Seminar
Speaker: Ravi Raghunathan, IIT Bombay
Title: The Weak Law, the Strong Law and the Central Limit Theorem or "What I tried to learn last semester".
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, March 02
Venue: Ramanujan Hall
Abstract: If one already knows the language of measure theory, the main theorems of a first course in probability are quite easy to state (and prove). This is what I will do in the lecture. The main difficulty for the uninitiated is the translation of terms used by probabilists into standard terms in measure theory. Thus, this will be a lecture on basic probability shorn of all probabilistic intuition. The lecture should be accessible to anyone with a basic knowledge of measure theory.

Online seminar
Speaker: Dr. Tuhin Majumder
Host: Koushik Saha
Title: Fitting Sparse Markov Models to Categorical Time Series Using Convex Clustering
Time, day and date: 10:00:00 AM – 11:00:00 AM, Thursday, March 05
Venue: Online (https://meet.google.com/wjs-deag-ysh)
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 m 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.

Commutative algebra seminar
Speaker : R.V.Gurjar, TFR  Bombay (Retd.)
Host : Tony Puthenpurakal
Title : Singularities of plane curves II
Time, day and date : 03:00:00 PM – 04:00:00 PM, Thursday, March 05
Venue : Ramanujan Hall
Abstract : We will consider reduced power series f(X,Y) over an algebraically closed field k of characteristic 0. Following topics about R=k[[X,Y]]/(f) will be discussed. Algebraicity results of Artin, Hironaka-Rossi, Samuel, Basic intersection theory of place curves, Conductor ideal, Value semi-group of R, Gorenstein's result, Dedekind's Conductor Formula, Jung's formula connecting the Milnor number and conductor, Applications of Dedekind's formula. We will give Abhyankar's proofs of Dedekind and Gorenstein results. If time permits, Abhyankar-Moh result about generators of the value semi-group will be discussed. 

Algebraic groups seminar
Speaker: Ankita Parashar, IIT Bombay
Host: Shripad Garge
Title: Bruhat decomposition and its applications
Time, day and date: 4:00:00 PM – 5:00:00 PM, Thursday, March 05
Venue: Room No 215
Abstract: We begin the third chapter of the book by Digne and Michel.

Finance Seminar
Speaker: Dr. Himalaya Senapati, AVP at HSBC
Host: Keshav Aggarwal
Title: Introduction to Mathematical Finance: Theory and Practice
Time, day and date: 05:15:00 PM – 06:15:00 PM, Friday, March 06
Venue: Ramanujan Hall
Abstract: We will begin by discussing expectation versus arbitrage-based pricing and introduce the notion of risk-neutral probabilities. We will then present the binomial tree model and use it to price a vanilla call option. We will conclude with a brief overview of the industry landscape and possible career opportunities in mathematical finance.

Statistics and Probability seminar
Speaker: Sagnik Nandy, The Ohio State University, Columbus
Host: Koushik Saha
Title: Degree-Heterogeneous Networks: Optimality, Inference, and Privacy
Time, day and date:  10:00:00 AM – 11:00:00 AM, Tuesday, March 10
Venue: https://meet.google.com/wjs-deag-ysh Online
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 develop a statistical theory for degree-heterogeneous $r$-uniform hypergraphs using the hypergraph $\beta$-model. First, I 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. I then prove that these rates are minimax optimal. Next, I study inference where I characterize minimax detection thresholds for testing the presence of degree heterogeneity.
Finally, I turn to privacy. Using edge differential privacy, I quantify the statistical price of privacy for estimation in $\beta$ models and show a sharp separation between local and central privacy regimes. Simulations illustrate the predicted privacy–utility tradeoffs, and an application to the Enron email hypergraph demonstrates the impact of different regimes of privacy on link prediction in a real organizational communication network.
(Work based on joint papers with B. Bhattacharya and B. Mandal.)

