Konstantin Jakob will speak on

The Deligne-Simpson problem for isoclinic $G$-connections

Classically, the Deligne-Simpson problem asks for the existence of local systems on a punctured $P^1$ with prescribed local monodromy. Via Riemann-Hilbert, such a local system corresponds to a regular singular ordinary differential equation, and one equivalently asks for the existence of a differential equation with prescribed local monodromy. In this talk, I will report on joint work with Zhiwei Yun on the generalization of the Deligne-Simpson problem to a certain class of irregular G-connections, which we call isoclinic. Using ideas from non-abelian Hodge theory and Springer theory, we formulate an existence criterion involving an algebraic invariant attached to an isoclinic G-connection. I will explain how to use this criterion to solve the Deligne-Simpson problem in many cases, including exceptional types, via the representation theory of finite Weyl groups.