Mathematics

he asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice. While it was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis, the stationary distribution of the ASEP and its variants has surprising connections to combinatorics. I will explain how the study of the ASEP on a ring leads to new formulas for Macdonald polynomials, a remarkable family of multivariate polynomials which generalize Schur polynomials. In a different direction, the inhomogeneous ASEP on a ring is closely connected to Schubert polynomials, which represent classes of Schubert varieties in the flag variety. This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and R\"odl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

This is joint work with Shoham Letzter and Alexey Pokrovskiy.

Fusion rings are a special class of associative unital rings with nonnegative integer structure constants and a notion of duality. For example, the group ring of a finite group is a fusion ring. We study fusion rings mainly because they arise as Grothendieck rings of categories associated to Hopf algebras, semisimple Lie algebras, vertex operator algebras, etc. In turn, these categories have application to topological quantum field theory, invariants of knots and links, and quantum computation, to name a few. In this talk we will discuss the brief history of the classification of categorifiable fusion rings and how number-theoretic properties of fusion rings dictate the existence of, or properties of, their categorifications.

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:

An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.