Analysis Seminar
Speaker: Dr. Pritam Ganguly, ISI Kolkata
Host: Santanu Dey
Title:  From Uncertainty to Rellich-type estimates: A journey through $ mathbb{R}^n$, the Heisenberg group, and symmetric spaces.
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: In this talk, we explore two classical yet evolving themes in analysis: uncertainty principles and Rellich-type estimates for eigenfunctions of the Laplacian. We begin with a theorem of Ingham, which characterizes the optimal decay of the Fourier transform of a compactly supported function on the real line. This result captures a sharp form of the (qualitative) uncertainty principle. We then examine an analogue in the setting of the Heisenberg group by investigating the optimal operator-valued decay of the group Fourier transform of compactly supported functions.

In the second part, we turn to a classical result of Rellich concerning solutions of the Helmholtz equation, a fundamental equation in mathematical physics. In his 1943 paper, Rellich proved that in the exterior region $\Omega = \{x \in \mathbb{R}^n \mid |x| > R_0\},$ there are no nontrivial solutions $ f $ of the Helmholtz equation $\Delta f +\lambda f=0, \qquad \lambda > 0,$ such that $ f \in L^2(\Omega) $. His proof is based on establishing the $L^2$-asymptotic growth of solutions over annuli. We discuss this result and its extension to $L^p$ in the context of rank one Riemannian symmetric spaces of non-compact type, highlighting how exponential volume growth and the dependence of the $L^p$-spectrum of the Laplace--Beltrami operator on $p$ lead to genuinely non-Euclidean phenomena.

Statistics and Probability seminar
Speaker: Sourish Das, Chennai Mathematical Institute
Host: Parthanil Roy
Title: Jacobi Prior: An Alternative Bayesian Method for Supervised Learning
Time, day and date: 02:30:00 PM – 03:30:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: The Jacobi prior offers an alternative Bayesian framework for predictive modelling, designed to achieve superior computational efficiency without compromising predictive performance. This scalable method is suitable for image classification and other computationally intensive tasks. Compared to widely used methods such as Lasso, Ridge, Elastic Net, uniLasso, the MCMC-based Horseshoe prior, and non-Bayesian machine learning methods including Support Vector Machines (SVM), Random Forests, and Extreme Gradient Boosting (XGBoost), the Jacobi prior achieves competitive or better accuracy with significantly reduced computational cost. The method is well-suited to distributed computing environments, as it naturally accommodates partitioned data across multiple servers. We propose a parallelisable Monte Carlo algorithm to quantify the uncertainty in the estimated coefficients. We establish that the Jacobi estimator is asymptotically close to, and asymptotically equivalent to, the posterior mode under the Jacobi prior.  To demonstrate its practical utility, we conduct a comprehensive simulation study comprising seven experiments focused on statistical consistency, prediction accuracy,  scalability, sensitivity analysis and robustness study. We further present three real-data applications: credit risk modelling using U.S. Small Business Administration (SBA) loan default data, multi-class classification of stars, quasars, and galaxies using Sloan Digital Sky Survey data, and spinal degeneration classification using sagittal MRI scans from the RSNA 2024 Lumbar Spine Degenerative Classification Challenge. In the spine classification task, we extract last-layer features from a fine-tuned ResNet-50 model and evaluate multiple classifiers, including Jacobi-Multinomial logit regression, SVM, and Random Forest. The Jacobi prior achieves state-of-the-art results in recall and predictive stability, especially when paired with domain-specific features. In addition, it substantially outperforms competing methods in runtime by large margins, leading to significant reductions in computational cost. This highlights its potential for scalable, high-dimensional learning in medical image analysis.

All code and datasets used in this paper are available at: 
https://github.com/sourish-cmi/Jacobi-Prior/ 
 

Mathematics Colloquium
Speaker: Sabyasachi Mukherjee, Tata Institute of Fundamental Research
Host: Parthanil Roy
Title: Fatou-Sullivan dictionary: Where rational dynamics meets Kleinian groups
Time, day and date: 04:00:00 PM – 05:00:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: Two central themes in holomorphic dynamics are the iteration of rational maps and the action of Kleinian groups on the Riemann sphere. Although these theories developed along largely independent paths, they exhibit striking conceptual parallels—a phenomenon often referred to as Sullivan’s Dictionary. As early as the 1920s, Fatou envisioned that these two classes of dynamical systems could be unified within the framework of iterated algebraic correspondences.

