Calendar

Posted September 27, 2024
Last modified October 16, 2024

Mathematical Physics and Representation Theory Seminar

Xinchun Ma, University of Chicago
Cherednik algebras, Torus knots and flag commuting varieties

In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, arising from cuspidal character D-modules. In describing this family of coherent sheaves, the geometry of nilpotent flag commuting varieties naturally emerges, closely related to the compactified regular centralizer in type A.

Posted September 27, 2024
Last modified October 24, 2024

Mathematical Physics and Representation Theory Seminar

Nikolay Grantcharov, University of Chicago
Infinitesimal structure of BunG

Given a semisimple group G and a smooth projective curve X over an algebraically closed field of arbitrary characteristic, let Bun_G(X) denote the moduli space of principal G-bundles over X. For a bundle P without infinitesimal symmetries, we describe the n^th order divided-power infinitesimal jet spaces of Bun_G(X) at P for each n. The description is in terms of differential forms on the Fulton-Macpherson compactification of the configuration space, with logarithmic singularities along the diagonal divisor. We also briefly discuss applications into constructing Hitchin's flat connection on the vector bundle of conformal blocks.

Posted March 9, 2025

Mathematical Physics and Representation Theory Seminar

David Boozer, Indiana University
Student Seminar on Instanton Homology and Foam Evaluations

This is to help prepare graduate students for David Boozer's talk at 2:30pm on the same day. He will discuss some of the basic definitions behind his 2:30pm talk and take questions from graduate students on the objects of study in his talk.

Posted February 10, 2025
Last modified February 24, 2025

Mathematical Physics and Representation Theory Seminar

David Boozer, Indiana University
The combinatorial and gauge-theoretic foam evaluation functors are not the same

Kronheimer and Mrowka have outlined a new approach that could potentially lead to the first non-computer based proof of the four-color theorem. Their approach relies on a functor J-sharp, which they define using gauge theory, from a category of webs in R^3 to the category of finite-dimensional vector spaces over the field of two elements. They have also suggested a possible combinatorial replacement J-flat for J-sharp, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor J-sharp to the subcategory of planar webs is not the same as J-flat.

Posted February 10, 2025
Last modified March 9, 2025

Mathematical Physics and Representation Theory Seminar

Maarten Mol, University of Toronto
Constructibility of momentum maps and variation of singular symplectic reduced spaces

Proper maps in various categories studied in singularity theory (for example, the real analytic category) are known to be constructible, in the sense that the image of the map can be stratified in such a way that the map is a topological fiber bundle over each stratum. Such stratifications provide insight into how the fibers of the map vary. In this talk we will discuss the existence of such a stratification for momentum maps of Hamiltonian Lie group actions (a natural class of maps studied in symplectic/Poisson geometry), which provides insight into how the so-called symplectic reduced spaces of the Hamiltonian action vary. Along the way we will also try to give an overview of some more classical results on the geometry of such maps.

Posted February 10, 2025
Last modified March 13, 2025

Mathematical Physics and Representation Theory Seminar

Sam Gunningham, Montana State University
Geometric Satake Revisited

The geometric Satake equivalence is a fundamental result in the geometric Langlands program. It can be understood as a kind of Fourier transform, relating different flavors of sheaves on a dual pair of spaces. Just like the usual Fourier transform, the equivalence exchanges the structures of convolution and pointwise product on each side. In this talk, I will discuss a circle of ideas relating pointwise tensor product of sheaves on the affine Grassmannian, the Knop-Ngo action for the group scheme of regular centralizers, and Moore-Tachikawa varieties. This builds on past joint work with D. Ben-Zvi and some current work in progress with D. Ben-Zvi and S. Devalapurkar.

Posted March 16, 2025
Last modified April 2, 2025

Mathematical Physics and Representation Theory Seminar

Justin Lanier, University of Sydney
Twisting cubic rabbits

A polynomial can be viewed as a branched cover of the sphere over itself that is compatible with a complex structure. If handed a topological branched cover of the sphere, we can ask whether it can arise from a polynomial, and if so, which one? In 2006, Bartholdi and Nekrashevych used group theoretic methods to explicitly solve this problem in certain special cases, including Hubbard’s twisted rabbit problem. Using a combinatorial topology approach that draws from the theory of mapping class groups, we solve an infinite family of twisted polynomial problems that are cubic generalizations of Hubbard’s twisted rabbit problem. This is joint work with Becca Winarski.

Posted February 10, 2025
Last modified April 14, 2025

Mathematical Physics and Representation Theory Seminar

Joshua Mundinger, University of Wisconsin
Hochschild homology of algebraic varieties in characteristic p

Hochschild homology is an invariant of noncommutative rings. When applied to a commutative ring, the Hochschild-Kostant-Rosenberg theorem gives a formula for Hochschild homology in terms of differential forms. This formula extends to the Hochschild-Kostant-Rosenberg decomposition for complex algebraic varieties. In this talk, we quantitatively explain the failure of this decomposition in positive characteristic.

Posted October 6, 2025
Last modified October 15, 2025

Mathematical Physics and Representation Theory Seminar

John O'Brien, Louisiana State University
The Splitting-Rank Derived Satake Equivalence

This talk is based on joint work with Tsao-Hsien Chen, Mark Macerato, and David Nadler. We discuss a generalization of Bezrukavnikov-Finkelberg's Derived Satake Equivalence from complex reductive groups to certain real reductive groups--or equivalently, from compact Lie groups to the corresponding symmetric spaces. We use Nadler's Real Geometric Satake to compute the equivariant cohomology of the based loop space of a splitting-rank symmetric space, then use Achar's parity-vanishing machinery to establish the equivalence of derived categories.

Posted October 15, 2025

Mathematical Physics and Representation Theory Seminar

Paul Sobaje, Georgia Southern University
A Geometric Model For Steinberg Quotients

Let G be a reductive algebraic group over a field of characteristic p > 0. Over the last decade, the longstanding search for a character formula for simple G-modules has been replaced (subsumed even) by the same problem for characters of tilting G-modules. In recent years I began studying "Steinberg quotients" of certain tilting characters. These are formal characters with good combinatorial properties straightforwardly derived from the representation theory of G. In some ways they are also the best candidates to be described by a characteristic p version of Weyl's famous formula. In joint work with P. Achar, we prove that these formal characters are in fact actual characters of a natural class of objects coming from geometric representation theory.