### Effective Methods for Arithmetic Groups

Arithmetic groups are groups that can be written as the set of all matrices with integer entries in a suitable linear group.

They are objects of fundamental interest in many areas of mathematics, including geometry, topology and number theory.

The aim of this course is to develop tools to find a fundamental domain for the action of a given arithmetic group of isometries of a (real or complex) hyperbolic space.

This will allow us to write explicit finite presentations, to solve the word problem, and to get a list of conjugacy classes of finite subgroups, for instance.

### Representation theory and homological algebra (2020-2021)

Roughly speaking, representation theory is the study of symmetries of linear spaces. Leaving aside its beauty, it has many applications in combinatorics, geometry, number theory, probabibility, and even quantum theory, to mention a few. The first part of the course will deal with basic notions of representation theory and category theory. The second part will cover the basics of homological algebra, which is, grosso modo, a pervasive set of tools to build invariants allowing to distinguish and study quite complicated objects in algebra, topology and geometry. From these invariants stem the main importance of homological algebra: its possibility to produce tools that allow to prove statements outside the scope of homological algebra. Some major examples of this phenomenon are the proof of the central theorem in algebraic geometry stating that the localisation of a regular local ring at a prime ideal is also regular, or that every regular local ring is an UFD.

Contents:

Part I. Representation theory

- Module theory

- Categories, functors, and natural transformations

- Simple, free, projective and injective modules

- Artin-Wedderburn theorem

- Maschke’s theorem

- Krull-Schmidt theorem

Part II. Homological algebra

- Complexes and homology

- Snake lemma

- Projective and injective resolutions

- Derived functors

- Tor and Ext, Künneth formula

- Group (co)homology

- Eckmann-Shapiro lemma

References:

Assem, Ibrahim. Algèbres et modules: cours et exercices. Presses Université Ottawa, 1997. 330 pp.

Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp.

Prerequisites:

- A first course on abstract algebra (covering group theory and ring theory).

- A course on basic topology.

Possible internships:

-Cohomology of Hopf algebras

-Koszul property of quiver algebras

-Representation theory of twisted group algebras