# Archive 2022 2021

All recorded talks are available on our YouTube-clannel

### 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

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.

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".

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.

### 2021

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).

The sphere packing problem asks for the densest configuration of non-overlapping unit balls in space. In this talk I shall speak about the sphere packing problem in various spaces and its generalisations. The talk will focus on linear programming and semidefinite programming methods as powerful tools for analysing and, in some cases, completely solving geometric optimisation questions.

Vinberg's theory of reflection groups has wide applications. We discuss some of these, to monodromy groups of hypergeometric and Painlevé equations. The nonlinear case is intimately connected to affine Markoff surfaces and it is a central ingredient in the Diophantine analysis of these surfaces.

In the early 1960s, Vinberg gave a description of homogeneous convex cones as cones of Hermitian positive de finite matrices in a matrix T-algebra $M_n$ of $(n \times n)$-matrices whose diagonal entries are just real numbers, but off-diagonal elements belong to different vector spaces. It turns out that rank 3 special Vinberg cones (corresponding to Clifford algebras) have important applications to Supergravity. No special background is required. The talk is based on joint works with V. Cortes; and with A. Marrani and A. Spiro.

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.

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 aims to summarize the status and goals of a broad research project. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

- We extend some basic algorithms of convex optimization from Euclidean space to the far more general setting of Riemannian manifolds, capturing the symmetries of non-commutative group actions. The main tools for analyzing these algorithms combine central results from invariant and representation theory.
- One main motivation for studying these problems and algorithms comes from algebraic complexity theory, especially attempts to separate Valiant’s algebraic analogs of the P and NP. Symmetries and group actions play a key role in these attempts.
- The new algorithms give exponential (or better) improvements in run-time for solving algorithmic many specific problems across CS, Math and Physics. In particular, in algebra (testing rational identities in non-commutative variables), in analysis (testing the feasibility and tightness of Brascamp-Lieb inequalities), in quantum information theory (to the quantum marginals problem), optimization (testing membership in “moment polytopes”), and others. This work exposes old and new connections between these diverse areas.