A quantized Gieseker variety is an associative algebra quantizing  the global
functions on a Gieseker moduli space. This algebra arises as a quantum Hamiltonian 
reduction of the algebra of differential operators on a suitable space. It depends on
one complex parameter and has interesting and beautiful representation theory. 
For example, when it has finite dimensional representations, there is a unique 
simple one and all finite dimensional representations are completely reducible. 
In fact, this is a part of an ongoing project with Pavel Etingof and Vasily Krylov,
one can explicitly construct the irreducible finite dimensional representation using 
a cuspidal equivariant D-module on sl_n and get an explicit dimension (and character) 
formula. This formula gives a “higher rank” version of rational Catalan numbers. 
I’ll introduce all necessary definitions, describe the resuls mentioned above and, 
time permitting talk about open problems.

 Mirzakhani proved two theorems about the asymptotic growth of the number of curves in a mapping class group orbit on a surface: one for simple curves and another for general curves. In this talk I will explain how to derive her second theorem from the one about simple curves. Time permitting, I will explain why similar methods can be used to also give a proof for the theorem about simple curves, hence giving a new (and very different) proof of both theorems.  

Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to an interesting class of cycles on the special fiber of Shimura varieties. In this talk we give a description of the top-dimensional irreducible components of ADLV’s modulo the action of a natural symmetry group, thus verifying a conjecture of Miaofen Chen and Xinwen Zhu. We obtain as a corollary, a description of the irreducible components of the basic locus of certain Shimura varieties in terms of a class set for an inner form of the structure group, generalizing classical results of Deuring and Serre. A key input for our approach is an analysis of certain twisted orbital integrals using techniques from local harmonic analysis. This is based on joint work in progress with X. He and Y. Zhu.