An Interview with Christophe Soulé

This interview with Christophe Soulé, CNRS Emeritus Research Director at IHES and member of the French Academy of Sciences, was conducted in May 2025 by Pierre Pansu, Professor of Mathematics Emeritus at Université Paris-Saclay.

Pierre Pansu (P.P.): Let’s start at the beginning. Tell us a little about your family background.

Christophe Soulé (C.S.): My father graduated from the École Polytechnique and worked as a mathematical physicist at the CEA (French Alternative Energies and Atomic Energy Commission). He was always a role model for me and took an interest in my career and research. This undoubtedly played a part in my beginning to obtain original mathematical results at a very early stage.

P.P.: You began your career as a research associate at the CNRS before you had even written your thesis.

C.S.: Yes, that was possible at the time. In my graduating class, six of us were fourth-year students at the École Normale Supérieure in that situation. As for me, I hadn’t even found a thesis topic yet!

With Serre’s help, however, I had already proved my first theorem. This was during my third year at the École Normale. I had begun asking myself questions following a postgraduate course taught by Godement. He had a gift for slipping research questions accessible to young students into his lectures. I had given a presentation on Fuchsian groups, and he asked me whether, as with $SL_2(\mathbb{Z})$, there was also a tree associated with these groups. After reading Trees, Amalgams, $SL_2$, I wrote to Serre proposing an idea for constructing a tree in higher dimensions. He replied with a letter that I have kept, beginning: “Your statement is quite natural…” The letter also contained an idea for a proof, which I was able to develop.

P.P.: Was this your work on the cohomology of $SL_3(\mathbb{Z})$?

C.S.: Yes, exactly!

P.P.: Did your computation of the cohomology follow from describing $SL_3(\mathbb{Z})$ as an amalgamated product?

C.S.: No, it followed from defining a suitable space for $SL_3(\mathbb{Z})$, analogous to the tree for $SL_2(\mathbb{Z})$. This space has dimension 3, whereas the symmetric space has a higher dimension. It can be viewed as a skeleton of the symmetric space, made up of three-dimensional simplices glued together in a nontrivial way. The space carries a group action whose stabilizers are finite because they are all contained in a compact group. This allowed me to compute the entire cohomology of $SL_3(\mathbb{Z})$.

P.P.: And what happens for $SL_4(\mathbb{Z})$?

C.S.: Two years after my work, R. Lee and R. H. Szczarba obtained the first results in that direction, although they remained incomplete.

Around that time, Jean-Luc Brylinski had asked me about the characteristic classes of $SL_n(\mathbb{Z})$. I hadn’t really understood the question because I knew nothing about characteristic classes. Nevertheless, this was precisely the direction I subsequently decided to explore.

There were characteristic classes in $SL_n(\mathbb{R})$ or $SL_n(\mathbb{C})$ that could be restricted to $SL_n(\mathbb{Z})$. I wondered what cohomology classes one would obtain.

Following an idea suggested by my doctoral supervisor Max Karoubi, I was able to establish lower bounds for the orders of certain Chern classes. Indeed, these classes lie in finite abelian groups and are torsion classes (they have finite order because the cohomology of a finite group is torsion). By restricting these Chern classes to diagonal subgroups, one can obtain lower bounds for the orders of the characteristic classes.

Later, while reading a paper by Grothendieck, I realized that an upper bound for the orders of these Chern classes was already known. In his article, he showed that characteristic classes in étale cohomology are finer than ordinary characteristic classes. The example he gave was precisely the upper bound for the orders of the Chern classes I was studying! I then wrote to Loday, telling him: “Whether we like it or not, we’re going to have to learn algebraic geometry.” And that is what I did.

P.P.: How did you go about learning Grothendieck’s mathematics?

