Combinatorics Seminar
Speaker: Himanshu Gupta (University of Regina, SK, Canada)
Host: Krishnan Sivasubramanian
Title: On the eigenvalues of the graphs D(5, q)
Time, day and date: 11:00:00 AM – 12:00:00 PM, Friday, August 1
Venue: Ramanujan Hall
Abstract: In 1995, Lazebnik and Ustimenko introduced the family of q-regular graphs D(k, q), which is defined for any positive integer k and prime power q. The connected components of the graph D(k, q) have provided the bestknown general lower bound on the size of a graph
for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D(k, q) is always less than or equal to 2√q, indicating that the graphs D(k, q) are almost Ramanujan graphs. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of D(5, q) is less than or equal to 2√q when q is an odd prime power. This is joint work with Vladislav Taranchuk.

Mathematics Colloquium
Speaker: Bhaswar B. Bhattacharya (Associate Professor, University of Pennsylvania)
Host: Parthanil Roy
Title: Higher-Order Graphon Theory: Fluctuations, Inference, and Degeneracies
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, August 06
Venue: Ramanujan Hall
Abstract: Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct joint confidence sets for the motif densities. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs. (Joint work with Anirban Chatterjee, Soham Dan, and Svante Janson)

Number theory seminar
Speaker: Hariom Sharma (IIT Roorkee)
Host: Ravi Raghunathan
Title: On representations of $GL(n,D)$ with a symplectic model
Time, day and date: 10:00:00 AM - 11:00:00 AM, Thursday, August 07
Venue: online (https://meet.google.com/pqd-fyod-xqi)
Abstract: Let $F$ be a non-Archimedean local field of characteristic zero, and let $D$ be the unique quaternion division algebra over $F$. For $n \in \mathbb{N}$, let $G_n = GL(n,D)$. The subgroup $H_n = Sp(n,D)$ of $G_n$ denotes the unique non-split inner form of the symplectic group $Sp(2n, F)$.
A smooth admissible complex representation $(\pi,V)$ of $G_n$ is said to have a symplectic model (or to be $H_n$-distinguished) if there exists a non-zero linear functional $\phi$ on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $h \in H_n$ and $v \in V$.
In this talk, we provide a complete list of irreducible admissible representations of $G_3$ and $G_4$ having a symplectic model. We demonstrate that induced representations from finite-length representations preserve the symplectic model. Furthermore, we classify those ladder representations of $G_n$ that admit a symplectic model. In addition, we prove a part of Prasad's conjecture which provides a family of irreducible unitary representations with a symplectic model.

Commutative Algebra seminar
Speaker: Tony Puthenpurakal (IIT Bombay)
Title: Lyubeznik's theory of F-modules-I
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, August 07
Venue: Ramanujan Hall
Abstract: In the first lecture we show Kunz result that if $R$ is regular of characteristic $p > 0$ then the Frobenius is flat. We also define F-modules introduced by Lyubeznik.

Speaker: Subho Majumdar (Head of AI at VIJIL, a US-based startup)

Host: Radhendushka Srivastava

Date: 8 Aug 2025

Time: 3:00 to 4:00 pm

Venue: Ramanujan Hall

Title: Towards Statistical Foundations for Reliable and Defendable Large
Language Models

Abstract: The emergence of Large Language Models (LLMs) has brought in
concomitant concerns about the security and reliability of generative AI
systems. While LLMs promise powerful capabilities in diverse real-world
applications, ensuring that their outputs are resilient to malicious
attacks and consistent across similar inputs has significant methodological
and computational challenges. This situation calls for the revisiting of
modern deep learning architectures through a statistical lens.

I will present on two interconnected themes in this area. First, I will
introduce Representation Noising (RepNoise), a defense mechanism that
protects the weights of open-source LLMs against malicious uses. RepNoise
achieves this through controlled noise injection in the knowledge
representations inside a model that makes it harder to recover harmful
information later. Second, I will discuss my work on the consistency
problem—the equivalent of robustness in LLMs concerned with measuring and
minimizing the sensitivity of LLM outputs to input variations through a
combination of controlled synthetic data generation and fine-tuning.

I will conclude by discussing ongoing work at the intersection of AI
security and statistics, including the development of statistical bounds
for the strength of defense mechanisms like RepNoise, and robustness
frameworks for ensuring AI system reliability in high-stakes applications.
 

Partial Differential Equations seminar
Speaker: Vikram Giri (ETH Zurich)
Host: Harsha Hutridurga
Title: Non-uniqueness for the transport equation with incompressible Sobolev vector fields
Time, day and date: 4:00:00 PM - 5:00:00 PM, Friday, August 08
Venue: Ramanujan Hall
Abstract: After recalling the DiPerna-Lions theory for the transport equation with Sobolev vector fields, we will review recent works that construct non-unique solutions using convex integration techniques.

