Abstract Philip Boalch

Philip Boalch will speak on

Modular differential equations and wild mapping class groups

Abstract: In celebrated work Gauss and Manin explained how to attach a system of linear differential equations to a family of varieties. The same procedure works if we take the first nonabelian cohomology, but now yields nonlinear differential equations. The simplest case (four-punctured Riemann spheres, GL_2 coefficients) yields the sixth Painleve equation, discovered in this way by R. Fuchs. In general this yields (nonlinear) flat connections on bundles over the moduli stack of curves (no interesting examples are known beyond families of curves). In this talk I will explain a generalisation of this story (initiated by Garnier) which yields a natural generalisation of the moduli of curves. The wild mapping class groups are the fundamental groups of these new moduli spaces, simple examples of which are the G-braid groups.