The CIM Colloquium is a Mathematical Sciences event taking place before the annual council of the CIM Associates. This year the lecture will be delivered by Peter Gothen, Professor at Departamento de Matemática da Faculdade de Ciências da Universidade do Porto on the 17th of March, at 03:00 pm (Lisbon time).

**TITLE:** *Surface groups, Higgs bundles and moduli spaces*

**ABSTRACT:**

*Any closed topological surface of genus greater than two has a metric of constant negative curvature. Such a metric can be realised by a homomorphism of the fundamental group of the surface into the isometry group G=PSL(2,R) of the hyperbolic plane. A homomorphism arising in this way is called a Fuchsian representation of the surface group. The subspace of Fuchsian representations (also known as the Teichmüller space of the surface) forms a connected component in the space of all representations of the surface group. I will review this beautiful chapter of mathematics. I will then explain how one can find analogous subspaces for some more general Lie groups G and how this connects to an emerging area of geometry known as Higher Teichmüller theory. Gadgets called Higgs bundles, and their moduli spaces, will play an important role in the story.*

**Video archive at** http://www.mat.uc.pt/~clcmat/videos/CIM_Colloquium_2023.mp4.

This event is supported by CIM.