Abstract: Quasi-isometric classification of groups is one of the central problems in geometric group theory. In this talk, we focus on quasi-isometries induced by homeomorphisms between the boundaries of hyperbolic groups and relatively hyperbolic groups. We will also discuss results related to Cannon’s Conjecture.
Abstract: Mean Field Games (MFGs) are systems of coupled partial differential equations consisting of a backward Hamilton–Jacobi–Bellman (HJB) equation and a forward Fokker–Planck-Kolmogorov (FPK) equation, which arise in the study of the mean-field limit of Nash equilibria for certain classes of large-population differential games. In this talk, I will discuss the well-posedness, qualitative properties, and numerical approximation of HJB equations and MFG systems, with particular emphasis on problems involving nonlocal/fractional operators.
I will also discuss a second direction of my research concerning boundary blow-up problems for nonlinear elliptic equations, including local and nonlocal problems. I will present results on the existence, uniqueness, and asymptotic behaviour of large solutions.
I will conclude with some perspectives on my ongoing and future work in these areas.
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Abstract: The tensor product problem, a classical question in representation theory, concerns the decomposition of the tensor product (or Kronecker product) of two irreducible representations into a direct sum of irreducible constituents. This problem arises naturally in several areas of mathematics and has been extensively studied for polynomial representations of complex numbers, as well as for symmetric and alternating groups. Although multiplicity-free tensor products have been characterised in certain cases, the tensor product problem remains widely open even for symmetric groups.
In this talk, we study the tensor product problem for general linear groups of degree two over finite principal ideal local rings. Restricting attention to regular representations, we extend several known results from the finite field setting to this broader context. In particular, we classify pairs of regular irreducible representations whose tensor product is multiplicity free. We use the structure of the representations of groups over finite local rings with a few character-theoretic techniques. We also discuss related problems of self-dual representations of these groups and further open questions.
This is based on joint work with Archita Gupta, Tejbir Lohan and Hassain M.
Abstract: The Poincaré metric on the unit disc is invariant under every biholomorphic self-map and turns the disc into a model of hyperbolic geometry. The Kobayashi pseudodistance extends this idea to complex domains and manifolds, providing a natural meeting point between complex analysis, metric geometry and holomorphic dynamics.
In this talk, I will introduce the basic ideas of Kobayashi geometry and discuss three aspects of my recent work. The first is the visibility property: quasigeodesic curves that join neighbourhoods of distinct boundary points are forced to pass through a compact region of the domain. I will describe some classes of visibility domains, local-to-global phenomena, and applications to boundary extension problems and holomorphic dynamics. For unbounded domains, the relevant boundary is obtained using the end compactification of the Euclidean closure; in the bounded case, this reduces to the usual Euclidean closure.
The second aspect concerns big and small horospheres, which encode the asymptotic geometry of a domain near its boundary. I will explain how their geometry can be used to study the boundary behaviour and continuous extension of biholomorphisms. Finally, I will discuss the metric, or horofunction, compactification associated with the Kobayashi distance and its application to a Denjoy–Wolff theorem for holomorphic self-maps.
I will conclude with some ongoing problems and future directions concerning horospheres and boundary regularity, multiply connected planar domains, the relationship between convexity, visibility and geodesics, and holomorphic dynamics under geometric conditions weaker than visibility.
Abstract: I will briefly introduce noncommutative geometry, its origins, and the motivation behind its development. I will then discuss the different approaches in studying noncommutative geometry, with particular emphasis on spectral triples and spectral geometry. Finally, I will briefly present our results on κ-Minkowski space, as well as our recent work on the asymmetric noncommutative 2-torus.
Abstract: In number theory, one of the central objects of study are representations of Galois groups, for example, $G_{\mathbb{Q}_p}$ the absolute Galois group of the field of $p$-adic numbers. In this talk, we will consider $p$-adic crystalline representations of $G_{\mathbb{Q}_p}$, for example, the $p$-adic Tate module of an elliptic curve with good reduction over $\mathbb{Q}_p$. We will first describe the classification of $p$-adic crystalline representations of $G_{\mathbb{Q}_p}$ in terms of certain $(\phi, \Gamma)$-modules known as Wach modules. Then, we will define syntomic complexes with coefficients in a Wach module, and relate its cohomology to the Galois cohomology of the associated crystalline representation. If time permits, we will look at a generalisation of these results to the relative case, i.e. $p$-adic representations of the etale fundamental group of a “small" affinoid algebra.
