From EWM Newsletter no. 27 (2016)

Claire Voisin is Professor at Collège de France (Algebraic geometry chair), and member of the Académie des sciences (Paris). In September 2016 she was awarded the CNRS Gold medal, the highest scientific award in France. She has been plenary speaker at the International Congress of Mathematicians. She has been distinguished visiting professor at IAS (Princeton, 2014-2015) and invited Senior fellow of ETH (Zurich 2017). She is foreign associate of the Accademia Nazionale dei Lincei and the National Academy of Sciences.

Voisin is recognized for her work on Hodge theory and algebraic cycles. She is known particularly for her construction of compact Kähler manifolds not homeomorphic to complex projective manifolds, for her proof of the generic Green conjecture on syzygies of canonical curves, and for her contribution to the stable Lüroth problem

EWM: How would you explain your research to a non-specialist?

I am working on algebraic geometry. In algebraic geometry, we are given varieties in a very algebraic way, via their spaces of algebraic functions: if a variety is defined by a number of polynomial equations in affine ambient space, its space of algebraic functions is the quotient of the space of polynomial functions on the ambient space by the ideal generated by these equations. So an important part of algebraic geometry is in fact commutative algebra. On the other hand, if one works over the complex numbers, the varieties one gets can also be considered as topological spaces. One of my main interests is the topology of these spaces: the question is not only computing it, but relating it to intrinsic properties of our varieties as algebraic varieties. Consider topological invariants like cohomology: they depend only on the underlying topological space, not on the precise equations defining our variety in ambient space. There are now enhanced structures like Hodge structures on cohomology which depend usually on the complex structure and sometimes determine it. One can say that they reflect the properties of the variety as a complex variety or manifold. But in algebraic geometry there are still more intrinsic and algebraic invariants constructed via algebraic K-theory or the theory of algebraic cycles. Relating the three sets of data is what I call studying the topology of algebraic varieties. Note that I also worked very much on a sort of natural extension of algebraic geometry, namely compact Kaehler geometry, and proved that there are more topological types of compact Kähler manifolds than complex projective manifolds.

EWM: What made you choose maths as a career?

I would not say that I chose math as a career; I got interested, so I started, then I continued and it was a sort of addiction. I never really ‘thought’ of doing this, it’s like this was simply obvious and also the easiest way.

EWM: So it just happened?

How can I say; once I started seriously doing maths, there was no alternative. I got used to it, I had to do this. Since I started, I never wanted to do something different. I would even say I find it more and more interesting over time.

EWM: Did your family encourage you?

The fact is that my family did not care so much, because I come from a very large family: I have eight sisters and three brothers. My parents were very happy if we were independent and earned money. I left my family’s home when I was 17, I got a scholarship and, starting from this point, I never had to ask money from my parents. I should say that when I was a child I had some contacts with maths, especially geometry, but my parents did not care so much about our future careers; if I had been a teacher in high school they would have been happy.

EWM: The standard question now: how does it feel to receive such a prestigious prize, the CNRS medal?

It was a very big surprise. You can see the list of the previous medals, it is hard to believe that I am in the same league. But of course, it is very nice that my work gets this recognition.

EWM: Is the prize for a particular result or for your career?

I think for my whole career, but of course, I have some results which are more important than the rest of my research and more deserving of recognition, for example on the Hodge theory of projective and Kaehler manifolds, and also, in a different direction, on the syzygies of projective curves.

EWM: Do you recall any particular difficulty you faced in your work, in your career?

What is hard are the moments when you lack inspiration to formulate new ideas, new problems. Also some times it happened that I did some research which was unsuccessful. It is important to be able to stop something which does not work, not to spend too much time and energy on an idea that you drive by force. You need to change. I always found travelling very useful for this, because if you are alone you tend to stay stuck on a subject, while if you travel you get some distance and you can try something new, a new subject, your mind has a new drive, a new energy.

EWM: So far you referred to difficulties in maths. Did you have any other type of difficulties? Did you have good conditions to work all the time?

I had excellent working conditions, because I had no teaching, I could teach only when I wanted to, and in high level courses. I had a CNRS position, so I was able to work at home, no time and energy wasted in public transportation. Life was made very easy by my CNRS position; and you know the French system of child care, so I had no excuse for not working full-time.

