AGNT Ical Atom

In a series of recent papers, Minhyong Kim defined an arithmetic analogue of topological Chern-Simons theory. In this talk, I will introduce this arithmetic Chern-Simons theory and then explain how to compute the arithmetic Chern-Simons invariant for finite, cyclic gauge groups. I will then give some recent applications of these computations.

My work in this talk is based on joint works with Tomer Schlank and Eric Ahlquist.

A classical Springer theory is an important ingredient in the classification of representations of finite groups of Lie type, completed by Lusztig.

The first result of this theory is the assertion that the so-called Grothendieck-Springer sheaf is perverse and is equipped with an action of the Weyl group. Our main result asserts that an analogous result also holds in the affine (infinite-dimensional) case.

In the first of my talk I will recall what are perverse sheaves, and why the Grothendieck-Springer sheaf is perverse. In the rest of the talk I will outline how to extend all this to the affine setting.

We believe that this should have applications to the representations theory of p-adic groups.

This is a joint work with Alexis Bouthier and David Kazhdan

I will discuss a somewhat mysterious connection between singularity theory and cluster algebras, more specifically between the topology of isolated singularities of plane curves and the mutation equivalence of quivers associated with their morsifications. The talk will assume no prior knowledge of any of these topics. This is joint work with Pavlo Pylyavskyy, Eugenii Shustin, and Dylan Thurston.

Let $V$ be a representation of a connected reductive group $G$. A representation is cofree if $k[V]$ is a free $k[V]^G$ module. There is a long history of work studying and classifying cofree representations of reductive groups. In this talk I present a simple conjectural characterization of cofree representations in terms of geometric invariant theory. Matt Satriano and I have proved the conjecture for irreducible representations of SL_n as well as for torus actions. I will give motiviation for the conjecture and explain the techniques which can be used for its verification. This talk based on joint work with Matt Satriano.

In this second talk we prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital A_\inftycategories with respect to the corresponding classes of quasi-equivalences are all equivalent. As an application, we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital A_\inftyfunctors between them. This is a joint work with Prof. A. Canonaco and Prof. P. Stellari arXiv:1811.07830.

I will describe recent threads in the study of number theory in function fields, the different techniques that are used, the challenges, and if time permits the applications of the theory to other subjects such as probabilistic Galois theory.