After surveying fundamental results in rational dynamics and Kleinian groups, we will present concrete realizations of Fatou’s vision by constructing algebraic correspondences that unite the behaviors of complex polynomials and Fuchsian groups in a single dynamical plane. We will also discuss examples of rational Julia set realizations of Kleinian limit sets, such as classical Apollonian-like gaskets. Timepermitting, we will outline the main analytic and algebraic ideas underlying this program, and highlight several applications to other areas of mathematics.

Topology seminar
Speaker: Bittu Singh, IIT Bombay
Host: Rekha Santhanam
Title: Rational Homotopy theory
Time, day and date: 11:45:00 AM – 1:00:00 PM, Thursday, March 12
Venue: Room 215
Abstract: Dennis Sullivan introduced a framework in which the rational homotopy type of a topological space is encoded by commutative differential graded algebras (CDGAs). Earlier, Daniel Quillen established an equivalence between the rational homotopy category and an algebraic category, although this equivalence arises through a sequence of intermediate categorical constructions. In their paper “On PL De Rham Theory and Rational Homotopy Type”, A. K. Bousfield and Victor K. A. M. Guggenheim gave a precise formulation of Sullivan’s equivalence using the language of model category theory.
In this talk, we outline this approach and discuss the relationship between topological spaces and CDGAs in rational homotopy theory.

Commutative algebra seminar
Speaker: Prof. R. V. Gurjar
Host: Tony J P
Title: Singularities of plane curves III
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, March 12
Venue: Ramanujan Hall
Abstract: We will consider reduced power series f(X,Y) over an algebraically closed field k of characteristic 0. Following topics about R=k[[X,Y]]/(f) will be discussed. Algebraicity results of Artin, Hironaka-Rossi, Samuel, Basic intersection theory of place curves, Conductor ideal, Value semi-group of R, Gorenstein's result, Dedekind's Conductor Formula, Jung's formula connecting the Milnor number and conductor, Applications of Dedekind's formula. We will give Abhyankar's proofs of Dedekind and Gorenstein results. If time permits, Abhyankar-Moh result about generators of the value semi-group will be discussed.

Algebraic groups seminar
Speaker: Hariom Sharma, IIT Bombay
Host: Shripad Garge
Title: Subgroups of maximal rank
Time, day and date: 04:00:00 PM – 05:00:00 PM, Thursday, March 12
Venue: Room no. 215
Abstract: We continue with the third chapter of Digne Michel and study quasi-closed subsets of root systems.

Geometry and Topology seminar
Speaker: Dr. Gorapada Bera, Simons Center for Geometry and Physics, Stony Brook University
Host: Sandip Singh
Title: Associative submanifolds of G₂-manifolds and adiabatic limits
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, March 16
Venue: Online (Google Meet: https://meet.google.com/jzy-kqch-nzy)
Abstract: G₂-manifolds are 7-dimensional Riemannian manifolds with special holonomy, analogous to Calabi–Yau 3-folds, and associative submanifolds are a special class of volume-minimizing 3-dimensional submanifolds, analogous to holomorphic curves or special Lagrangians. Inspired by the counting of holomorphic curves or special Lagrangians in Calabi–Yau 3-folds, Joyce, Doan, and Walpuski have proposals about defining enumerative invariants of G₂-manifolds by counting closed associative submanifolds. In this talk, I will focus on three topics connected to this enumerative theory. First, I will discuss a key ingredient of these proposals: the appearance of conically singular associative submanifolds and their desingularizations. Afterwards, I will describe a method for constructing associative submanifolds in a class of G₂-manifolds known as twisted connected sums. Finally, I will discuss Donaldson’s program on the adiabatic limit of K3-fibered G₂-manifolds, along with conjectures regarding associative submanifolds and potential directions for future research.

Geometry and Topology seminar
Speaker: Dr. Ishan Banerjee, Ohio State University
Host: Sandip Singh
Title: Monodromy of curves in an algebraic surface
Time, day and date: 9:30:00 AM - 10:30:00 AM, Tuesday, March 17
Venue: Online (Google Meet: https://meet.google.com/jzy-kqch-nzy)
Abstract: Given an algebraic surface X embedded into P^N, we can consider the family of smooth curves in X arising from intersecting a degree d hypersurface in P^N with X. This is a fiber bundle with fibers Riemann surfaces of genus g for some g.