C.S.: At the time, the main reference was the EGA. Studying it required an enormous effort, and I never undertook that task. I learned the theory through an introductory text on scheme theory by Jean Dieudonné, and above all I was fortunate enough to be able to ask Pierre Deligne questions. I particularly remember asking him: “Is étale cohomology, like K-theory, invariant under taking the product with the affine line?” In fact, I learned as I went along, relying on first-hand sources such as Deligne and Breen.

P.P.: You just mentioned Jean-Louis Loday and K-theory. How did you come to work on this subject?

C.S.: I met Jean-Louis Loday at the ICM in Vancouver in 1974. We became friends, and he introduced me to K-theory and the remarkable conjectures of Lichtenbaum, which predicted that the orders of K-theory groups were determined by Bernoulli numbers.

In my thesis [2], I thus showed that $K_2(\mathbb{Z})$ contains an element of order 691, which is the numerator of the twelfth Bernoulli number.

Moreover, the K-theory of $\mathbb{Z}$ is related to the homology of $GL_n(\mathbb{Z})$, which brings us back to the homology of $SL_n(\mathbb{Z})$. The K-theory of $\mathbb{Z}$ was therefore a continuation of my work on $SL_3(\mathbb{Z})$, $SL_4(\mathbb{Z})$, and so on, and the question of stable homology for $SL_n(\mathbb{Z})$ arose naturally.

I later showed that K-theory groups were at least as large as predicted by Lichtenbaum’s conjecture (see below).

P.P.: Many of the results you mention appear in your doctoral thesis, for which you were awarded the CNRS Bronze Medal.

C.S.: Yes… Lichtenbaum was actually one of the referees for my thesis, and his report was rather positive. I don’t know whether that played a part in my receiving the CNRS medal. I met Lichtenbaum later at Cornell (where I had been invited by Brown), and I have excellent memories of that visit and my interactions with him.

P.P.: Let’s return to the beginning of your career. How did you come to write your thesis under Karoubi?

C.S.: Godement had given me my first research topic, but it wasn’t something that really interested him. He worked on automorphic forms, and in fact I attended his seminar with Jean-Pierre Labesse for a long time.

I became interested in discrete groups because they were classical objects that appeared very early in courses at the École Normale (free products, amalgams, the van Kampen theorem, etc.). As I said, Godement wasn’t interested in these questions, and although Serre tended to guide young students, he took on very few doctoral students. So I had to find another thesis supervisor, and Karoubi agreed.

P.P.: Could you tell us a little more about your fascination with Bernoulli numbers?

C.S.: For me, it all began with the connection between Bernoulli numbers and finite subgroups of $SL_n(\mathbb{Z})$. The orders of these groups are related to the denominators of Bernoulli numbers. This observation goes back to Minkowski and Burnside and also appears as an exercise in Bourbaki [1].

By restricting the cohomology of $SL_n(\mathbb{Z})$ to its finite subgroups, one can use the connection with Bernoulli numbers to say something about characteristic classes. In fact, this is how one obtains the result that the denominators of Bernoulli numbers provide lower bounds for the orders of the torsion Chern classes, a result we discussed earlier.

Bernoulli numbers are also related to the values of the Riemann zeta function at negative odd integers. Later, while working on Arakelov theory, I encountered the values of the derivative of the zeta function at negative odd integers.

P.P.: I imagine that’s no coincidence…

C.S.: Not at all. There is actually a rather deep connection. Indeed, the formal power series defining the Todd class in the Riemann–Roch theorem involves all the Bernoulli numbers. The fact that these numbers reappear through the zeta function in Arakelov theory shows that this theory is finer than ordinary topological theory: it is not merely the values of the zeta function that appear, but the values of its derivative.

P.P.: Let’s return to your study of étale cohomology. Did this theory ultimately prove useful in your research?