I should mention that what made my life so very easy is that my husband is also a mathematician, so not only the every day schedule is much softer, but we understood both that we needed time for us. At the weekends, I used to work in the morning and he in the afternoon. That was nice, we both agreed that we should do things this way.

EWM: What do you love most in your job? What do you dislike most?

I like very much the moment I start a new research, I like very much the moment I have something in my mind: sometimes it is barely an idea, sometimes it’s just the beginning of something. But there is this quality of the dream, and the fact that your mind works alone, you do not need to force it.

I also like to give talks; this is a bit different, but I like it very much. I have to challenge myself to discuss, because I am what in French we call ’introvertie’. There is a lot of introversion in our work, because we are contemplating something. But there is also a part of our work that is different, discussing, giving talks, attending conferences, which is also nice. Still, for me, the very nice part of my job is when I work on something new by myself.

EWM: And the bad part?

The bad part….there is some bad part, some suffering, when you are trying to do something which is difficult. There are some moments when you spend much energy, and moments in which the dynamics of research is a little lost. You don’t feel you are inside of mathematics. But I am afraid this is especially bad for my family….

EWM: What kind of advice would you give to a student wanting to work in mathematics?

I don’t like much giving advice; I think doing mathematics is a very personal choice and I am not sure I would like to encourage anyone doing mathematics because it is not the best way of earning money, not the best way of having an easy life. I don’t think it’s an easy job. I don’t want to interfere, but if a person has this idea of doing maths I would suggest not being too specialised. It is not a good idea, learning only those things that you believe are interesting for your own research. Especially when you are young, you can learn a lot of things, you have memory and capacity. When you get older, you know more, but you lose part of the ability to learn new things. Young people should really try to learn a lot, not only on their thesis topics. It’s very important. A career in mathematics is long, it’s for forty years, if you start with a too small quantity of knowledge, it will be hard to make it substantial over a long time.

EWM: Why do you think there are so few women in certain areas of mathematics, in particular at high levels?

I don’t know. In my domain, algebraic geometry, they are fewer then men, still less than in other fields. I find that my male colleagues are very pleasant, not sexist, I never had a problem with that, so it cannot be because of that. I don’t see when the problem arise.

EWM: Well, for example there is a common belief that many girls leave after their PhD or even before because they get married, have children and they don’t have time for research. For example in many European countries most of the women work part time, or don’t work at all.

In France many women work, the child care system is excellent. It may be different in other countries but it does not concern France. But still there are a few women in maths. It is true that there are more women in biology, for example, but maybe for maths you need more free time, a lot of room to let your mind free, a lot of freedom is necessary in research. I have been very lucky with my CNRS position.

EWM: Do you have a suggestion to change things or since you don’t see the reasons, you don’t think there are solutions?

In France of course, as in many countries, we hear a lot about the parity issue, the government is making laws about parity. Personally, I think that these things are dangerous. The problem is that, actually, this will not improve this specific problem of having few women wanting to work in mathematics. I think the parity action is going the wrong way. I don’t think there are more women in pure maths than during the Seventies in France, after all that has been done to try improve the situation.

EWM: How do you succeed in balancing family and career and do you have time for hobbies, do you have other passions apart from maths?

I actually used to think the other way: I had five children, I was very passionate about educating them, about spending time with them and I think that family life helped me a lot. This job can be hard psychologically, with a lot of uncertainties related to research, the work sometimes goes very slowly. For me having to spend a lot of time with my children, having family duties, made me better balanced and that was in some sense very helpful.

You know, it’s hard to convince yourself that you are doing something very useful, if you do pure maths. I don’t ask myself a lot of these questions anymore but sometimes it is not clear if you are doing something very important. With a family, you don’t have to ask yourself this question. I think I would have had much more psychological difficulty in my research if I had not had my family.

For the rest, I don’t really have free time between my family, my research, some travels. Of course, I like to read literature, in my ‘free time’ at night but I don’t have time to do something more substantial.

EWM: Can I ask you what you are reading now?

I have just read “Purity” by Jonathan Franzen, and then I am reading a very recently published book by Laurent Mauvigner, a French author, which I appreciate very much.

EWM: What do your children think of your work, are they happy you are so busy or resent you not having much for them?

My children are grown up now, so no reason to complain anymore, but in any case I never felt my work was a problem for my children. I have always spent much time with them as children, especially on vacation. I also did research during vacations, but I have five children, they played, they had time on their own and we spent a lot of time doing things together. I never felt they were unhappy with me doing research.