A senior researcher at the CNRS, Vincent Lafforgue was invited as a plenary speaker at the 2018 International Congress of Mathematicians in Rio de Janeiro. He answered our questions.
What are your research interests?
My current research interest is arithmetic geometry. In fact, not only does this subject mix, as its name goes, algebraic geometry and arithmetic, it also incorporates many ideas from topology as well as category theory. Furthermore, the Langlands program, on which I work, links objects from arithmetic geometry, that is, representations of Galois groups, to objects of an analytic nature, that is, automorphic forms. Among the great current questions, Langlands functoriality and generalized Riemann hypothesis, concerning automorphic forms on number fields, are of an analytic nature. On the other side, the topoï defined by Grothendieck are closely linked to logic. Arithmetic also provides many questions in algorithmic complexity and stands at the foundation of a good part of cryptography. I therefore have the feeling to be standing at a central point of mathematics.
Would you tell us about mathematicians who influenced you, or whom you particularly admire, whether they be historical figures or contemporaries?
I'll first mention d’Alexander Grothendieck (1928-2014). Together with his students, he refounded algebraic geometry in the context of categories. In particular, his vision of topï and motives has already had grand consequences, and more of them, as important, are certainly yet to come. He also had a considerable influence outside his school, as exemplified by the spread of higher categories and the works of Beilinson, Drinfeld, Gaitsgory, Kontsevich, Lurie, Voevodsky (who unfortunately recently passed away) and many others. Grothendieck not only changed mathematics, but also the way to think mathematics.
In my work, I was also very much influenced by Drinfeld, who invented chtoucas and initiated with Laumon the geometric Langlands program.
What is it that you like as a mathematician?
On top of mathematics itself, I like freedom, and working at the CNRS has a lot to do with it. It is extraordinay fortunate to be able to freely browse the whole of mathematical domain. Let me specify that that freedom has had very positive effects on my research, as without the CNRS, I would never have been able to move from my initial subject, operator algebra, to arithmetic geometry, and I do hope to study yet more subjects.
Do you already know what you will be talking about in Rio?
I'll talk about the recent advances on function fields in Langlands program, and on geometric Langlands program.
What does this congress mean for you?
Firstly, there are the Proceedings that will come out of it. Introductory texts and abstracts, such as those written for such congresses, prove very useful when one tackles a new subject. But there is more to it. ICM is, with the European congress, the only international congress gathering mathematicians from all disciplines. Somehow, it brings the mathematical community to self-awareness and it makes it aware of all its potential. I personnally became aware of the urgency of contributing to limit the seriousness of the present ecological crisis. I usually prefer to develop mathematics without looking for immediate applications, but the urgency justifies an exception to this principle.
Vincent Lafforgue is a senior researcher at the CNRS. He is a member of the Institut Fourier (a joint CNRS & Grenoble Alpes University maths lab).