm attaining operators defined on H∞ in the space B(H∞), it is called the Bishop-Phelps-Bollob´as property for H∞. We also give a representation for a subclass of norm attaining operator, namely hyponormal absolutely norm attaining operators on a Hilbert space.

Human fascination for symmetries in crystals and honeycombs gave rise to the study of tilings. This talk discusses the mathematical theory behind tilings of spaces and some recent results about affine tilings.

The talk explains birational equivalence of algebraic varieties and the aim of the Minimal Model Program, or Mori Program: to find a good and minimal model in each birational equivalence class.

The talk addresses the large-time behaviour of solutions to a reaction-diffusion system with diffusion degeneracies and mass-action kinetics. It explains an indirect diffusion effect and uses the entropy method to derive explicit convergence rates. This is joint work with Saumyajit Das.

The study of stochastic differential equations provides a framework for stochastic flows. The talk begins with Brownian motion and SDEs, then explores Gaussian flows arising from SDEs and the existence and uniqueness of associated stochastic PDEs, with further extensions to Lévy flows.

The classical theorem of Abel and Jacobi characterises zeroes and poles of meromorphic functions using the Abel–Jacobi homomorphism. This talk discusses refinements in higher dimensions, motives, and conjectural refinements due to Bloch and Beilinson.

Sup-norm bounds of eigenfunctions are a classical problem in spectral theory with connections to number theory. The talk generalizes a technique for uniform sup-norm bounds of cusp forms to Siegel upper half-space via the weighted Siegel–Maass Laplacian and the corresponding heat kernel.

The talk addresses existence, uniqueness, and boundary behaviour of positive weak-dual solutions for nonlocal elliptic equations with singular nonlinearities. The operators include several fractional Laplacians, and the nonlinearities include purely singular terms, source terms, and absorption terms.

We study random walks on a d-dimensional torus by affine expanding maps. Under an irrationality condition on translation parts, Haar measure is the unique stationary measure, implying uniform distribution of almost every orbit in certain self-similar sets. This is joint work with Yiftach Dayan and Barak Weiss.

Higher category theory simultaneously generalizes homotopy theory and category theory. In these lectures, I will introduce higher categories and illustrate the theory with examples from geometry and physics. I will discuss applications of higher category theory to the construction and study of moduli spaces in algebraic geometry, and to the classification of topological quantum field theories. We will see that the latter leads to a type of geometry where certain higher categories play the role of geometric spaces. This "categorical geometry" is inspired by ideas from string theory, and provides a natural language for studying mirror symmetry, which relates algebraic geometry to symplectic geometry.