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

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.

## Time & date

In order to obtain the correct time and date of forthcoming talks at your geographic location, please consult our schedule at ResearchSeminars.org

## Zoom & recordings

All recorded videos will appear in YouTube Playlist

### Upcoming talks

### Past Talks: Spring 2022

In the ocean of combinatorial inequalities, two islands are especially difficult. First, Mason's conjectures say that the number of forests in a graph with k edges is log-concave. More generally, the number of independent sets of size k in a matroid is log-concave. Versions of these results were established just recently, in a remarkable series of papers by Huh and others, inspired by algebro-geometric considerations.

Second, Stanley's inequality for the numbers of linear extensions of a poset with value k at a given poset element, is log-concave. This was originally conjectured by Chung, Fishburn and Graham, and famously proved by Stanley in 1981 using the Alexandrov–Fenchel inequalities in convex geometry. No direct combinatorial proof for either result is known. Why not?

In the first part of the talk we will survey a number of combinatorial inequalities. We then present a new framework of combinatorial atlas which allows one to give elementary proofs of the two results above, and extend them in several directions. This talk is aimed at the general audience.

On results and (mostly) conjectures on automorphic functions on moduli spaces of 2-dimensional vector bundles on curves over local fields.

### Past talks

Wall-crossing structures appeared several years ago in several mathematical contexts, including cluster algebras and theory of generalized Donaldson-Thomas invariants. In my lecture I will describe the general formalism based on a graded Lie algebra and an additive map from the grading lattice to an oriented plane ("central charge").

A geometric example of a wall-crossing structure comes from theory of translation surfaces. The number of saddle connections in a given homology class is an integer-valued function on the parameter space (moduli space of abelian or quadratic differentials), which jumps along certain walls. The whole theory can be made totally explicit in this case. Also, I'll talk about another closely related example, which can be dubbed a "holomorphic Morse-Novikov theory".

Arithmeticity of locally symmetric spaces is an old and important area of study in which Vinberg proved some central results. I will discuss the history of the area, some open questions and then focus on recent joint work with Bader, Miller and Stover. We prove that non-arithmetic real and complex hyperbolic manifolds cannot have infinitely many maximal totally geodesic submanifolds.

Hilbert's tenth problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be. The second half of the lecture will explore some of the techniques from arithmetic geometry that have been used towards answering this question and the related question for the ring of integers of a number field.

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).