Dynamical Systems Seminar

Abstract:

Studying the existence of fixed points for holomorphic maps on Teichmüller spaces serves as a framework for proving ‘geometrization’ theorems, such as Thurston’s topological characterisation of hyperbolic 3-manifolds. In this talk, we will prove a general theorem addressing the existence of fixed points for holomorphic maps on Teichmüller spaces and a more general class of complex domains by focusing on the intrinsic shape of these domains and leveraging a novel idea that combines arguments from (complex) geometry and elementary combinatorics. We’ll also discuss a geometric generalisation and possible applications, obtained by a distillation of the ideas and arguments involved in the proofs.