Laurent Lafforgue is Permanent Professor at the Institut des Hautes Études Scientifiques.

In 2002, he received the Fields Medal for his fundamental contributions to the Langlands program.

## Schedule

October 3rd, 2017 - December 19th, 2017

Each Tuesday, 11:00 - 12:30 and 14:30 - 16:00

room 2.1 Via Castelnuovo

## Programme

- The notions of symmetry groups, group actions and invariants
- Emmy Noether's theorem on symmetries and invariants
- Examples of applications of Noether's theorem
- The notion of group of symmetries as a group of transformations of a set provided with a certain structure or, more generally, as a group of isomorphisms of an object of a category
- The notion of group-object of a certain category, in particular that of group bundle or gauge group
- The notion of group action
- The notion of invariant: what do we mean by invariants of a group or of an action of a group on an object?

- Typology of groups
- The notion of real or complex Lie group
- The splitting of Lie groups in the discrete part and the continuous (connected) part and then the splitting of connected Lie groups into simple groups
- The algebraic nature of simple group
- The splitting of algebraic groups into linear groups and abelian varieties
- The splitting of linear algebraic groups into unipotent radical and reductive quotient
- Structure of reductive groups and parametrization by discrete parameters
- Groups of points of an algebraic group with values in:
- a finite field (which defines the Lie-type finite groups)
- the fields of reals or complex numbers (which defines the main Lie groups)
- the p-adic fields (which defines p-adic groups)
- the ring of integers (which defines arithmetic groups)

- Other families of discrete subgroups of continuous groups
- Weyl groups and more generally Coxeter groups
- Crystallographic groups

- Typology of group actions
- Categories of discrete sets endowed with actions of a group. Grothendieck's characterization of these categories. Application to the theory of the Poincaré fundamental group and to Galois theory
- First examples of homogeneous spaces: projective spaces, varieties of flags, Grassmann varieties
- The parametrization of abelian varieties by homogeneous spaces
- The categories of finite-dimensional vector spaces provided with the action of an algebraic group. Grothendieck's characterization of these categories

- Regular representations
- The action of a group G by translation on the spaces of functions on homogeneous spaces G/H. The special case of automorphic functions
- In particular, the double action of a group G by left and right translation on the space of functions on G = (G✕G)⁄G
- Spectral decomposition of the regular representation in the case of finite groups. The central algebra of invariant functions by conjugation and its characters.
- The special case of finite Lie groups. Introduction to Langlands' duality in this case
- Spectral decomposition in the case of compact groups. The special case of Fourier series
- Spectral decomposition in the case of a locally compact commutative groups. The special case of the additive group and the Fourier transform
- What can be said in the general framework of a locally compact group
- The case of p-adic groups. The notion of Bernstein center

## Lecture notes

Preliminary version - last updated on December, 11th 2017

By Laurent Lafforgue, translated into italian by Elias Megier e Simone Noja