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Abstract: Classical Higgs bundles on a compact Riemann surface are organized by the Hitchin fibration, whose fibers can be described through spectral curves and rank-one torsion-free sheaves via the Beauville-Narasimhan-Ramanan correspondence. In this talk, we discuss an extension of this picture to Higgs bundles twisted by a rank-two vector bundle \(V\). We describe a spectral correspondence for \(V\)-twisted Higgs bundles in terms of rank-one torsion-free sheaves on an \(S\)-spectral curve, equipped with an additional \(L\)-twisted Higgs field satisfying Hecke-type boundary conditions. Here \(S\) and \(L\) are holomorphic line bundles that appear by the Hecke transformation of \(V\). This talk is based on joint work with David Alfaya and Indranil Biswas (arXiv2506.06573).
Abstract: Higher-order Markov chains are widely used to model categorical time series. However, a major challenge in fitting such models is the exponentially growing number of parameters as the model order increases. Sparse Markov Models (SMMs) provide a parsimonious framework in which all possible histories of order are partitioned into groups such that histories within the same group share identical transition probability vectors. In this paper, we develop a novel method for fitting SMMs based on convex clustering with regularization. The regularization parameter is selected using the Bayesian Information Criterion (BIC). We establish model selection consistency of the proposed estimator under increasing sample size. Extensive simulation studies under diverse settings demonstrate strong finite-sample performance and consistently improved cluster recovery compared to existing competing approaches. Applications to real data on modeling and classifying disease sub-types further highlight the practical advantages of our method, showing superior classification performance.
Abstract: Networks arising in real relational datasets exhibit strong degree heterogeneity: the propensity of nodes to participate in interactions varies widely. When the full set of edges is unavailable or too sensitive to release, degree summaries often become the primary data object, motivating the classical $\beta$-model and its higher-order extension to hypergraphs.
In this talk, I shall develop a statistical theory for degree-heterogeneous $r$-uniform hypergraphs using the hypergraph $\beta$-model. First, I shall characterize sharp estimation rates for the maximum likelihood estimator (MLE) of the model parameters in both $\ell_2$ and $\ell_\infty$ losses, highlighting an effective sample size scaling of order $n^{r-1}$ per node parameter. Then I shall prove that these rates are minimax optimal. Next, I shall study inference where I characterize minimax detection thresholds for testing the presence of degree heterogeneity.
Finally, I shall turn to privacy. Using edge differential privacy, I shall quantify the statistical price of privacy for estimation in $\beta$ models and show a sharp separation between local and central privacy regimes.
Through simulations I shall illustrate the predicted privacy–utility tradeoffs and explore an application to the Enron email hypergraph demonstrating the impact of different regimes of privacy on link prediction in a real organizational communication network.
zoom link: https://zoom.us/j/92846433145?pwd=atEH8YcdEY55pFTKBUhJmcDTkbOL7U.1
Abstract: In the interconnected and interdependent world of the twenty-first century, causal inference is complicated by interference, because the treatment assignments of units can affect the outcomes of other units via treatment and outcome spillover. Since outcome spillover induces dependence among outcomes, closed-form expressions for causal effects and convergence rates for causal effect estimators are challenging and unavailable. In thistalk, I will provide insight into causal mechanisms under interference, by presenting closed-form expressions for causal effects in the presence of treatment and outcome spillover, which help disentangle the contributions of treatment, treatment spillover, and outcome spillover into the causal effects. The main results do not make assumptions about the joint probability law of treatment assignments, outcomes, and connections beyond linearity of conditional expectations of outcomes and the standard assumptions of ignorability and positivity, thus allowing complex dependence among outcomes and connections. Building on the closed-form expressions, I will present causal effect estimators along with rates of convergence, obtained by controlling dependence and characterizing a high-probability subset of data that addresses collinearity issues. Finally, I will discuss generalizability of causal inference under interference, which concerns how causal conclusions based on dependent observations of outcomes can be generalized from a sample to the population.
