Urshita Pal

Research interests

I am interested in studying (co)homology of arithemetic groups like SL_nZ, and (co)homological stability patterns in families of groups and topological spaces. Surprisingly, a lot of these questions often boil down to proving high-connectivity of suitably defined simplicial complexes, which can then be done using combinatorial descriptions of these complexes. Overall I tend to enjoy any math that involves topology, algebra and combinatorics (and pretty pictures!)

Talks

1. Configurations, Graphs and Trees, Student Combinatorics, Winter 2024 (notes, abstract)
   
   Configuration Spaces give good prototypical examples to understand the (co)homology of certain groups like braid groups, and the phenomenon of homological stability. In this talk we will explore the topology of configurations of n points on the plane. Through pictures and examples, we will first see how the motion of the particles generates (co)homology classes of these spaces. We will then see how to associate trees and graphs to these classes, and describe a combinatorial pairing between trees and graphs that is analogous to the pairing between cohomology and homology.
2. The Nerve Lemma and Spectral Sequences, Student Topology, Winter 2024 (notes, abstract)
   
   The Nerve Lemma is a useful tool that, under some hypotheses, lets one compute the homology of a space in terms of a simplicial complex built from an open cover of the space. In this talk we shall see the statement of this lemma, a proof glimpse by analyzing a certain double complex, and if time permits, a proof sketch via (a gentle introduction to) spectral sequences.
3. Rational Duality Groups and the Cohomology of SL_nZ, Fall 2023 (notes, abstract)
   
   Certain classes of groups, such as the special linear groups over integers, satisfy a (co)homological duality property that allows one to study their rational cohomology groups. I'll discuss a criterion to test when a group satisfies this duality, and see how in the case of SLnZ this comes down to the fact that the Solomon-Tits building is homotopy equivalent to a wedge of spheres.
4. Configurations, Graphs, and Trees, Student Topology, Fall 2023 (notes, abstract)
   
   In this talk we will explore the topology of configurations of n points in Euclidean space. Through pictures and examples, we will first see how the motion of the particles generates (co)homology classes of these spaces. We will then see how to associate trees and graphs to these classes, and describe a combinatorial pairing between trees and graphs that is analogous to the pairing between cohomology and homology.
5. Introduction to Group (Co)homology, Student Commutative Algebra, Fall 2023 (notes, abstract)
   
   (Co)homology groups are important algebraic invariants associated to a group G. In this talk we will look at how to define group (co)homology algebraically and topologically, prove the equivalence of the two definitions, and briefly see examples where having both perspectives can help study various finiteness properties and duality properties of a group.
6. High Dimensional Cohomology of SLnZ - Part 2, Student Topology, Winter 2023 (notes, abstract)
   
   This the second of a 2 part series on the cohomology of the discrete group SLnZ - In the second talk, we’ll see the virtual cohomological dimension of SLnZ, and a duality result analogous to Poincaré duality, namely Borel-Serre duality, that helps compute higher dimensional cohomologies of SLnZ.
7. High Dimensional Cohomology of SLnZ - Part 1, Student Topology, Winter 2023 (notes, abstract)
   
   This is the first of a 2 part series on the cohomology of the discrete group SLnZ and topological tools used to study it - In this first talk, we'll define the notions of (co)homology of a group and cohomological dimension, and explore these notions in the specific case of SLnZ.
8. Grassmannian Cohomology and Symmetric Polynomials, Student Combinatorics, Winter 2023 (notes, abstract)
   
   Surprisingly, the cohomology ring of a complex grassmannian is isomorphic to a quotient of the ring of symmetric polynomials. While a great result in its own right, understanding this cohomology ring can also help solve problems in enumerative geometry.
   In this talk I will outline a proof of this result, and time permitting, we will also see an application of this in enumerative geometry.
9. A Gentle Introduction to Representation Stability, Student Topology, Fall 2022 (notes, abstract)
   
   Representation Stability is a phenomenon observed in many families of spaces - such as the pure braid groups, flag varieties, etc. - where the (co)homologies of a growing family of spaces stabilize as representations of a group.
   In this talk I will explain this phenomenon by focussing on (pure) braid groups as an example, with an emphasis on lots of pictures and minimal prerequisites.
10. Combinatorial Nullstellensatz and its applications, Student Combinatorics, Winter 2022 (notes, abstract)
    
    The Combinatorial Nullstellensatz is a statement about zeroes of a multi-variable polynomial over a field. It has seen a remarkable number of applications to number theory, enumerative combinatorics, and graph theory. Roughly speaking, it gives quantitative information on how a polynomial of a certain degree cannot vanish over a large enough set of values.
    In this talk I will briefly explain the statement of this theorem, and then illustrate through examples a general technique for using it to prove a variety of powerful results.
11. Braid groups, Student Topology, Fall 2021 ( abstract)
    
    Braid groups are an interesting class of finitely generated groups that come up naturally in a wide range of areas in Mathematics - topology, dynamics, representation theory, cryptography, to name a few.
    Part of their importance is based on the many ways in which they can be defined. We will look at a few different interpretations of braid groups, and discuss some interesting structures on them as well as their applications.

Notes

1. Course Notes for Math 636, on Out(F_n), taught by Alex Wright, Fall 2023. These notes were jointly written by the students and the instructor.
2. Exercises on Combinatorial Topology; I wrote these to supplement the content in A. Bjorner's Chapter on Topological Methods in Combinatorics, specifically Pg 1853-1856. (more)
   This note covers topics like collapsibility, shellability, Cohen-Macaulayness, etc. Some exercises supplement the material, whilst others give sketches of facts that are stated without proof in the notes.
3. Cohomology of the Complex Grassmannian, my term paper from David Speyer's class on Representation Theory of GLnC, Fall 2022.
4. On (Co)homology with Twisted Coefficients. (more)
   I wrote these while studying A. Hatcher's "Algebraic Topolology", Ch3.H, on Local Coefficients. The notes follow Hatcher's treatment in defining (co)homology with twisted coefficients, and give an outline of an analogue of Poincare duality for non-orientable manifolds, which comes at a cost of a "twisting of coefficients".
5. On Bundles of Groups. (more)
   I wrote these while studying A. Hatcher's "Algebraic Topolology", Ch3.H, on Local Coefficients, and these are intended to fill in some details missing from the book. They define a bundle of groups (different from a principal bundle!), give examples and non-examples, and describe pullback bundles. These notes also have a solution sketch to Exercise 3H.3 in the textbook, showing that bundles of groups over a space X with fiber group G are in correspondence with maps from X into B(Aut G) upto homotopy.
6. On K(G,1)-spaces and their uniqueness. I wrote these as I was studying this topic.
7. Branched Covers of the Sphere and Plane, from Malavika Mukundan's summer Minicourse, 2022.