Talk
Speaker: Dr. Madhab Mondal, Dept of Maths, IIT Guwahati
Host: S. Krishnan
Title: On Incidence and Distance Matrices of Graphs with Matrix Weights
Time, day and date: 11:00:00 AM – 12:00:00 PM, Monday, March 30
Venue: Ramanujan Hall (https://meet.google.com/dkp-hbsd-rdd)
Abstract: Throughout all graphs are assumed to be simple and finite. A graph with matrix weights is a graph in which each edge is assigned a nonzero matrix from Ms, the set of all square matrices of order s. The main objective of this study is to investigate certain algebraic properties of matrices associated with matrix-weighted graphs. In particular, we consider the incidence matrix, the q-distance matrix, and the 2-Steiner distance matrix. Let G be a oriented weighted graph with n vertices and m edges, where the edge weights are matrices from Ms, and let Q(G) be the vertex-edge incidence matrix. We observed that when the edge weights are singular matrices, the rank of Q(G) depends on the structure of the graph and the selection of the weights on the edges of G. In particular, we showed that rank of Q(G) can be used to characterize trees and unicyclic graphs. Moreover, for large values of s, we proved that there exists rank one weights from Ms such that every integer between n − 1 and m can be realized as rank of Q(G). We also investigated the smallest possible values of s for which these ranks can be attained. Surprisingly, the smallest value of s for which m can be achieved as rank of Q(G) is the arboricity of the graph. This observation led us to introduce a new graph invariant, the k-semi-arboricity, which generalizes the classical notion of the arboricity. In 2006, Bapat, Lal and Pati introduced a q-analogue version of the distance matrix (called the q-distance matrix) of a tree. We consider the q-distance matrix Dq of a weighted tree, where the edge weights are matrices from Ms. We deduce a formula for the determinant of Dq. Subsequently, we present a necessary and sufficient condition for Dq to be invertible and derive an expression for the inverse whenever it exists. The expression for D−1 q leads us to introduce the q-analogue of the Laplacian matrix (named as the q-Laplacian matrix Lq) for a matrix-weighted tree. A formula for the determinant of Lq is also provided. In 2022, Azimi and Sivasubramanian studied the 2-Steiner distance matrix D2 for an arbitrary tree T. They showed that if T has n vertices and p pendant vertices, then rank D2 = 2n − p − 1. They also provided a class BT of bases for the row space Row(D2). More interestingly, it was shown that for each B ∈ BT , | det(D2[B, B])| = n − 1. Thus, just like the determinant of the distance matrix, | det(D2[B, B])| depends only on n. We consider the 2-Steiner distance matrix D2 of a weighted tree, where the edge weights are positive definite matrices of order s. We show that rank D2 is independent of the specific choice of the positive definite matrix weights from Ms. We provide a combinatorial construction of a much larger class B∗ T of bases for Row(D2) which contains BT with strong linearalgebraic structure (that is, every non-basic row is an affine combination with integer coefficients of a B∗ T -basic rows of D2). We prove a more general result: let T be a weighted tree on n vertices with p pendant vertices, where the edge weights are positive definite matrices of order s. Assume that T has weight matrices W1, . . . , Wp for the pendant edges and Wp+1, . . . , Wn−1 for the remaining edges. Then for each B, C ∈ B∗ T , the absolute value of the determinant | det(D2[B, C])| = det(W1 + · · · + Wn−1) det(W1 · · · Wn−1) det(Wp+1 · · · Wn−1). Thus, unlike the usual distance matrix, where the determinant remains the same for any tree on n vertices with edge weight matrices W1, . . . , Wn−1, this determinant remains the same for any tree on n vertices as long as the determinants of the products of the weight matrices of the non-pendant edges of the trees are equal.

