## Topology of function spaces

*By Dimas Tejada and Ingrid Martínez, Universidad de El Salvador.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.

## An introduction to Geometric Group Theory

*By Luis Sánchez, Universidad Nacional Autónoma de México.Language: Spanish.*

The main goal of this mini course is to provide an introduction to geometric group theory, that is, the study of groups via their actions on geometric and topological objects. Among the topics we will define and study are: the word metric, the Cayley graph of a group, quasi-isometries, the Schwartz-Minor lemma, and a brief introduction to some quasi-isometry invariants of groups.

Bibliography: *Geometric group theory, an introduction*; Clara Löh. *Office hours with a geometric group theorist*; ed. by Matt Clay & Dan Margalit.

## Categories and Modules

*By Nadia Romero, Universidad de Guanajuato.Language: Spanish.Format: streaming.*

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.