I would like to explain what the global Langlands correspondence over function fields is, to present Vincent Lafforgue's results about it and to give an idea of his methods. In the most famous case, that of the general linear group GL_n, Vincent Lafforgue recovers the results of his brother Laurent Lafforgue (and, for n=2, of Drinfeld), but there are in my opinion a few advantages to his methods:
- Even in this case, they are totally new and technically simpler.
- They work for general reductive groups, for example symplectic groups or orthogonal groups.
- They have been inspiring new ideas to approach the local Langlands correponsdance over local fields of any characteristic.