Online zoom link for research talk: https://zoom.us/j/92848188339?pwd=ArX46c8kgY5GadX3EFELkPjS6pQftZ.1
Meeting ID: 928 4818 8339
Passcode: 763240
Abstract: Piecewise deterministic Markov processes (PDMPs) have emerged as promising alternatives to traditional Markov chain Monte Carlo methods, particularly in large-scale statistical applications. An important example is the Zig-Zag process, for which principled subsampling strategies have been empirically shown to deliver essentially independent samples at a computational cost that need not grow with the size of the data. While much of the theoretical development of PDMP-based algorithms has focused on their behaviour in high dimensions, in Bayesian statistics it is equally important to understand how their complexity scales with the data size $n$. This talk presents new theoretical results on this question. The discussion is divided into three parts, with separate results presented in each case to build up a broader understanding of the large-$n$ behaviour of the Zig-Zag process. Overall, the talk uses a rigorous scaling limit analysis as $n \to \infty$ to quantify the algorithmic complexity of the Zig-Zag process as a function of $n$, and to assess its scalability for large datasets. Based on joint works with Gareth Roberts, Joris Bierkens, and Sebastiano Grazzi.
Abstract: The density of points visible from the origin in sets such as the Ammann–Beenker point set has recently attracted attention. These sets can also be viewed as cut-and-project sets. In this talk, we will present an error estimate for the density of visible points for some classes of cut-and-project sets, along with related results. This is joint work with Ilya and Barak.
Abstract: After reviewing some classical examples, I will state a conjecture and present new results on the value at s=0 of the algebraic p-adic L-function attached to an ordinary Galois representation, in the criticial and non-crtical case.
Abstract: One of the objectives of $p$-adic Langlands program is to parametrize $p$-adic Galois representations and study their properties. A natural source of such representations arises from geometry - as subquotients of $p$-adic cohomology of algebraic varieties of a $p$-adic field $K$. We will discuss recent advances in studying Galois cohomology of such objects viewing them through new 'prisms'.
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Abstract: After brief introduction to the general differential equation governing fluid dynamics, I would like to discuss a model arising from sinking of a rigid solid into a thin film of fluid, surrounded by air. This leads to movement of the contact point, that is where the air, liquid and solid meet, and the formation of a meniscus. This free boundary problem, together with the no-slip (Dirichlet) condition at the fluid-solid interface, gives rise to a fourth order quasilinear parabolic equation. An interesting observation shows that this particular thin film equation is well-posed and also the contact point can possibly move, contrary to the classical thin film equation for a droplet arising from no-slip condition.
Abstract: We shall first outline cloaking and then inverse homogenization for Maxwell equations. In the following part, we will discuss how relative Homogenization and Tartar's H-measure play a role in this
Abstract: In this talk, I will discuss the zero set of solutions to partial differential equations. A natural question is whether a nontrivial solution can vanish at a point together with all of its derivatives, and more generally, how rapidly solutions can approach zero. After a brief historical overview, I will present a new sharp estimate for the order of vanishing of solutions to parabolic equations with variable coefficients. In the case of real-analytic coefficients, I will show how this estimate 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 talk is based on joint work with Agnid Banerjee and Nicola Garofalo.
Abstract: In this seminar, I will be presenting high-order accurate entropy-stable ADER (Arbitrary high-order DERivative) predictor–corrector numerical schemes as an efficient numerical method for solving Hyperbolic Conservation laws. Traditional high-order methods such as WENO, TVD, and RKDG typically rely on multi-stage Runge-Kutta timestepping to maintain the temporal accuracy. However, these approaches become inefficient beyond the third order because of the Butcher barriers and the high memory traffic they impose on modern CPU and GPU hardware. ADER methods overcome these limitations using a predictor–corrector strategy that achieves arbitrarily high-order accuracy within a single-step update in both space and time, making them well-suited for parallel architectures.
Moreover, for a general nonlinear hyperbolic system, the solutions may break down in a finite amount of time, leading to infinitely many weak solutions. While conservation laws are necessary, they do not guarantee physical admissibility. Entropy stability is essential to enforce the second law of thermodynamics and select the physically admissible weak solutions at shocks. In this work, we construct high-order accurate, robust, and efficient entropy stable ADER schemes by combining entropy-conservative fluxes with suitable dissipation and limiter corrections inside the ADER framework. The designed numerical schemes are entropy stable and show high-order convergence, sharp shock capturing and robustness over various multidimensional test problems.
