Seminar
Speaker: Udit Mavinkurve
Host: Swapneel Mahajan
Title: The Fundamental Groupoid in Discrete Homotopy Theory
Time, day and date: 2:25:00 PM, Monday, September 08
Venue: Room 105
Abstract: In classical homotopy theory, graphs are treated as 1-dimensional CW complexes. But since the classical notions of continuous maps and their homotopies do not respect the discrete nature of graphs, this fails to capture the full combinatorial richness of graph theory. Discrete homotopy theory, introduced around 20 years ago by Barcelo et al., building on the work of Atkin from the mid-seventies, is a homotopy theory specifically designed to study discrete objects like graphs. This theory has found a wide range of applications, including in matroid theory, hyperplane arrangements, and more recently, in topological data analysis.

In this talk, based on joint work with Chris Kapulkin, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert¨Cvan Kampen theorem, generalizing a previous result of Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction.

Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. We propose a new line of attack, by breaking it into more tractable problems comparing the homotopy theories of the respective n-types, for each integer n ¡Ý 0. We also solve this problem for the first nontrivial case, n = 1.
 

Statistics/Probability Seminar
Speaker: Promit Ghoshal, University of Chicago
Host: Parthanil Roy
Title: Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond
Time, day and date: 4:00:00 PM – 5:00:00 PM, Tuesday, September 09
Venue: Ramanujan Hall
Abstract: Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near optimal rates with polynomial dimensional dependence and extend these results to bilinear enhanced kernels.

Seminar
Speaker: Dr. Ramesh Mete (IIT Bombay)
Host: Saikat Mazumdar
Title: Reading seminar on the Yamabe flow
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, September 10
Venue: Room 113
Abstract: In the 1980s, Hamilton proposed a heat flow approach to the Yamabe problem (uniformization theorem in dimension >2). The goal is to start from a given initial metric and deform it to a metric with constant scalar curvature by means of an evolution equation. Hamilton showed that the flow exists for all time t > 0; however, convergence turns out to be highly nontrivial. 
For conformally flat metrics, Ye [1994] showed that the flow converges to a metric of constant scalar curvature. Subsequently, Schwetlick-Struwe [2003] showed that the flow converged in dimensions 2<n<6 under the assumption that the Yamabe energy of the initial metric was "not too large". The energy assumption was then removed by Brendle [2005] to establish unconditional convergence of the flow in low dimensions 2<n<6. Convergence in dimensions n>5 was also shown by Brendle [2007] under a technical hypothesis on the conformal class. 
Our aim in particular is to discuss the work of Schwetlick-Struwe and Brendle in low dimensions.

Mathematics Colloquium
Speaker: Manas Rachh, IIT Bombay
Title: Complex scattering makes for simple numerics
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, September 10
Venue: Ramanujan Hall
Abstract: Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to derive explicit error bounds for these methods. Many of these methods also require a large computational domain, and so require a large number of discretization points to accurately solve the problem.
In this talk, we will present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integral equations can then be analytically continued into the complex plane, where they can be safely truncated with controllable accuracy. We demonstrate this procedure for the example of two dielectric waveguides meeting at an interface. For this problem, we show that the fundamental solutions and densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with exponential accuracy.

Commutative Algebra seminar
Speaker: Tony Puthenpurakal
Title: F-modules IV
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 11
Venue: Ramanujan Hall
Abstract: We continue our study of F-modules

Geometry and Topology Seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Geometric Kernels of Proper Maps Between Non-Compact Surfaces
Time, day and date: 11:30:00 AM, Friday, September 12
Venue: Ramanujan Hall
Abstract: A map between 2-manifolds is said to have a geometric kernel if it sends a non-contractible simple loop to a null-homotopic loop. For compact 2-manifolds without boundary, every map that is not injective on the fundamental group admits such a kernel. In contrast, maps between compact 2-manifolds with boundary, or between non-compact 2 manifolds, need not admit a geometric kernel. In this talk, I will present a sufficient condition, formulated in terms of Brown’s proper fundamental group, which guarantees that a degree-one map between non-compact 2-manifolds without boundary admits a geometric kernel.

Statistics and Probability seminar
Speaker: Dr. Kaartick Adhikari, IISER Bhopal
Host: Koushik Saha
Title: The spectrum and local weak convergence of sparse random uniform hypergraphs.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Friday, September 12
Venue: Ramanujan Hall
Abstract: The notion of the local weak convergence, also known as Schramm convergence, for a sequence of graphs was introduced by Benjamini and Schramm. It is well known that the local weak limit of the sparse Erdos-Renyi graphs is the Galton-Watson measure with Poisson offspring distribution almost surely. Recently, Adhikari, Kumar, and Saha showed that the local weak limit of the line graph of the sparse Linial-Meshulam complexes is the d-block Galton-Watson measure almost surely. In this talk, we discuss the local weak convergence of a unified model, namely, the weighted line graphs of sparse k-uniform random hypergraphs on n vertices.