L6

In this talk, I will talk about the category of coherent sheaves on the affine Flag variety of a simply-connected reductive group over $\mathbb{C}$. In particular, I'll explain how the convolution product naturally leads to a construction of the Iwahori-Hecke algebra, and present some combinatorics related to computing duals in this category.

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance. 

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

The speakers will be giving short presentations introducing topics in algebraic number theory, arithmetic topology, random matrix theory, and analytic number theory.

Undergrads and Master's students are encouraged to come and sample a taste of research in these areas.

The $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another, and the toroidal $n$-queens problem asks the same question where the board is considered on the surface of a torus. Let $Q(n)$ denote the number of $n$-queens configurations on the classical board and $T(n)$ the number of toroidal $n$-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that $T(n)>0$ if and only if $n \equiv 1,5 \mod 6$. Much more recently Luria showed that $T(n)\leq ((1+o(1))ne^{-3})^n$ and conjectured equality when $n \equiv 1,5 \mod 6$. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) $n \equiv 1,5 \mod 6$. We also show that $Q(n)\geq((1+o(1))ne^{-3})^n$ for all $n \in \mathbb{N}$ which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both $Q(n)$ and $T(n)$. 

In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a $4$-partite $4$-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.

This is joint work with Peter Keevash.

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $A$ is singular with probability at most $e^{-cn}$ for some absolute constant $c>0$, thereby resolving a well-known conjecture. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.