N3.12

Let $X$ denote an open subset of $\mathbb{C}^d$, and $\mathcal{O}$ its sheaf of holomorphic functions. In the 1970’s, Ishimura studied the morphisms of sheaves $P\colon\mathcal{O}\to\mathcal{O}$ of $\mathbb{C}$-vector spaces which are continuous, that is the maps $P(U)\colon\mathcal{O}(U)\to\mathcal{O}(U)$ on the sections are continuous. In this talk, we explain his result, and explore its analogues in the non-Archimedean world.

The character table of GL(2,Fq), for a prime power q, was constructed over a century ago. Many of these characters were determined via the explicit construction of a corresponding representation, but purely character-theoretic techniques were first used to compute the so-called discrete series characters. It was not until the 1970s that Drinfeld was able to explicitly construct the corresponding discrete series representations via l-adic étale cohomology groups. This work was later generalised by Deligne and Lusztig to all finite groups of Lie type, giving rise to Deligne-Lusztig theory.

In a similar vein, we would like to construct the representations affording the (smooth) characters of compact groups like GL(2,Zp), where Zp is the ring of p-adic integers. Deligne-Lusztig theory suggests hunting for these representations inside certain cohomology groups. In this talk, I will consider one such approach using a non-archimedean analogue of de Rham cohomology.

Auslander-Reiten theory provides lots of powerful tools to study algebras of finite representation type. One of these is Auslander correspondence, a well-known result establishing a bijection between the class of algebras of finite representation type and their corresponding Auslander algebras. I will present these classical results in a key example: the class of algebras associated to quivers of type A_n. I will talk about well-known results regarding their derived equivalence with another class of algebras, and I will present a more recent result regarding the perfect derived category of the Auslander algebras of type A_n.

The canonical dimension is a fundamental integer-valued invariant that is attached to mod p representations of p-adic Lie groups. I will explain why it is both an asymptotic measure of growth, and an algebraic quantity strongly related to Krull dimension. We will survey algebraic tools that can be applied in its calculation, and describe results spanning the last twenty years. I'll present a new theorem and suggest its possible significance for the mod p local Langlands programme. 

We can learn a lot about an integral domain by studying the Galois group of its fraction field. These groups are generally quite complicated and hard to understand, but their representations, so-called Galois representations, contain more easily accessible information. These also play the lead in many important theorems and conjectures of modern maths, such as the Modularity theorem and the Langlands programme. In this talk we give a quick introduction to Galois representations, motivated by lots of examples aimed at a general algebraist audience, and talk about some open problems.

We will look at the branching of irreducible representations of symmetric groups from the perspective of Okounkov-Vershik, and then look at Hecke algebras, affine Hecke algebras and cyclotomic Hecke algebras, in particular how the graded Grothendieck groups of their module categories “are” irreducible highest weight modules for affine $sl_l$, where $l$ is the “quantum characteristic”, and the branching graph is a highest weight crystal (for affine $sl_l$). The Fock space realisation of the highest weight crystal will get us back to  the Young graph for in the case of the symmetric group that we considered at the beginning.

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

I will explain in what sense the theory of finite tensor categories is a categorification of the theory of finite dimensional algebras. In particular, I will introduce finite module categories, review a key result of Ostrik, and present Morita theory for finite categories. I will give many examples to illustrate these ideas. Then, I will explain the elementary properties of finite braided tensor categories. If time permits, I will also mention my own work, which consists in categorifying these ideas once more!