Commutative Algebra seminar
Speaker: Vaibhav Pandey (Purdue University, West Lafayette, USA)
Host: Sudhir R. Ghorpade
Title: The optimal number of equations needed to define a variety: Connections with invariant theory
Time, day and date: 5:0:00 PM – 6:00:00 PM, Friday, August 08
Venue: Ramanujan Hall
Abstract: The arithmetic rank of a variety is the smallest number of equations needed to define it, i.e., the number of hypersurfaces needed to cut out the given variety. In general, this number turns out to be notoriously difficult to compute.

This talk will shed light on the interplay between classical and modern techniques in algebra. We will begin with a quick introduction to classical invariant theory and its role in laying some of the groundwork for modern algebra. We will focus on the classical representations of linear algebraic groups and explicitly compute the arithmetic ranks of their `nullcone variety' (introduced by Hilbert) in all characteristics.

This is ongoing work with Jack Jeffries, Anurag Singh, and Uli Walther.

Mathematics Colloquium
Speaker: Keerthi Madapusi (Boston College)
Host: Dipendra Prasad
Title: A motivated introduction to Shimura varieties
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, August 13
Venue: Ramanujan Hall
Abstract: The aim of this talk is to present some motivation for the theory of Shimura varieties. These objects, which have played a central role in many advances in number theory and arithmetic geometry in the past four decades, are in fact rich tapestries where geometry, arithmetic and representation theory all get woven together in a still somewhat mysterious fashion. I will first explain some fundamental problems where their use was essential and then try to give some ways of thinking about them in informal terms.

Commutative Algebra seminar
Speaker: Tony Puthenpurakal (IIT-Bombay)
Title: F-modules II
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, August 14
Venue: Ramanujan Hall
Abstract: In the first lecture we show Kunz result that if $R$ is regular of characteristic $p > 0$ then the Frobenius is flat. We also define F-modules introduced by Lyubeznik.
We continue with our discussion on F-modules

Mathematics Colloquium
Speaker: Vinayak Vatsal (University of British Columbia)
Host: Dipendra Prasad
Title: Special values of the Riemann zeta function
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, August 20
Venue: Ramanujan Hall
Abstract: I will give formulae for values of the Riemann zeta function at negative odd integers, originally found by Euler. These values are rational numbers, which — despite their apparently elementary nature — are deeply connected with hard number-theoretic problems. I will describe some of these connections, and try to describe a few modern results on this topic.

Combinatorics seminar
Speaker: Venkata Raghu Tej Pantangi
Host: Niranjan Balachandran
Title: Erd\H{o}s-Ko-Rado Combinatorics
Time, day and date: 11:45:00 AM - 12:45:00 PM, Thursday, August 21
Venue: Ramanujan Hall
Abstract: The eponymous Erdős-Ko-Rado (EKR) theorem is a central result in extremal combinatorics, which determines the size and the structure of the largest possible collections of pairwise intersecting $k$-subsets of a fixed $n$-set. This seminal result spawned a family of analogous results for a wide range of mathematical objects that possess a notion of
intersection, such as vector spaces, permutations, perfect matchings, spanning trees, and many more. In this talk, I will
(i) introduce the EKR theorem along with some of its interesting generalizations and analogues; and
(ii) give an overview of some algebraic techniques that built a unifying methodology to prove many EKR-type results.

Seminar
Speaker: Dr Deep Makadiya (IIT Bombay)
Host: Dipendra Prasad
Title: Triangular and Unitriangular Decompositions of the Odd Unitary Group $SU_{2n+1}(R)$.
Time, day and date: 2:30:00 PM, Thursday, August 21
Venue: Room 105
Abstract: The existence of triangular and unitriangular factorizations has been extensively studied for classical groups over commutative rings. These decompositions are well understood for groups such as $\mathrm{SL}_n (R), \mathrm{SO}_n (R), \mathrm{Sp}_n(R)$, and $\mathrm{SU}_{2n}(R)$. However, the case of the odd unitary group $\mathrm{SU}_{2n+1}(R)$ has long posed difficulties, particularly over general commutative rings.
In this talk, I will present recent progress on this problem. I will begin by introducing two new classes of commutative rings—those satisfying the \emph{special stable range one} condition and those that are \emph{$\theta$-complete}. After outlining some of their key properties and giving illustrative examples, I will explain how these conditions allow us to establish triangular and unitriangular factorizations for the group $\mathrm{SU}_{2n+1}(R)$.
This is joint work with Prof. Shripad M. Garge.
 