Associated to this family of curves we have a monodromy group contained in the mapping class group of a genus g surface. We prove under some ampleness conditions that if X is simply connected this monodromy group is of finite index (we explicitly determine what it is). This is joint work with Nick Salter.

Talk
Speaker: RUDDARRAJU AMRUTHA
Host: Manoj Kumar Keshari
Title: Elementary Symplectic Groups and their Generalizations
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A.A. Suslin proved that the elementary linear group of size atleast 3 is a normal subgroup of the group of invertible matrices. Suslin also proved a relative version of this result w.r.t. an ideal. V.I. Kopeiko proved a symplectic analogue of this result: the elementary symplectic group is a normal subgroup of the symplectic group, where both are defined with respect to the standard skew-symmetric matrix. I will talk about a generalization of Kopeiko's result with respect to any invertible skew-symmetric matrix of Pfaffian 1.

A stronger result holds when the ring is Euclidean or semi-local. Over a Euclidean ring, the elementary symplectic group with respect to an invertible skew-symmetric matrix is equal to the symplectic group with respect to the skew-symmetric matrix. This result also holds for a semi-local ring. I will talk about these equality results.

Finally, I will talk about a generalization of the elementary symplectic group with respect to an invertible alternating matrix, in the case of a projective module, which is the Vaserstein group of a symplectic module.
This Vaserstein group is a normal subgroup of the group of isometries of the symplectic module.

Statistics and Probability seminar
Speaker: Dr. Abhinek Shukla, Centre for Biomedical Data Science, Duke-NUS Medical School
Host: Koushik Saha
Title: Improving Inference in Stochastic Gradient Descent Based on Equal Batch-Size Batch-Means Estimator
Time, day and date: 12:00:00 PM - 1:00:00 PM, Wednesday, March 18
Venue: Online (https://meet.google.com/wjs-deag-ysh)
Abstract: Stochastic gradient descent (SGD) is a widely used technique for solving various optimization problems ranging from regression on a large dataset employed in statistics, to training deep neural networks for high dimensional models in machine learning. Performing tasks such as inference in SGD is a challenging problem due to its time-inhomogeneous Markovian nature. An underlying asymptotic normality of the averaged SGD (ASGD) estimator allows for the construction of a batch-means estimator of the asymptotic covariance matrix. In contrast to the existing increasing batch-size (IBS) strategy proposed for reducing correlation between far-apart batches, we propose a memory efficient equal batch-size (EBS) estimator and show consistency of the proposed batch-means estimator under mild conditions. The proposed EBS technique offers bias-correction of the variance at no additional cost to memory and is also shown to outperform the IBS estimator in extensive simulations. Further, since joint inference for high dimensional problems may be difficult, we present marginal-friendly simultaneous confidence intervals, and demonstrate improved predictions based on the proposed covariance estimators of ASGD.

Monge-Ampere equation
Speaker: Sooraj A P, IIT Bombay
Host: Harsha Hutridurga
Title: On the Monge-Ampere equation
Time, day and date: 2:00:00 PM – 3:00:00 PM, Wednesday, March 18
Venue: Room 215
Abstract: We shall show the existence and Hölder regularity of classical solutions to the Monge-Ampère equation using tools from elliptic regularity theory.

Topology Seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Mapping Class Groups: Classical Foundations
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, March 18
Venue: Room 215
Abstract: This talk provides an introduction to the mapping class group, an object that algebraically captures the topological symmetries of a surface by considering its orientation-preserving homeomorphisms up to isotopy. The study of these groups naturally bridges fields such as low-dimensional topology, algebraic geometry, algebraic topology, and dynamical systems. In this series of talks, we will discuss some of the connections between mapping class groups and these areas. The first talk will cover the formal definition, various fundamental properties, and the generation of mapping class groups.

