Jean-Baptiste Teyssier
Homotopy theory and the derived moduli of Stokes data
Stokes data are algebraic data that can be extracted from the asymptotic analysis of solutions for flat bundles at singular points. Due to works of Deligne-Malgrange in dimension 1 and Mochizuki in higher dimension, they were shown to realize the irregular Riemann-Hilbert correspondence for flat bundles. In these lectures, we will explain how recent advances in stratified homotopy theory obtained in joint works with Peter Haine and Mauro Porta provide new perspectives on Stokes data and enable to construct their derived moduli in any dimension. This is joint work with Mauro Porta.
Lecture notes can be found here.
Valerio Toledano Laredo
Stokes phenomena, quantum groups and Poisson-Lie groups
Quantum groups have long been known to be related to Conformal Field Theory through the Knizhnik-Zamolodchikov (KZ) equations. This Betti role as natural receptacles of monodromy has been significantly expanded in recent years by including the Casimir equations which are dual to the KZ ones. This has led to a novel construction of quantum groups from the dynamical KZ (DKZ) equations. Unlike their precursors, these have irregular singularities and therefore exhibit Stokes phenomena, which describe the discontinuous change of asymptotic of solutions near singular points. In particular, the Stokes matrices of the simplest DKZ equations are R-matrices of the corresponding quantum group.
In a parallel development, Boalch constructed the Poisson structure on the dual G^* of a complex reductive group G by using Stokes phenomena for the simplest irregular connection on the trivial G-bundle over P^1. This transcendental linearization of G^* is particularly tantalizing in that it is very close in spirit to the above construction of quantum groups.
The main goal of these lectures will be to explain how quantum groups arise from the dynamical KZ equations, describe Boalch’s construction, and obtain a precise link between these two uses of Stokes phenomena, by showing that the latter construction can be obtained as a semiclassical limit of the former.
Zhiwei Yun
Affine Springer fibers and wildly ramified Langlands on P^1
I will explain how certain affine Springer fibers (called homogeneous) can be realized as central fibers of completely integrable systems. These integrable systems are slices of Hitchin systems, and they admit non-abelian Hodge companions. I will also explain how these spaces show up in formulating a particular case of the wildly ramified geometric Langlands conjecture on P^1. This is joint work with Roman Bezrukavnikov, Pablo Alvarez-Boixeda and Michael McBreen.