Talk
Speaker: Prof. B. K. Das, IIT Bombay
Title: Dirichlet series kernels and their multiplier algebras
Time, day and date: 2:30:00 PM – 3:30:00 PM, Monday, March 30
Venue: Room 215
Abstract: Hilbert spaces of Dirichlet series associated with Dirichlet series kernels form a rich class of reproducing kernel Hilbert spaces, connecting operator-theoretic and function-theoretic phenomena. In this talk, we shall classify all normalized CNP Dirichlet series kernels in terms of their underlying weight and frequency data and study their multiplier algebras. The structure of the associated multiplier varieties plays a central role in understanding these algebras. We shall determine these varieties explicitly and use them to analyze algebraic and isometric isomorphisms between multiplier algebras arising from a broad class of CNP Dirichlet series kernels. In particular, we shall resolve an open problem posed by McCarthy and Shalit.

Special Session
Speaker: Madhusudan Manjunath, IIT Bombay
Title: Mathematics for Well-Being II
Time, day and date: 4:00:00 PM - 5:30:00 PM, Monday, March 30
Venue: Ramanujan Hall
Abstract: Mathematics as seen today is a dazzling array of branches each interacting in an intricate, dynamic manner weaving out seemingly fresh patterns. While there are about four-five major branches of Mathematics, each of these contains an innumerable array of subareas, sub-subareas and so on. Questions arise: How will it evolve eventually? Is there an ultimate?
Will it bring ``good" to humanity? Is there a sense in which all of Mathematics can be ``known"? We carefully explore these questions taking guidance from an ancient school of Indian philosophy. We pay special attention to the practical aspects. 

Talk
Speaker: Prof. Monika Bhattacharjee, IIT Bombay
Title: Testing High-Dimensional Means under Sparse Missingness
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, March 30
Venue: Room 215 (https://meet.google.com/dkp-hbsd-rdd)
Abstract: We propose a new testing procedure for the population mean vector in high-dimensional settings with missing observations. Existing approaches typically require finite fourth moments and impose restrictive assumptions on the missingness mechanism, often effectively limiting attention to dense missing patterns.
Our method overcomes these limitations by standardizing the test statistic using the conditional variance given the observed missingness structure. This removes the need for the missingness probabilities to be bounded away from zero and allows for genuinely sparse missing patterns. Moreover, the proposed procedure requires only slightly stronger than second moments, making it applicable to heavy-tailed distributions where the fourth moment may not exist.
The framework further accommodates weak dependence across variables through geometric alpha-mixing and does not rely on the assumption that the observations are identically distributed, thereby substantially broadening its applicability. We establish the asymptotic normality of the test statistic, along with consistency and correct asymptotic size under suitable conditions.

Algebraic groups seminar
Speaker: S. Velmurugan, IISc, Bengaluru
Host: Shripad Garge
Title: On the Dimensions of Special p-Groups
Time, day and date: 4:00:00 PM - 5:30:00 PM, Monday, March 30
Venue: Mathematics Conference Room
Abstract: In this talk, we consider a problem raised in a recent paper of A. Ayyer and D. Prasad concerning the distribution of conjugacy class sizes of a finite group G and the squares of the dimensions of its irreducible representations. In particular, we study the classification of groups for which the multiset of conjugacy class sizes coincides with the multiset of squares of irreducible character degrees. Abelian groups provide a basic example where this property holds. We then show that there exist many infinite families of non-abelian nilpotent groups exhibiting the same phenomenon. Finally, we conclude by stating a conjecture toward a classification of such groups. This is a joint work in progress with D. Prasad.

Talk
Speaker: Prof. Sourav Pal, IIT Bombay
Title: Rational dilation for operators associated with various domains, von Neumann's inequality and distinguished varieties
Time, day and date: 10:30:00 AM – 11:30:00 AM, Tuesday, March 31
Venue: Ramanujan Hall (https://meet.google.com/dkp-hbsd-rdd)
Abstract: We define spectral set, complete spectral set and rational dilation and study rational dilation of commuting operator tuples associated with various domains in $\mathbb C^n$. We discuss representations of distinguished varieties in several families of domains and find conditions under which von Neumann's inequality holds on such sets.

Seminar
Speaker: Chandan Bhaumik, IIT Bombay
Host: Manoj Keshari
Title: Computations in the Steinberg group
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, April 01
Venue: Ramanujan Hall
Abstract: The Steinberg symbol {-,-} is a fundamental tool for studying the K_2 group. In this talk, I will discuss some computational approaches to computing the Steinberg symbol. If A is a finite field or if A is the ring of integers modulo a power of an odd prime, I will show that all Steinberg symbols are trivial in K_2(A).

