210319 Nikita Nikolaev

Meromorphic connections on Riemann surfaces (i.e., an invariant point of view on complex singular ODEs) can be understood as representations of relevant holomorphic Lie algebroids or equivalently (by integration, called the Riemann-Hilbert correspondence) as Lie groupoid representations. I will describe an approach to studying these objects called abelianisation. It uses simple combinatorial data attached to the Riemann surface (certain topological graphs) to recast the representations into the nonabelian group GL(n,C) instead as the rather more amicable representations into the abelian group C* but on a suitable n-fold covering Riemann surface. This technique has its origins in the works of Fock-Goncharov (2006) and Gaiotto-Moore-Neitzke (2013), as well as the WKB analysis. I will explain this correspondence in the case n=2 as an equivalence of categories of representations.

This presentation is based on arXiv:1902.03384 and the more recent work in progress extending those results.