Algebra/Topology seminar

Speaker: Lorenzo Ruffoni.

Title: Projective structures, representations, and ODEs on surfaces.

Abstract: In one of its easiest formulations, Hilbert's XXI problem deals with the relationship between linear ODEs on a surface and representations of its fundamental group. When a complex structure on the surface is fixed, a classical theory is available. However, not much is understood in the complementary case, i.e. when the type of the ODE is fixed, while the complex structure is allowed to vary. Projective structures have been known since Poincare's times to be a geometric bridge between the analytic and the algebraic side of this picture. In this talk we will present how their geometric deformation theory can be used to explore the space of ODEs associated with a fixed representation, including some recent results.