# Pierre Deligne: Early Years

Some quotes from  Interview with Abel Laureate Pierre Deligne, by Martin Raussen (Aalborg, Denmark) and Christian Skau (Trondheim, Norway), European Mathematical Society  Newsletter, September 2013, pp. 15-23.

You were born in 1944, at the end of the Second World War in Brussels. We are curious to hear about your first mathematical experiences: In what respect were they fostered by your own family or by school? Can you remember some of your first mathematical experiences?

I was lucky that my brother was seven years older than me. When I looked at the thermometer and realized that there were positive and negative numbers, he would try to explain to me that minus one times minus one is plus one. That was a big surprise. Later when he was in high school he told me about the second degree equation. When he was at the university he gave me some notes about the third degree equation, and there was a strange formula for solving it. I found it very interesting.

When I was a Boy Scout, I had a stroke of extraordinary good luck. I had a friend there whose father. Monsieur Nijs, was a high school teacher. He helped me in a number of ways; in particular, he gave me my first real mathematical book, namely Set Theory by Bourbaki, which is not an obvious choice to give to a young boy. I was 14 years old at the time. I spent at least a year digesting that book. I guess I had some other lectures on the side, too.

Having the chance to learn mathematics at one’s own rhythm has the benefit that one revives surprises of past centuries. I had already read elsewhere how rational numbers, then real numbers, could be defined starting from the integers. But I remember wondering how integers could be defined from set theory, looking a little ahead in Bourbaki, and admiring how one could first define what it means for two sets to have the “same number of elements”, and derive from this the notion of integers.

I was also given a book on complex variables by a friend of the family. To see that the story of complex variables was so different from the story of real variables was a big surprise: once differentiable, it is analytic (has a power series expansion), and so on. All those things that you might have found boring at school were giving me a tremendous joy.

Then this teacher, Monsieur Nijs, put me in contact with professor Jacques Tits at the University of Brussels. I could follow some of his courses and seminars, though I still was in high school.

It is quite amazing to hear that you studied Bourbaki, which is usually considered quite difficult, already at that age.

Can you tell us a bit about your formal school education? Was that interesting for you, or were you rather bored?

I had an excellent elementary school teacher. I think I learned a lot more in elementary school than I did in high school: how to read, how to write, arithmetic and much more. I remember how this teacher made an experiment in mathematics which made me think about proofs, surfaces and lengths. The problem was to compare the surface of a half sphere with that of the disc with the same radius. To do so, he covered both surfaces with a spiralling rope. The half sphere required twice as much rope. This made me think a lot: how could one measure a surface with a length? How to be sure that the surface of the half sphere was indeed twice that of the disc?

When I was in high school, I liked problems in geometry. Proofs in geometry make sense at that age because surprising statements have not too difficult proofs. Once we were past the axioms, I enjoyed very much doing such exercises. I think that geometry is the only part of mathematics where proofs make sense at the high school level. Moreover, writing a proof is another excellent exercise. This does not only concern mathematics, you also have to write in correct French – in my case – in order to argue why things are true. There is a stronger connection between language and mathematics in geometry than for instance in algebra, where you have a set of equations. The logic and the power of language are not so apparent.

You went to the lectures of Jacques Tits when you were only 16 years old. There is a story that one week you could not attend because you participated in a school tip…?

Yes. I was told this story much later. When Tits came to give his lecture he asked: Where is Deligne? When it was explained to him that I was on a school trip, the lecture was postponed to the next week.

He must already have recognised you as a brilliant student. Jacques Tits is also a recipient of the Abel Prize. He received it together with John Griggs Thompson five years ago for his great discoveries in group theory. He was surely an influential teacher for you?

Yes; especially in the early years. In teaching, the most important can be what you don’t do. For instance, Tits had to explain that the centre of a group is an invariant subgroup. He started a proof, then stopped and said in essence: ‘An invariant subgroup is a subgroup stable by all inner automorphisms. I have been able to define the centre. It is hence stable by all symmetries of the data. So it is obvious that it is invariant.”

For me, this was a revelation: the power of the idea of symmetry. That Tits did not need to go through a step-by-step proof, but instead could just say that symmetry makes the result obvious. has influenced me a lot. I have a very big respect for symmetry, and in almost every of my papers there is a symmetry-based argument.

Can you remember how Tits discovered your mathematical talent?

That I cannot tell, but I think it was Monsieur Nijs who told him to take good care of me. At that time, there were three really active mathematicians in Brussels: apart from Tits himself, Professors Franz Bingen and Lucien Waelbroeck. They organised a seminar with a different subject each year. I attended these seminars and I learned about different topics such as Banach algebras, which were Waelbroeck’s speciality, and algebraic geometry.

Then, I guess, the three of them decided it was time for me to go to Paris.Tits introduced me to Grothendieck and told me to attend his lectures as well as Serre’s. That was an excellent advice.