Mathematics Colloquium
Speaker: Sudarshan R. Gurjar, IIT Bombay
Title: Belyi-type theorems for vector bundles
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A classical theorem of Belyi states that a non-singular, irreducible, complex projective curve is defined over number field if and only if it admits a non-constant morphism to CP^1 which is branched over at most 3 points. I will discuss analogues theorems for vector bundles and vector bundles with connections and Higgs fields. 

This is based on some joint work with Indranil Biswas.

CACAAG seminar
Speaker: Madhusudan Manjunath, IIT Bombay    
Title: A Gentle Introduction to Non-Archimedean Analytic Geometry à la Berkovich
Time, day and date: 05:30:00 PM – 6:30:00 PM, Wednesday, March 18
Venue: Ramanujan Hall
Abstract: A gentle introduction to Berkovich spaces with an emphasis on intuition, motivations and examples. 

Talk
Speaker: Terrence George, MIT
Host: Niranjan Balachandran
Title: Limit shapes beyond classical tilings
Time, day and date: 9:30:00 AM – 10:30:00 AM, Friday, March 20
Venue: Online (meet.google.com/dyx-hzmn-cbj)
Abstract: The dimer model refers to the study of random dimer covers, or perfect matchings, of a bipartite graph. A remarkable feature of these models is the emergence of limit shapes: in large periodic graphs, a random matching typically concentrates around a deterministic shape. Although limit shapes arise quite generally in dimer models, they are explicitly understood only in two classical cases, namely the hexagonal and square lattices, corresponding to lozenge and domino tilings. In this talk, I will explain how ideas from integrable systems make it possible to compute limit shapes for arbitrary periodic dimer models. This is based on joint work in progress with Tomas Berggren and Alexei Borodin.

Algebraic groups seminar
Speaker: Sabyasachi Dhar, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Parabolic subgroups and Levi subgroups
Time, day and date: 4:00:00 PM - 5:30:00 PM, Friday, March 20
Venue: Room No .215
Abstract: We continue with the remaining part of Chapter 3 of Digne-Michel.

Algebraic groups seminar
Speaker: Patrick Polo, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Preparations for the Harish-Chandra induction
Time, day and date: 5:30:00 PM - 6:30:00 PM, Friday, March 20
Venue: Room No .215
Abstract: We cover preliminaries for the Harish-Chandra induction.

IPDF Talk
Speaker: Dr. Kanoy Kumar Das
Host: Ananthnarayan Hariharan
Title: Square-free powers of Cohen--Macaulay simplicial forests
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, March 23
Venue: Online (https://meet.google.com/kof-mymh-vim)
Abstract: In this talk, we study square-free powers of facet ideals of simplicial complexes, a notion that lies at the intersection of combinatorial commutative algebra and matching theory. A fundamental contrast motivates our work: while ordinary powers of a graded radical ideal are rarely Cohen--Macaulay (except in the complete intersection case), square-free powers exhibit significantly different behavior.

Given a simplicial complex $\Delta$, let $I(\Delta)^{[k]}$ denote the $k$-th square-free power of its facet ideal in a polynomial ring $R$. In this talk, we show that if $\Delta$ is a Cohen--Macaulay simplicial forest, then the quotient $R/I(\Delta)^{[k]}$ is Cohen--Macaulay for all $k \geq 1$.

Our approach is primarily combinatorial, complemented by tools from homological algebra. For a Cohen--Macaulay simplicial complex $\Delta$, we derive explicit formulas for $\mathrm{depth}(R/I(\Delta)^{[k]})$ and $\dim(R/I(\Delta)^{[k]})$ for all $k \geq 1$. We introduce a new structural notion, called a \emph{special leaf}, which enables us to control the behavior of the depth of square-free powers. As a consequence, we show that the normalized depth function associated to a Cohen--Macaulay simplicial forest is non-increasing.

This talk is based on a recent work with A. Roy and K. Saha.

