A fundamental problem in the theory of meromorphic connections on P^1 is to understand the space of such systems with given local behavior. Here, the local behavior of a connection at a singular point means the "formal type" there--the isomorphism class of the induced formal connection. Given a collection of singular points and corresponding formal types, there are several natural questions one might ask :
- Does there exist a connection with these formal types ?
- If such a connection exists, is it unique up to isomorphism ?
- Can one construct an explicit moduli space of such connections ?
Classically, these questions were studied under the assumption that all singularities are regular singular (i.e. simple poles). For example, in 2003, Crawley-Boevey solved the Deligne-Simpson problem for Fuchsian connections (a variant of question 1) by reinterpreting the problem in terms of quiver varieties. Later, mathematicians including Boalch, Hiroe, and Yamakawa investigated these questions when "unramified" irregular singularities are allowed. (Unramified means that the formal types can be expressed in upper triangular form without introducing roots of the local parameter.)
In recent years, there has been increasing interest in meromorphic connections (and G-connections where G is a reductive group) with ramified singularities due to developments in the geometric Langlands program. In this talk, I will give an overview of recent progress on the ramified version of these problems due to myself and various collaborators.
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