# Mathematics

## Large orbit closures of translation surfaces are strata or loci of double covers V

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.

## A Snapshot of Early 20th Century Women Mathematicians

Over the millennia, from Theano (born c. 546 B.C.), the wife of the Greek mathematician Pythagoras and herself a mathematician, to Maryam Mirzakhani (May 1977 – July 2017), who in 2014 became the first woman to win the Fields medal, the most prestigious award in mathematics, there have been many brilliant female mathematicians working in all areas of math. I will mention a few who were active in the late 19th and the first half of the 20th centuries, and discuss the work and impact of one of them in greater depth.

## Large orbit closures of translation surfaces are strata or loci of double covers III

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.

## Large orbit closures of translation surfaces are strata or loci of double covers IV

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.

## New lower bounds for van der Waerden numbers

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

## Graph Density Inequalities, Sums of Squares and Tropicalization

Establishing inequalities among graph densities is a central pursuit in extremal graph theory. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares or as a rational sum of squares. In this talk, we will explore how one does so and we will then identify simple conditions under which a graph density expression cannot be a sum of squares or a rational sum of squares. Tropicalization will play a key role for the latter, and will turn out to be an interesting object in itself. This is joint work with Greg Blekherman, Mohit Singh, and Rekha Thomas.

## An improvement on Łuczak's connected matchings method

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.

## Vector Copulas and Vector Sklar Theorem

This talk introduces vector copulas and establishes a vector version of Sklar’s theorem. The latter provides a theoretical justification for the use of vector copulas to characterize nonlinear or rank dependence between a finite number of random vectors (robust to within vector dependence), and to construct multivariate distributions with any given non-overlapping multivariate marginals. We construct Elliptical, Archimedean, and Kendall families of vector copulas and present algorithms to generate data from them. We introduce a concordance ordering for two random vectors with given within-dependence structures and generalize Spearman’s rho to random vectors. Finally, we construct empirical vector copulas and show their consistency under mild conditions.

## Optimal Coffee shops, Numerical Integration and Kantorovich-Rubinstein duality

Suppose you want to open up 7 coffee shops so that people in the downtown area have to walk the least amount to get their morning coffee. That’s a classical problem in Optimal Transport, minimizing the Wasserstein distance between the sum of 7 Dirac measures and the (coffee-drinking) population density. But in reality things are trickier. If the 7 coffee shops go well, you want to open an 8th and a 9th and you want to remain optimal in this respect (and the first 7 are already fixed). We find optimal rates for this problem in ($W_2$) in all dimensions. Analytic Number Theory makes an appearance and, in fact, Optimal Transport can tell us something new about $\sqrt{2}$ . All of this is also related to the question of approximating an integral by sampling in a number of points and a conjectured extension of the Kantorovich-Rubinstein duality regarding the $W_1$ distance and testing of two measures against Lipschitz functions.

## Deep kernel-based distances between distributions

Optimal transport, while widespread and effective, is not the only game in town for comparing high-dimensional distributions. This talk will cover a set of related distances based on kernel methods, in particular the maximum mean discrepancy, and especially their use with learned kernels defined by deep networks. This set of distance metrics allows for effective use in a variety of applications; we will cover foundational properties and develop variants useful for distinguishing distributions, training generative models, and other machine learning applications.