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.