Calendar

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
Last modified October 16, 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.

Posted November 30, 2025

Mathematical Physics and Representation Theory Seminar

Iain Moffatt, Royal Holloway, University of London
Graph Theory Graduate Student Q&A and Lunch

Bring your lunch to the third floor lounge to interact with Iain Moffatt before his talk (at 1:30pm). All are welcome--professors, students, postdocs--but this question and answer period is mainly for graduate students. Iain will speak about the basic objects of his talk--hypermaps. These are combinatorial structures that encode embeddings of (generalized) graphs on surfaces, and they appear naturally in areas ranging from topological graph theory to algebraic geometry and low-dimensional topology. In this informal session he will introduce hypermaps, draw plenty of pictures, and explain how they arise in current research. The goal is to build intuition for the main talk, so questions are encouraged and no prior background in hypermaps or surface embeddings will be assumed.

Posted November 12, 2025

Mathematical Physics and Representation Theory Seminar

Iain Moffatt, Royal Holloway, University of London
Hypermap minors

As mathematicians we conventionally model networks as graphs. In a graph, each edge has exactly two ends, each lying on a vertex. Hypergraphs generalise graphs by allowing an edge to have any number of ends. As the edges of a hypergraph can connect any number of vertices, not just two, they offer a way to model higher-order interactions in networks. Graphs often arise in applications with the additional structure of an embedding in a surface. This is also happens for hypergraphs: a hypermap is a hypergraph embedded in a closed surface. This talk is about hypermaps. I'll begin by reviewing the basics of hypermaps, including various ways to describe them. I'll go on to present a theory of hypermap minors based upon a smoothing operation in cubic graphs. I'll discuss various aspect of this theory such as commutativity, duality and Tutte's triality, polynomials, and relations with Farr's theory of alternating dimaps. This is joint work with Jo Ellis-Monaghan and Steven D. Noble.

Posted February 3, 2026

Mathematical Physics and Representation Theory Seminar

Karl-Hermann Neeb, Universität Erlangen-Nürnberg
Coadjoint orbits carrying Gibbs ensembles

Coadjoint orbits are orbits for the action of a Lie group on the dual of its Lie algebra. They carry a natural symplectic structure and are models for homogeneous systems in classical mechanics. Gibbs measures on these orbits provide a natural setting for models of thermodynamic systems. We say that a coadjoint orbit carries a Gibbs ensemble if the set of all $x$, for which the function $\alpha \mapsto e^{-\alpha(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $\Omega_\lambda$. We describe a classification of all coadjoint orbits with this property. In the context of Souriau's Lie group thermodynamics, the subset $\Omega_\lambda$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $\Omega_\lambda$ (modulo central shifts) diffeomorphically onto the interior of the convex hull of the coadjoint orbit $\cO_\lambda$. This provides an interesting perspective on the underlying information geometry.

Posted February 10, 2026

Mathematical Physics and Representation Theory Seminar

Milo Moses, Caltech
Four-colorings, recoverability, and topological quantum matter

In ongoing joint work with A. Kitaev, D. Ranard, I. Kim, we have been developing a formalism for exploring topologically ordered quantum states using recovery maps. In this talk, I will explain the potential applicability of these recovery maps to the study of 4-colorings of planar graphs. Conditioned on an unproven technical lemma (which perhaps someone in the audience can demonstrate!), I can show that bridgeless planar graphs quasi-isometric to the Euclidean plane obey a medley of properties reminiscent of topological quantum order.