Combinatorics seminar
Speaker: Jagdeep Singh, Mississippi State University
Host: Niranjan Balachandran
Title: From Cographs to Comatroids: Forbidden Structures and Apex Classes
Time, day and date: 10:00:00 AM - 11:00:00 AM, Wednesday, March 25
Venue: Online (https://meet.google.com/pcb-jarm-pvq)
Abstract: Complement reducible graphs, or cographs, form one of the most well-understood hereditary classes of graphs: they admit linear-time recognition, possess multiple elegant characterizations, and have a simple forbidden subgraph description (no induced path of length three). In this talk, I will describe two natural generalizations of cographs: 2-cographs and sesquicographs. I will then present results showing that apexing (adding a single vertex or edge) preserves the finiteness of forbidden induced subgraphs, along with explicit bounds on their size. Finally, I will extend this framework from graphs to binary matroids, introducing comatroids as the natural matroid analogue of cographs. Analogous extension results hold in this setting. This work includes joint work with James Oxley, Thomas Zaslavsky and Vaidy Sivaraman.

Link for the talk: https://meet.google.com/pcb-jarm-pvq

Talk
Speaker: Chandan Bhaumik, IIT Bombay
Host: Manoj Kumar Keshari
Title: The Steinberg Group and the K_2 Functor
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, March 25
Venue: Ramanujan Hall
Abstract: This series of talks focuses on the computation of the K_2 group for the ring of integers. In the first talk, we will cover the definition of the Steinberg group and the K_2 functor, along with some fundamental properties.

PDE Seminar
Speaker: Saroj Si, IIT Roorkee
Host: Harsha Hutridurga
Title: On the well-posedness of the growth–coagulation models with singular coagulation kernels
Time, day and date: 2:30:00 PM – 3:30:00 PM, Wednesday, March 25
Venue: Ramanujan Hall
Abstract: Attached (https://drive.google.com/drive/folders/1SlhC3HhNTTBJxX2b1Sbvkbck7EKwLUyJ)

Mathematics Colloquium
Speaker: Keshav Aggarwal, IIT Bombay
Title: Cancellations in sums of arithmetic functions
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, March 25
Venue: Ramanujan Hall
Abstract: Many important number theoretic problems relate to the rate of growth of sums of arithmetically interesting functions. Some of these problems include bounding the least quadratic non-residue, Quantum Unique Ergocidity conjecture of Rudnick and Sarnak, bounding moments of L-functions, simultaneous non-vanishing of L functions and the subconvexity bound problem. We will give an overview of how various such problems relate to obtaining cancellations in certain sums, and the recent methods developed in this direction.

Combinatorics seminar
Speaker: Himanshu Gupta, University of Regina
Host: Niranjan Balachandran
Title: Expanders, Ramanujan Graphs, and Inverse Eigenvalue Problems for Graphs
Time, day and date: 10:00:00 AM - 11:00:00 AM, Thursday, March 26
Venue: Online (https://meet.google.com/cua-knbe-fzr)
Abstract: Spectral graph theory studies the relationship between the structure of a graph and the eigenvalues of matrices associated with it. Expanders are a special class of sparse yet highly connected graphs that play an important role in computer science and mathematics. The spectral gap quantifies expansion, and Ramanujan graphs are those achieving the optimal bound. In the first part of this talk, we discuss expanders, spectral gap, Ramanujan graphs, and spectra of Cayley graphs. We then turn to the family of graphs introduced by
Lazebnik and Ustimenko in 1995, which play a central role in extremal graph theory providing the best known examples for general lower bounds on the size of a graph for given order and girth. An important conjecture by Ustimenko asserts that this family of graphs is almost Ramanujan. We present recent progress on this conjecture in joint work with Taranchuk.
In the second part, we focus on the inverse eigenvalue problem for
graphs, which seeks to determine all spectra realizable by matrices whose off-diagonal zero/nonzero pattern matches edges/nonedges of a given graph. The Laplacian matrix of a graph is a fundamental object in combinatorics. We introduce the inverse eigenvalue problem for Laplacian matrices of a graph and present results on realizable spectra for several graph families, based on joint work with Catral, Fallat, and Lin.