Commutative Algebra seminar
Speaker: Tony Puthenpurakal
Title: F-modules III
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, August 21
Venue: Ramanujan Hall
Abstract: We continue our investigation of F-modules

Speaker: Dr. Raghavendra Tripathi, New York University, UAE
Title:On Erdos matrices and related problems
Time, day and date: 2:00 PM - 3:00 PM, Monday, August 25
Venue: Department Conference Room

Talk by Dr Shubham Jaiswal, IITB Post Doc at 4 pm on Monday, 25.08.2025 in Ramanujan Hall .

Host : Prof. Dipendra Prasad

Title : Root clusters and number fields

Abstract, as given by Dr Shubham Jaiswal :

I will introduce the notion of root clusters of a polynomial and related
notions of root capacity and intersection indicium of field extensions and
discuss our results on the inverse problems for these concepts over number
fields. Further I'll introduce the notions of minimal generating sets of
Galois closure and compositum feasible triplets and discuss our main
results on these concepts over number fields. Our methods for all these
problems are Galois theoretic in nature and heavily rely on the known
cases of the inverse Galois problem.

The talk is based on my recent solo work as well as my earlier joint works
with Prof Vanchinathan and Prof Bhagwat.

Date-Day-Time : 26th August, Tuesday, 4 pm

Host:  Sudarshan R. Gurjar

Venue: Ramanujan Hall

Speaker: Ankur Sarkar

Title- Smooth Structures on the product of a 4 manifold with a
standard sphere.

Abstract- The study of exotic smooth structures on manifolds is one of
the fundamental problems in topology. In particular, the
classification of smooth structures on a given smooth manifold M is
connected to the determination of a subgroup of the group of homotopy
spheres, namely, the concordance inertia group of M. In this talk, we
compute the concordance inertia group of the product of a 4 manifold M
with the standard k-sphere using the stable homotopy type of M, where
k varies between 1 to 10. Using the above computations of the
concordance inertia group, we classify all smooth manifolds
homeomorphic to the product of M with the standard k-spheres, up to
concordance. As an application of the above computations, we give a
complete diffeomorphism classification of closed, oriented, smooth
manifolds homeomorphic to the product of complex 2 projective space
with standard k-sphere, where k lies between 4 and 6. This is a joint
work with Samik Basu and Ramesh Kasilingam.
 

Special Colloquium
Speaker: Ved Datar (Indian Institute of Science)
Host: Anusha Mangala Krishnan
Title: The Kobayashi-Hitchin principle in complex geometry
Time, day and date: 11:30:00 AM - 12:30:00 PM, Thursday, August 28
Venue: Ramanujan Hall
Abstract: A very general principle lies at the heart of complex differential geometry - existence of canonical differential geometric objects on projective manifolds (metrics, connections, solutions to non-linear PDEs etc) must be equivalent to a purely algebro-geometric notion called “stability”. The first example of such a correspondence was the famous Narasimhan-Seshadri theorem. Since then, this general principle has become ubiquitous in all of complex differential geometry and is now the single most important way to arrive at new conjectures. I will illustrate this principle by way of two examples that have seen impressive progress over the last two decades and yet continue to remain active areas of research - existence of K\”ahler-Einstein metrics on Fano manifolds and existence of
solutions to some fully non-linear PDEs such as the $J$-equation.
 

Special Colloquium
Speaker: James Eldred Pascoe (Drexel University)
Host: Sourav Pal
Title: Matrix convex functions
Time, day and date: 2:30:00 PM - 3:30:00 PM, Thursday, August 28
Venue: Ramanujan Hall
Abstract: Given a function on some subset of the real numbers, there is a natural way to evaluate it on self-adjoint matrices with eigenvalues in the domain known as the functional calculus. For a simple function like a polynomial, power series or a rational function, such agrees with substituting the matrix into the expression and evaluating. Kraus analyzed the so-called matrix convex functions which are the functions such that $f_v(X) = \langle f(X)v, v\rangle$ is convex as a function of matrix inputs $X.$ For classical convex functions, we have the simple condition that some weak version of the second derivative must be nonnegative, but for matrix convex functions it turns out they must automatically be analytic and have a very particular form which can be codified in terms of an integral representation formula. (That is, matrix convex functions are essentially moment generating functions.) Helton, McCullough and Vinnikov generalized the Kraus representation to rational expressions of several noncommuting variables as their so-called Butterfly realization. With Ryan Tully-Doyle, we generalized their result to arbitrary functions of noncommuting matrix inputs leveraging a powerful technique known as the ``royal road" to lift the automatic analyticity from the one variable Kraus theorem to several variables for free and thus reduce to an algebraic problem. 
In this talk, we will give a gentle introduction to the functional calculus, its rapidly maturing multivariate generalization known as free noncommutative function theory, and the theory of matrix convex functions in both contexts.