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.

In this series of two lectures I will review what we know about moduli space of instanton bundles on projective spaces. We will see that the situation drastically change when the dimension of the projective space increase. At the end of the two talks I will state some interesting open problems.