Interview with Marie-France Vigneras, invited to give the Emmy Noether Lecture at ICM2022

Link to the virtual ICM 2022 talks

What is your field of research?

Representations of p-adic reductive groups.

What made you take up mathematics?

My teachers: of elementary maths at the Van-Vollenhoven high school in Dakar, and then at the University of Bordeaux.

How would you describe your profession ?

Fascinating, it’s like a drug.

Are there places or encounters that were decisive in your career?

My career has been easy, back then there was no recruitment problem at universities. I thought many times that I should choose to do something else in my life, but certain things were decisive in my not doing so: at Bordeaux, there was a stimulating atmosphere in the maths department, I met some fantastic visiting mathematicians, including Don Zagier, who made me dream of being a real mathematician, but the list is long. At Bonn, the great humanity of Fritz Hirzebruch; at a conference at Plans sur Bex, the charm of Ken Ribet; at the Ecole Normale Supérieure de Jeunes Filles, the joy of being with women mathematicians; at the Institut des Hautes Etudes Scientifiques, the friendship of Barry Mazur; at the Mathematics Research Institute at Berkeley, Jean-Pierre Serre giving a lecture on the  Galois representations of vector spaces over finite fields; and at the Radcliffe-Harvard Institute, the encounters with artists and researchers of all disciplines.

What findings have profoundly affected your ways of doing maths?

a) The Langlands programme. The famous book by Jacquet-Langlands resolved some questions that I struggled with while I was working on my PhD thesis; I also decided that I must understand the representation theory of reductive groups in complex vector spaces, and the theory of automorphic representations, which are part of that programme. I saw immediately the power of Langlands’ ideas. Using the arithmetic of quaternions, which was the subject of my thesis, and the Selberg trace formula, which is part of the Langlands programme, I proved that two different hyberbolic drums may have the same sound, providing a counter-example to a conjecture.

b) The conjectures that Jean-Pierre Serre presented in the lecture I  mentioned above: I decided to study the representations of p-adic reductive groups on vector spaces over finite fields. At that time, arithmeticians were not interested in the Langlands programme, and automorphic mathematicians were not interested in representations over finite fields. It was an advantage for me because it meant that I could work undisturbed, but the disadvantage was that my articles were not accepted for publication. For one of my referees, I had to write a book to justify my arguments about the Langlands correspondence for a p-adic linear group over fields of characteristic different from p.

c) Christophe Breuil’s classification of supersingular représentations of the linear group GL (2, Q_p) over a field of characteristic p:  I  should have seen it.   I started working on representations of reductive p-adic groups over vector spaces on fields of characteristic p. Unfortunately, despite two decades of work, neither I nor anyone else understands the supersingular representations of a general reductive p-adic group. I have simply shown, with Florian Herzig and Karol Koziol, that they exist.

What do you like about being a mathematician?

The constant learning and working, always surrounded by such intelligent young people as my students.

What is a “profound” result for you? Or an “elegant” solution?

A simple statement, with a difficult solution, and a new method that can be used to solve other problems.

An elegant solution is quick, simple, illuminating, and clever, and can use already-known substantive results.

How involved are you in disseminating ideas about mathematics to a broader audience?

For 30 years, it has been a cause of great stress to me to be obliged to give exams to students who know nothing about maths. They go to university after finishing high school because the engineering schools have a selection process but the universities don’t, and then they are going to become high school teachers of a subject that they neither like nor understand – and they will do considerable damage! It was suicidal to go on like this, but nothing has been done in 30 years to avoid this catastrophe.

What would you say to young people who want to take up a scientific career?

I can only speak about mathematics: even for students of the Ecole Normale Supérieure, it is difficult to find a position to do mathematical research in France and it is badly paid. 

Have you already decided what you will talk about at the ICM 2022?

The St Petersburg congress has been cancelled, but the planned lectures will be broadcast by Zoom and there will be published proceedings. I have just finished the text for those proceedings, which is different from the oral version, which is aimed at a broader audience.

What does this congress mean to you?

The ICM is very important. It results in great visibility, recognition of work done, and of the significance of a topic that I started, and is now  developed by many mathematicians, particularly in the remarkable work of Jean-Francois Dat.  Meetings among mathematicians are essential, so this ICM will be sorely missed. I would have preferred that a real ICM gathering be simply put off for a year.