Scientific

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.

Colour ${1,\ldots,N}$ red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about $\sqrt{N}$. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about $\exp{\left(\left(\log{N}\right)^{3/4}\right)}$, and in particular less than any fixed power of $N$.

I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.

I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.