Calendar

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 November 13, 2025
Last modified November 16, 2025

Colloquium Questions or comments?

Sky Cao, Massachusetts Institute of Technology
Yang-Mills, probability, and stochastic PDE

Originating in physics, Yang-Mills theory has shaped many areas of modern mathematics. In my talk, I will present Yang-Mills theory in the context of probability, highlighting central questions and recent advances. In particular, I will discuss the role of stochastic partial differential equations (SPDEs) in these developments and survey some of the recent progress in this field.

Posted November 15, 2025

Algebra and Number Theory Seminar Questions or comments?

Esme Rosen, Louisiana State University
TBA

TBA

Posted November 13, 2025
Last modified November 17, 2025

Colloquium Questions or comments?

Mengxuan Yang, Princeton University
Flat bands in 2D materials

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by these angles, the resulting material is superconducting and the so-called energy bands are flat and topological. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer--MacDonald's model, including the mathematical characterization of magic angles and flat bands, the generic existence of Dirac cones and how topological phase transitions occur at magic angles. I will also discuss some new mathematical discoveries in twisted multilayer graphene.

Posted November 13, 2025
Last modified November 23, 2025

Colloquium Questions or comments?

Peter Bradshaw, University of Illinois Urbana-Champaign
Toward Vu's conjecture

In 2002, Vu conjectured that graphs of maximum degree $\Delta$ and maximum codegree at most $\zeta \Delta$ have chromatic number at most $(\zeta+o(1))\Delta$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the "dense regime,'' when $\zeta$ is close to $1$, by Hurley, de Verclos, and Kang.

In this talk, I will discuss one of my recent results achieving the first major progress in the sparse regime where \zeta approaches 0, the case of primary interest to Vu. The result states that there exists $\zeta_0 > 0$ such that for all $\zeta \in [\log^{-32}\Delta,\zeta_0]$, the following holds: if $G$ is a graph with maximum degree $\Delta$ and maximum codegree at most $\zeta \Delta$, then $\chi(G) \leq (\zeta^{1/32} + o(1))\Delta$. This bound is derived from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. The more general result also extends to the list coloring setting, which is of independent interest.

This talk is based on joint work with Dhawan, Methuku, and Wigal.

Posted August 27, 2025
Last modified November 26, 2025

Informal Geometry and Topology Seminar Questions or comments?

Krishnendu Kar, Louisiana State University
Khovanov Homology

Wrapping up our discussion on Khovanov Homology from this semester.

Posted September 10, 2025
Last modified September 23, 2025

Geometry and Topology Seminar Seminar website

Corey Bregman, Tufts University
TBA

TBA

Posted November 12, 2025

Colloquium Questions or comments?

Iain Moffatt, Royal Holloway, University of London
Graphs in surfaces, their one-face subgraphs, and the critical group

Critical groups are groups associated with graphs. They are well-established in combinatorics; closely related to the graph Laplacian and arising in several contexts such as chip firing and parking functions. The critical group of a graph is finite and Abelian, and its order is the number of spanning trees in the graph, a fact equivalent to Kirchhoff’s Matrix--Tree Theorem.

What happens if we want to define critical groups for graphs embedded in surfaces, rather than for graphs in the abstract?

In this talk I'll offer an answer to this question. I'll describe an analogue of the critical group for an embedded graph. We'll see how it relates to the classical critical groups, as well as to Chumtov's partial-duals, Bouchet's delta-matroids, and a Matrix--quasi-Tree Theorem of Macris and Pule, and describe how it arises through a chip-firing process on graphs in surfaces.

This is joint work with Criel Merino and Steven D. Noble.