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

  1. The notions of symmetry groups, group actions and invariants
    1. Emmy Noether's theorem on symmetries and invariants
    2. Examples of applications of Noether's theorem
    3. 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
    4. The notion of group-object of a certain category, in particular that of group bundle or gauge group
    5. The notion of group action
    6. The notion of invariant: what do we mean by invariants of a group or of an action of a group on an object?
  2. Typology of groups
    1. The notion of real or complex Lie group
    2. 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
    3. The algebraic nature of simple group
    4. The splitting of algebraic groups into linear groups and abelian varieties
    5. The splitting of linear algebraic groups into unipotent radical and reductive quotient
    6. Structure of reductive groups and parametrization by discrete parameters
    7. 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)
    8. Other families of discrete subgroups of continuous groups
      • Weyl groups and more generally Coxeter groups
      • Crystallographic groups
  3. Typology of group actions
    1. 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
    2. First examples of homogeneous spaces: projective spaces, varieties of flags, Grassmann varieties
    3. The parametrization of abelian varieties by homogeneous spaces
    4. The categories of finite-dimensional vector spaces provided with the action of an algebraic group. Grothendieck's characterization of these categories
  4. Regular representations
    1. The action of a group G by translation on the spaces of functions on homogeneous spaces G/H. The special case of automorphic functions
    2. In particular, the double action of a group G by left and right translation on the space of functions on G = (G✕G)⁄G
    3. Spectral decomposition of the regular representation in the case of finite groups. The central algebra of invariant functions by conjugation and its characters.
    4. The special case of finite Lie groups. Introduction to Langlands' duality in this case
    5. Spectral decomposition in the case of compact groups. The special case of Fourier series
    6. Spectral decomposition in the case of a locally compact commutative groups. The special case of the additive group and the Fourier transform
    7. What can be said in the general framework of a locally compact group
    8. 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