The main objective of representation theory is the study of algebras via the study of their modules (i.e. representations). In this mini-course we will consider a combinatorial approach to this theory, using quivers (i.e. oriented graphs) and their representations (given by vector spaces associated to vertices and linear maps associated to arrows).

Higgs bundles appear in several guises including (a) as solutions to gauge-theoretic equations for connections and sections of a bundle (b) as holomorphic realizations of fundamental group representations or, equivalently, local systems and (c) as special cases of principal bundles with extra structure (principal pairs). Each point of view leads to a construction of a moduli space, i.e.

There will be 3 lectures:

- The first is devoted to explaining the main concepts and we will take the opportunity to talk about other statistical properties of dynamical systems, such as Central Limit Theorems, Laws of large numbers or Birkhoff's ergodic theorem, Poincaré's recurrence theorem and Kac's theorem.