The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.

Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an
Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.  This work is joint with Dan Barbasch and Dan Ciubotaru.
 

My research focuses on numerical techniques that help provide scalable computation within simulations of two-phase fluid flow problems. The efficient solution of the linear systems which arise is key to obtaining practical computation. I will motivate and discuss new methods which seek to generalise effective techniques for a single phase to the more challenging setting of two-phase flow where the governing equations have discontinuous coefficients.

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).

The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.