# Introductory Courses

## Topology of function spaces

Language: Spanish.

Starting at a basic level, for maps between topological spaces the compact and open topology will be analyzed as well as the exponential law. The concepts of smooth manifolds and maps will be introduced, and the main results on the Weak and Strong Topologies on the Space of $C^{r}$-Maps between manifolds will be described. Finally, we will introduce the degree modulo 2 and the Brouwer degree of smooth maps. The course will finish by discussing the celebrated Poincaré-Hopf Theorem.

Bibliography: Differential Topology; Hirsch, Morris. An introduction to differentiable manifolds and Riemannian Geometry; Boothby, William.

## Categories and modules

Language: Spanish.

We will start by recalling the basic notions and results of category theory, in particular those concerning the category of modules over a ring. Then we will apply these results to the theory of representations of finite groups and finally we will see some recent results in representation theory that make use of these categorical methods.

Bibliography: Rings and categories of modules; Anderson, F. y Fuller, K. Toposes, triples and theories; Barr M. y Wells C. Categories for the working mathematician; Mac Lane, S. Advanced modern algebra; Rotman, J.

## Algebraic topology from a homotopical viewpoint

By Alejandra Trujillo, Centro de Investigación en Matemáticas A.C.
Language: Spanish.

The objective of this course is to define homology and cohomology groups using Moore and Eilenberg-MacLane spaces. The study subjects are: function spaces, homotopy classes, homotopy groups, fibrations, cofibrations, CW complexes, Moore and de Eilenberg-MacLane spaces, and finally homology and cohomology groups.

Bibliography: Algebraic Topology from a Homotopical Viewpoint; Prieto, Carlos; Aguilar, Marcelo; Gitler, Samuel.