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.