### Projective structures, representations, and ODEs on surfaces.

Geometry and Topology Seminar

23rd November 2020, 4:00 pm – 5:00 pm

Zoom seminar, if interested, please email one of the organisers to gain access to the Zoom link

Hilbert's XXI problem deals with the relationship between linear ODEs on a surface and representations of its fundamental group. Deligne obtained a general existence result in the 1970s in the case the complex structure of the surface is fixed. However, not much is known 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 for a long time as 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. For a concrete case study, we will discuss the relationship between grafting, hypergeometric ODEs, and triangle groups (joint work with S. Ballas, P. Bowers, and A. Casella).

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