The Vinberg Distinguished Lecture Series aims at bringing together all mathematicians interested in wide dissemination of ideas from various domains of pure and applied mathematics.

During his lifetime, È. B. Vinberg has made outstanding contributions to many branches of mathematics, such as Lie groups and algebraic groups, representation theory, invariant theory, hyperbolic geometry, automorphic forms, and discrete subgroups of Lie groups.

We hope to carry on Vinberg's legacy and bring together a wider community of mathematicians interested in the above mentioned domains as well as other fields of study.

## Format

## and

## Registration

A usual lecture is about 90 minutes long: the first part is a survey talk, and the second part is aimed at experts. There is a short break (about 10 minutes) in between.

The Vinberg Lecture talks are online and will be held in Zoom. Please register here to receive a Zoom link.

## Zoom & recordings

Join us in Zoom via link or with

Meeting ID: 928472910

Password: Some kind of puzzle or in open

All recorded videos will appear on YouTube Channel

### Upcoming talks

Let $A$ and $B$ be two permutations in $\text{Sym}(n)$ that ``almost commute'' -- are they a small deformation of permutations that truly commute? More generally, if $R$ is a system of words-equations in variables $X = \{x_1, \ldots ,x_d\}$ and $A_1, \ldots, A_d$ are permutations that are nearly solutions; are they near true solutions?

It turns out that the answer to this question depends only on the group presented by the generators $X$ and relations $R$. This leads to the notions of ``stable groups'' and ``testable groups''.

We will present a few results and methods which were developed in recent years to check whether a group is stable or testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic, and with invariant random subgroups.

This talk will survey as well as discuss geometric and topological properties of arithmetic hyperbolic manifolds of simplest type. These are precisely the class of arithmetic hyperbolic manifolds that contain an immersed co-dimension one totally geodesic submanifolds.

### Spring 2022

### Past talks

The Two Families Theorem of Bollobas says the following: Let $(A_i,B_i)$ be a sequence of pairs of sets such that the $A_i$ have size a, the $B_i$ have size $b$, and $A_i$ and $B_j$ intersect if and only if $i$ and $j$ are distinct. Then the sequence has length at most $binom{a+b}{a}$.

This beautiful result has many applications and has been generalized in two distinct ways. The first (which follows from the original result of Bollobas) allows the sets to have different sizes, and replaces the cardinality constraint with a weighted sum. The second uses an elegant exterior algebra argument due to Lovasz and allows the intersection condition to be replaced by a skew intersection condition. However, there are no previous results that have versions of both conditions.

In this talk, we will explain and extend the exterior algebra approach. We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. As an application, we prove a new extension of the Two Families Theorem that allows both (some) variation in set sizes and a skew intersection condition.

This is joint work with Elizabeth Wilmer (Oberlin).