Urshita Pal

Invited Talks

1. Steinberg Modules and Hopf Algebras (expository); University of Pennsylvania; March 2025 (notes, abstract)
   This talk aims to give an exposition of recent work of Ash-Miller-Patzt, establishing a Hopf algebra structure on the cohomology groups H^k(GLnZ; Q). These groups are of interest in several areas of mathematics but have proven difficult to compute. I will discuss how they can be studied via "Steinberg modules", describe how this can be used to put a Hopf algebra structure on them, and how this helps to find new cohomology classes.
2. Steinberg Modules and the Cohomology of SL_nZ (expository); AIM Workshop; July 2024; Caltech, Pasedena (notes)

Workshop Talks

1. High Connectivity of Disordered Arc Complexes (expository); Talbot Workshop, May 2025 (notes, abstract)
   We discuss the full arc complex and the complex of disordered arcs on an oriented surface. We will use Hatcher's flow argument to show that the full arc complex is contractible, and from this deduce high connectivity of the disordered arc complex using a "badness argument" with simplicial complexes that is standard in the literature. This high connectivity can be used to deduce homological stability of surface mapping class groups.

Learning Seminar Talks

1. (Co)homology of Euclidean Configuration Spaces, a 5-day minicourse I taught in August 2025, as part of the grad student summer minicourse program at Michigan. (notes, syllabus)
   We shall introduce homological and representation stability by studying the topology of configuration spaces.
   Talk 1 and 2 give an in-depth look at configuration spaces of the plane, and a beautiful combinatorial description for their (co)homology using graphs and trees.
   Talk 3 discusses the phenomenon of homological stability and see why these configuration spaces exhibit such patterns.
   Talk 4 covers an extremely useful "scanning argument" which can help calculate the stable homology in such cases.
   And finally, Talk 5 gives an introduction to the little disks operad, its connection with configuration spaces, and a few remarks about the similarities and differences between configurations of points in the plane versus in higher dimensional Euclidean spaces.
2. Discrete Morse Theory, Student Combinatorics, Winter 2025 (notes, abstract)
   Discrete Morse Theory is a tool to study the homotopy type of a simplicial complex, via "nice" functions on it. It gives an algorithm for collapsing certain simplices without changing the homotopy type, thus simplifying the cell structure of the simplicial complex. This talk aims to give an overview of this technique, covering several examples along the way.
3. Hopf Algebras, Steinberg Modules, and Cohomology of GL_nZ, Student Topology, Winter 2025 (notes, abstract)
   This talk aims to give a gentle exposition of recent work establishing an algebraic structure on the groups H^k(GL_nZ; Q). These cohomology groups are of interest in several areas of mathematics, and computing them has proven extremely difficult. One approach to studying these groups is via "Steinberg modules", which are defined in terms of certain simplicial complexes. After giving a brief introduction to these objects, I will describe a combinatorial way to put a product and coproduct structure on the Steinberg modules, and discuss recent work of Ash-Miller-Patzt that used this to deduce a Hopf algebra structure on the cohomology H^k(GL_nZ; Q), and describe how this helps in finding new cohomology classes.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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 the symmetric groups.
    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.
13. 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.
14. 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. (Co)homology of Euclidean Configuration Spaces, notes from a 5-day minicourse I taught in August 2025, as part of the grad student summer minicourse program at Michigan. (Syllabus)
   We shall introduce homological and representation stability by studying the topology of configuration spaces.
   Talk 1 and 2 give an in-depth look at configuration spaces of the plane, and a beautiful combinatorial description for their (co)homology using graphs and trees.
   Talk 3 discusses the phenomenon of homological stability and see why these configuration spaces exhibit such patterns.
   Talk 4 covers an extremely useful "scanning argument" which can help calculate the stable homology in such cases.
   And finally, Talk 5 gives an introduction to the little disks operad, its connection with configuration spaces, and a few remarks about the similarities and differences between configurations of points in the plane versus in higher dimensional Euclidean spaces.
2. High Connectivity of the Curve Complex, a full exposition of Harer's proof (with one intermediate simplifying step due to Brendle-Broaddus-Putman) of high connectivity of the curve complex of an orientable surface.
3. (Co)homology of (Ordered) Euclidean Configuration Spaces, an exposition of Dev Sinha's article The Homology of the Little Disks Operad. (Description)
   This note first gives an overview of the combinatorial description of the (co)homology of these configuration spaces using graphs and trees, and then dives into some proofs and ideas behind these computations.
4. 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.
5. 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. (Description)
   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 Bjorner's notes.
6. Cohomology of the Complex Grassmannian, my term paper from David Speyer's class on Representation Theory of GLnC, Fall 2022.
7. On (Co)homology with Twisted Coefficients. (Description)
   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".
8. On Bundles of Groups. (Description)
   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.
9. On K(G,1)-spaces and their uniqueness. I wrote these as I was studying this topic.
10. Branched Covers of the Sphere and Plane, from Malavika Mukundan's summer Minicourse, 2022.
11. A Point-Set Topology Problem on the Ordered Square. This was a problem I proposed for a point-set course I TA-ed back in CMI in 2021. (Description)
    The problem was about constructing a subspace of the "ordered square" (i.e., the square [0,1]^2 equipped with the the lexicographical/dictionary order topology) that was path-connected. The proof of path connectivity mimicked the proof idea of Urysohn's Lemma. Both Urysohn's Lemma and the ordered square had been recently introduced in the class at the time, so I wanted to write a problem whose solution combined both these concepts.