Interview with Jean Fasel, invited speaker at ICM2022

Link to the virtual ICM 2022 talks

What is your field of research?

I’m interested in problems of an algebraic nature, such as, for example, understanding the structure of projective modules over a given ring. The techniques used to solve this kind of questions have greatly evolved in the last few years, especially with the introduction of the so-called motivic methods, which enable us to import ideas and techniques from algebraic topology to solve highly algebraic problems.

What are the origins of the problems you work on?

The origin of questions in mathematics is a complicated matter. There is an infinite number of interesting problems, and probably one of the ways to measure their interest is to see whether they have applications or significance in other areas of mathematics or fields close to it, such as physics, biology, or finance.

I’m in a field that is quite far from this kind of applications, although there are sometimes links to more concrete things such as information theory and coding. These questions are thus also quite “internal”, having been partially posed already in the 1970s and which we are only now beginning to answer partially.

What made you take up mathematics?

I never particularly liked maths at school. After finishing high school, I started studying mediaeval French and history. Quite quickly, I realized that that wasn’t really what I wanted to do. A physicist friend tried to convince me to do physics; I found it quite interesting but there was something that bothered me a bit: in physics, I had the impression of learning a profession, that is, something with which I could do something concrete afterwards, but I was not at all interested in that professional side of my training. At first, I chose mathematics as a challenge and then because I was absolutely sure that I would never have a job linked to maths. I took it up as a kind of training, a sort of mental gymnastics before going on to do something else. When I started studying seriously, I was immediately fascinated and I never left the gym! I seldom spend a whole day without thinking about maths, even on holidays.

What it is that you like about being a mathematician?

What I like the most is the creative freedom that is independent of everything. If I want to think about the theory of evolution in biology, I will need a laboratory, colleagues, funding, etc.  There is none of that in mathematics. We look into a question, and that’s all. If we are lucky enough, we will understand it. With a bit more luck, we will solve the problem. At the end of all that, then we can go on and think about something completely different and follow our interests.

Clearly, that description is a bit romantic because we are after all dependent on a few things: just to get a job, one must solve questions that are of interest to others. But once settled, one can think: now, I’m going to look into the relationships between maths and mediaeval literature, for example. This is an interesting point and sufficiently rare in the contemporary professional environment to be worth mentioning. That’s why I tell people who are leaning towards a scientific career, especially in mathematics, that this freedom remains for us despite of everything, and is perhaps greater than anywhere else.

What balance do you find between your teaching and your research?

We teach at all levels. The higher the teaching level, the closer the link between research and teaching. But this link is essential, it is difficult to do one without the other. To use the metaphor of mental gymnastics again, if one does research, one is sufficiently well trained to have a lot of insight into a subject and to know exactly where the difficulties lie, how to explain them and how to avoid the pitfalls.

What is a little frustrating, maybe, is the fact that mathematics is often taught as if it were something utilitarian: we teach not only people who will go on to practice mathematics, but also those who will choose IT or biology, and few of those choose maths out of pure interest. So, we always have to explain why we do things, while for us that is often only a secondary motivation. If we do maths, it’s because we are interested in them.
This is a problem that can be approached in terms of research and we can ask ourselves: what makes that particular question so fascinating to me? What aims can I achieve? Might this interest the students enough to get them through learning a difficult theorem? This is perhaps the strongest link between research and teaching.

How would you describe your profession?

My job is a passion; you have to love asking questions, be highly resistant to frustration, because even if we sometimes solve problems, the immense majority of the questions that we pose will not be solved by us, even possibly not during our lifetime, so it is a job in which one must be highly resilient in the face of frustration, and tough enough to think every day about problems that are beyond us. You must prefer seeking to finding. That’s what best describes mathematicians.

Are there any places or periods that were decisive for you?

I did my PhD in Switzerland in an institute in which there were only a few of us. There were fewer than a dozen teachers and very few students. The other PhD candidates were working on completely different subjects, and so it was very difficult to talk about maths apart from my PhD advisor. I had the feeling that I was doing my PhD in a kind of isolation.
I then did a postdoc at the Tata Institute for fundamental research in Mumbai, and I found myself in an institution with a lot of people, students and other people who were interested in what I had done in my thesis. Which surprised me a lot! I discovered that I was part of a greater whole, a world of mathematics that never stops, it was very stimulating. It was then that I really started to be a mathematician, when I realized that I was part of a big family rather than an isolated individual.

What, for you, is elegance in mathematics?

The aesthetic motivation is the primary motivation, for me at least. I think that the only measure of a demonstration or of a theory, other than the fact that it is true, is its aesthetic value. What is aesthetically pleasing? I tend to think that it is highly aesthetic to reveal a simple principle that can explain phenomena that are, a priori, very complicated.
To take an example from outside maths, I think that the theory of evolution is very aesthetically pleasing. Before that theory, there was a lot of confusion. Each species seemed to evolve according to its own rules. If one were a specialist in mice, for example, one might say: mice are like this and have evolved in this way, therefore it simply must be like this. And then if one thinks about giraffes, it is different. All of a sudden, Darwin comes along and say: there is a very simple principle which means that there are mutations, and the best mutations remain. And he explains things in a very clear manner. It is a discovery that I find beautiful.

Is there a strong link between mediaeval literature and mathematics?

The link that I saw as a student was the non-professional element. I did not see myself making a living with a degree in mediaeval literature. There was also the thing about amusement, or mental gymnastics, which might have been a first common feature.
And then the other common point is the aesthetic value. When one is interested in mediaeval poetry, one doesn’t really know how people pronounced things, it is difficult to reproduce the sounds. There was a transformation process from Latin to French and I think it was partly motivated by aesthetic values: if I have to choose among 100 words, maybe I will choose the one that I find the most pleasing, or pronounceable – we arrive again at the theory of evolution. And I think that it exists in maths too: if I must remember 100 different results, maybe I will choose the most aesthetically pleasing, or the one with the greatest significance.

What is your link to the dissemination of mathematics?

The most basic level is communication with colleagues in the same field of research. They sometimes want to know about details in a paper, or ask for opinions on the problems they face.  This is relatively easy, as there is a common ground, a common culture. The next level is the colloquium talk for an audience of mathematicians a bit further from our own area of research.  The idea is to present the state of affairs of a subject, the interesting problems, and what remains to be done. To my opinion, this is slightly harder than the previous level.
Then there is yet another layer, which is the lecture for a general audience. This is an exercise that I find very difficult. I have done a few, mostly for senior high school students. It is very hard to do something that is at the same time accurate, interesting and at a reasonable level of difficulty.

I think that there is a serious problem in mathematics, which is the lack of communication with the rest of the world and especially the political and professional worlds, even though many of my colleagues put much efforts into communicating interesting mathematics to pupils and high school students.

You can take any newspaper, one that you have been reading every day for years, you will have come across perhaps one article from time to time that talks about mathematics. It is a shame that it is not a bigger part of our culture because we live in a world where mathematics are present everywhere.

What’s more, there is a real interest in it. When I meet people and tell them that I’m a mathematician, they are immediately intrigued. My own limitations make communication quite hard, I have many problems to explain things in plain words. One can draw certain parallels, make analogies. They say, “Oh, so, is that what you do?” And I reply, “Er, no, not really, it’s an illustration.” They always leave a little disappointed!
The difficulty is that, to make an accurate statement, one needs some technical knowledge, and that knowledge is not widespread. A public lecture requires a great deal more preparation than chatting with a colleague in the office next door, which makes it difficult.