Topology Seminar
Speaker: Aparajita Karmakar, Postdoc IITB
Host: Rekha Santhanam
Title: Mackey Functors: Motivations from Equivariant Topology
Time, day and date: 11:45:00 AM – 12:45:00 PM, Thursday, March 26
Venue: Room 215
Abstract: Classical (co)homology theories, as studied in algebraic topology, assign algebraic invariants to spaces that capture their global structure. However, many spaces of interest come equipped with symmetries, encoded by a group action. This leads to the setting of equivariant topology. In this talk, we begin by explaining why equivariant theories cannot take values in ordinary abelian groups, but instead require richer structures encoding restriction and transfer maps between subgroups.This leads to the notion of Mackey functors, which arise naturally as coefficient systems for equivariant homology and cohomology theories. We will also talk about the Burnside category, highlighting its role as the correct domain governing equivariant phenomena.

Commutative Algebra seminar
Speaker: R. V. Gurjar, TIFR(retd)
Host: Tony Puthenpurakal
Title: Singularities of plane curves IV
Time, day and date: 3:00:00 PM - 4:00:00 PM, Thursday, March 26
Venue: Ramanujan Hall
Abstract: We will consider reduced power series f(X,Y) over an algebraically closed field k of characteristic 0. Following topics about R=k[[X,Y]]/(f) will be discussed.
Algebraicity results of Artin, Hironaka-Rossi, Samuel, Basic intersection theory of place curves, Conductor ideal, Value semi-group of R, Gorenstein's result, Dedekind's Conductor Formula, Jung's formula connecting the Milnor number and conductor, Applications of Dedekind's formula. We will give Abhyankar's proofs of Dedekind and Gorenstein results. If time permits, Abhyankar-Moh result about generators of the value semi-group will be discussed. 

Student Seminar
Speaker: Om Milind Joglekar, IIT Bombay
Host: Suman Kumar Sahoo
Title: Two Guesses Too Few: The Combinatorics of the NYT Connections Endgame
Time, day and date: 4:00:00 PM – 5:00:00 PM, Thursday, March 26
Venue: Ramanujan Hall
Abstract: You’ve played the New York Times staples like Wordle and the Crossword, but Connections introduces a unique mathematical hurdle. We’ve all been there - staring at the final eight tiles, knowing purple is the most annoying category ever created. In this talk we will demonstrate a few algorithms and strategies which help you solve the puzzle with optimal probability. We will also show that the given number of lives is just 2 shy of the number of lives you need to brute force the solution.

Algebraic groups seminar
Speaker: Patrick Polo, IIT Bombay, Mumbai
Host: Shripad Garge
Title: Harish-Chandra induction
Time, day and date: 4:00:00 PM - 5:30:00 PM, Thursday, March 26
Venue: Room No .215
Abstract: We introduce the Harish-Chandra induction.

Geometry and Topology seminar
Speaker: Dr. Sudipta Ghosh, University of Notre Dame
Host: Sandip Singh
Title: Rational homology spheres and SU(2)-SL(2,C) representations
Time, day and date: 5:30:00 PM - 6:30:00 PM, Thursday, March 26
Venue: Online (Google Meet: https://meet.google.com/jzy-kqch-nzy)
Abstract: The Poincaré Conjecture, proved by Perelman, states that every closed, connected three-manifold other than S³ has nontrivial fundamental group. A related question, Problem 3.105(A) in Kirby’s list, asks whether any three-manifold other than S³ admits a nontrivial homomorphism from π₁(Y) to SU(2); this problem remains open. Using instanton Floer homology, Zentner showed that for integer homology spheres, the analogous statement holds when SU(2) is replaced by SL(2,C). In this talk, I will discuss recent progress on the existence of irreducible SL(2,C) and SU(2) representations for rational homology spheres. Some of these results are joint with Steven Sivek and Raphael Zentner, others with Mike Miller Eismeier, and others with Zhenkun Li and Juanita Pinzón-Caicedo.

History of Mathematics Seminar
Speaker: Jasbir S. Chahal, Brigham Young University, Provo, Utah, USA
Host: Sudhir R. Ghorpade
Title: History and solutions of some number theory problems
Time, day and date: 4:00:00 PM - 5:00:00 PM, Friday, March 27
Venue: Ramanujan Hall
Abstract: This is mostly a historical talk. It is aimed at a general audience, in particular students aiming to specialize in number theory. Some of these problems go back all the way to antiquity. I will explain how they have attracted the best minds ever since and are still doing so.

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.