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).

# Courses

Tema 1. “Lenguajes” algebraicos en los siglos XVI y XVII

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.

16/04/2012. Introduction to Theory of Prime Ends

18/04/2012. Applications I: Index of iterates of planar homeomorphisms

20/04/2012. Applications II: Rotation number of planar attractors

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12-15 Setembro. Total de 8 horas.

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.

Lecture 1: Representations of surface groups and harmonic maps

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.

Graded manifolds are "manifolds where some coordinates anticommute".

Among other things, they provide a way to encode well-known

matematical structures in a geometric and concise fashion.

I plan to cover

- graded linear algebra, graded manifolds,

- how they encode $L_{\infty}$ algebras and Lie algebroids