Talk
Speaker: Sooraj A P, IIT Bombay
Host: Harsha Hutridurga
Title: Topics in Optimal Transportation
Time, day and date: 2:30:00 PM – 3:30:00 PM, Wednesday, April 01
Venue: Room 105
Abstract: This talk will be an introduction to the Optimal transportation problem and some of its formulations. We shall see some basic motivating examples and, if time permits, sufficient conditions for existence of solutions to these problems.

Commutative Algebra seminar
Speaker: Kaushik Khamari, IIT Bombay
Host: Tony Puthenpurakal
Title: On Cohen-Macaulay Property of Affine Semigroup Rings
Time, day and date: 3:00:00 PM - 4:00:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: In this talk, we discuss the Cohen-Macaulay property of affine semigroup rings. In the process, we describe the relationship between local cohomology modules of these rings and the homology of certain simplicial complexes. As an application, we present a proof of Hochster's Theorem, which asserts that normal semigroup rings are Cohen-Macaulay. We follow the paper of N.V. Trung and L.T. Hoa, "Affine Semigroups and Cohen-Macaulay Rings generated by monomials.

Algebraic groups seminar
Speaker: Patrick Polo, IIT Bombay
Host: Shripad Garge
Title: Properties of Harish Chandra induction and restriction
Time, day and date: 4:00:00 PM - 5:30:00 PM, Thursday, April 02
Venue: Room No .215
Abstract: We will explain how properties like transitivity of induction and Mackey formula rely on structural properties of parabolic subgroups and their unipotent radicals.

Statistics Seminar
Speaker: Savita Pareek, Post doctoral fellow MIT Sloan
Host: Siuli Mukhopadhyay
Title: Semiparametric Dynamic Copula Models using Rolling-window Portfolio Optimization
Time, day and date: 4:30:00 PM – 5:30:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: The mean-variance portfolio model, based on the risk-return trade-off for optimal asset allocation, remains fundamental in portfolio optimization. However, its reliance on restrictive assumptions about asset return distributions limits its applicability to real-world data. Parametric copula structures provide a novel way to overcome these limitations by accounting for asymmetry, heavy tails, and time-varying dependencies. Existing methods have been shown to rely on fixed or static dependence structures, thus overlooking the dynamic nature of the financial market. 
In this study, a semiparametric model is proposed that combines nonparametrically estimated copulas with parametrically estimated marginals to allow all parameters to dynamically evolve over time. A novel framework was developed that integrates time-varying dependence modelling with flexible empirical beta-copula structures. Marginal distributions were modeled using the skewed generalized t-family. This effectively captures asymmetry and heavy tails and makes the model suitable for predictive inferences in real-world scenarios. Furthermore, the model was applied to rolling windows of financial returns from the USA, India, and Hong Kong economies to understand the influence of dynamic market conditions. The approach addresses the limitations of models that rely on parametric assumptions. By accounting for asymmetry, heavy tails, and cross-correlated asset prices, the proposed method offers a robust solution to optimize diverse portfolios in an interconnected financial market. Through adaptive modeling, it allows for better management of risk and return across varying economic conditions, leading to more efficient asset allocation and improved portfolio performance.

Student Seminar
Speaker: Rajendra Meena, IIT Bombay
Host: Suman Kumar Sahoo
Title: The Brahmagupta-Pell Equation: Solutions and Structure
Time, day and date: 5:35:00 PM – 6:35:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: The Brahmagupta–Pell equation x 2 − dy2 = 1, where d is a non-square integer, is a classical Diophantine equation that has fascinated mathematicians for centuries due to its simple form yet surprisingly rich structure. A natural question is whether such an equation admits non-trivial integer solutions, and if so, how one can describe all of them. In this talk, we address these questions by proving the existence of non-trivial solutions and introducing the notion of a fundamental solution, from which all other solutions can be generated. We also describe all solutions explicitly and briefly discuss method for solving the generalized equation x 2 − dy2 = n, where n ∈ Z. The ideas will be illustrated with examples. If time permits, we will also mention the classical Indian” Chakravala method” for computing the fundamental solution.