Abstract: In this talk, I will discuss the reduced order modeling for parametric PDE eigenvalue problems and briefly discuss our contributions in this direction. First, I will discuss the reduced order modeling for PDE eigenvalue problems in general setup. I will discuss how to choose the snapshots for finding eigenvalues and eigenvectors in reduced space. The success of the projection-based order modeling lies on the assumption that the problem is affine parameter dependent. For non-affine parameter dependent problem, we have proposed data-driven model. So, I will discuss the data-driven model for parametric eigenvalue problems using Gaussian Process regression (GPR). Then, I will talk about image processing problems such as image inpainting and segmentation. I will discuss an inpainting model and present some theoretical results related to this model. Some numerical results will be presented to demonstrate the improved performance of our model.
Abstract: We investigate the average number of lattice points within a ball where the lattice is chosen at random from the set of unit determinant ideal or modules lattices of some cyclotomic number field. The goal is to consider the space of such lattice as a probabilistic space and then study the distribution of lattice point counts. This is inspired by the connections of this problem to lattice-based cryptography and sphere packings in a high dimensional Euclidean space. Based on joint work with Vlad Serban, Maryna Viazovska, Ilaria Viglino.
Abstract: This talk focuses on my recent research on grouped multiple hypothesis testing inan online setting. Classical multiple testing procedures are offline in nature, meaning thatthe entire collection of hypotheses and corresponding test statistics is available before thetesting procedure begins. This setting allows for efficient use of available resources, such as the overall error budget and auxiliary structural information about the hypotheses, leading to procedures with high statistical power while maintaining control of a global error measure.
In contrast, online multiple testing procedures are a relatively recent development in theliterature. In the online framework, hypotheses arrive sequentially over time, and decisionsmust be made in real time based only on past information, before future test statistics areobserved. The lack of knowledge about future test statistics makes the task of controlling an overall error measure substantially more challenging than in the offline setting.
The talk introduces the ‘Grouped Online Testing Algorithm (GOTA)’ , which integrates ideasfrom both online and offline multiple testing to address settings in which hypotheses arrivein groups over a potentially infinite sequence. Unlike most existing multiple testingprocedures that rely on p-values, GOTA is built using the local false discovery rate as itsfundamental building block. I will discuss the theoretical properties of the algorithm,including its guarantees for controlling an overall error measure, as well as its practicalperformance. Simulation studies demonstrate that the proposed method achieves substantially higher power than a comparable p-value–based procedure.
Given the current lack of multiple testing methods tailored to such grouped online settings,this work aims to fill an important methodological gap. The talk will also briefly reviewfoundational concepts in multiple hypothesis testing and highlight my related recentresearch. No prior background in multiple testing is assumed, and the talk is intended to be accessible to everyone interested.
Abstract: The study of statistics of random permutations is arguably the earliest result in probability. These statistics bring out deep connections with fields like combinatorics, number theory, and representation theory. The probability that a uniform random permutation has $k$ orbits/cycles is log-concave (in $k$). In fact, it was observed by Levy that the number of orbits of a random permutation has the same distribution as the sum of independent Bernoullis. This allows one to deduce a central limit theorem for the number of orbits of a uniform random permutation. The situation is more delicate for a random pair of commuting permutations. Consider a pair of commuting permutations drawn uniformly at random from the set of all commuting pairs of permutations. It was conjectured by (Nekrasov--Okunkov) Heim-Neuhauser that the probability it has $k$ orbits is (unimodal) log-concave. This problem remains widely open. In this talk, we will discuss some recent partial progress on this problem. In particular, we prove a CLT for the number of orbits of a random pair of commuting permutations.
Abstract: We describe a framework for Bayesian analysis of vector-valued time series of counts. The approach consists of a flexible level correlated model (LCM) framework for building hierarchical models that incorporate correlated latent level effects and temporal effects to model the multivariate data. This allows for faster computation than using the multivariate Poisson distribution, whose likelihood calculation can be slow as the vector dimension increases. The LCM framework is versatile and allows us to model many types of multivariate time series such as counts, positive-valued observations, etc. For count time series, this framework allows us to combine univariate distributions for counts (Poisson, negative binomial, ZIP, etc.) for each component series, while accounting for association among the components via an unobserved (latent) Gaussian random vector. We also allow for univariate autoregression (AR) or vector autoregression (VAR) evolution of the latent states. We employ the integrated nested Laplace approximation (INLA) setup for fast approximate Bayesian modeling via the R-INLA package, building custom functions to handle the VAR evolution. We illustrate our approach using intra-day financial data streams. We show an application to analyzing financial data streams. This flexible framework can be easily extended to other scenarios such as modeling multivariate positive-valued time series, with application in several domains including ecology, marketing, and transportation safety.