C.S.: The question I was asking myself was whether one could obtain more than Grothendieck had obtained for $H^0$. In particular, I was looking for an upper bound for $H^k$, with $k>0$. Quillen had advised me to consult Illusie, but he couldn’t help me. Eventually, I obtained a letter he had written about K-theory in the late 1970s, and this letter helped me establish the connection between K-theory and étale cohomology. I was thus able to define a map from K-theory to étale cohomology which, under certain conditions, is surjective. I therefore obtained an optimal lower bound and proved part of Lichtenbaum’s conjecture (this is the result I referred to earlier).

I had no idea how to prove the injectivity of this map, and with good reason: the complete proof only came much later, through Voevodsky’s work.

P.P.: Does your thesis contain other results that have had a lasting impact?

C.S.: In a way… I realized much later that I could have deduced from a result in my thesis ([2], Section 2.2.7.3) a nontrivial statement related to Vandiver’s conjecture. Lee and Szczarba’s computation of $K_4(\mathbb{Z})$ implies that the corresponding étale cohomology is small and, in fact, vanishes. Its vanishing means that a certain Tate twist of the class group of the cyclotomic field vanishes. This is consistent with Vandiver’s conjecture, which states that the “+” part of the class group vanishes. I had been using Chern classes to obtain lower bounds for K-theory in terms of étale cohomology, whereas here I needed to reverse the argument and obtain an upper bound for étale cohomology in terms of K-theory. Unfortunately, it never occurred to me to make this reversal. Fifteen years later, Kurihara finally had the idea and proved the result.

P.P.: Let’s turn to your work with Henri Gillet.

C.S.: Gillet and I were both part of the community working on K-theory. He had written a thesis connecting K-theory to cohomological theories satisfying a certain number of axioms. What brought us together was our interest in characteristic classes in K-theory. Later, I came up with a question and sent a letter to Thomason and Gillet asking whether they knew how to solve it. That must have been in 1980 or 1981. Gillet replied, and we began a collaboration that lasted until 2000. We were very good friends and complemented each other extremely well: he had a very abstract perspective, whereas I was highly intuitive, more inclined to guess how things worked. We got along wonderfully. Even today, some people think that Gillet–Soulé is a single person!

P.P.: So your collaboration began with a very specific question, and you ended up developing an entire theory together…

C.S.: We had almost answered my original question when Faltings proved the Mordell conjecture and developed Arakelov theory in his paper Calculus on Arithmetic Surfaces. Faltings was working in dimension 2, and Gillet and I wondered what happened in higher dimensions. There was also a 1984 paper by Manin suggesting that a general Riemann–Roch theorem should be proved for Arakelov theory. We therefore wanted to develop the theory in higher dimensions, formulate a Riemann–Roch-type statement, and prove it. This took us about ten years of work, but we succeeded. The clean, definitive formulation of 1992 owes a great deal to Bismut. It is the arithmetic analogue of the Grothendieck–Hirzebruch–Riemann–Roch formula, which computes the Euler characteristic of a vector bundle.

P.P.: How did you begin working with Bismut?

C.S.: I had many discussions with Quillen, and he kept telling me that I should go and talk to Bismut. Eventually, I did!

He had worked on the index theorem using probabilistic methods. Quillen had defined a metric for Riemann surfaces and had suggested, in a short paper that Gelfand had encouraged him to write, that analytic torsion could be used to define a metric on the determinant line bundle, and that the curvature of this object should yield the Riemann–Roch theorem at the level of differential forms. Bismut and Freed then developed the techniques needed to address this question in full generality, that is, in higher dimensions. This was part of a major project that Bismut himself pursued over many years. The technical difficulties were extremely considerable.

I have just mentioned the curvature of the determinant line bundle, but in fact, the arithmetic Riemann–Roch theorem in the Arakelov setting computes the determinant line bundle itself (together with its metric), not merely its curvature. To achieve this, one needs very precise control of analytic torsion. Gillet and I had developed machinery for producing a Riemann–Roch-type statement, but that statement depended on results from analysis. So we naturally turned to Bismut for these questions.

Recollections of my work with Christophe Soulé by Jean-Michel Bismut

P.P.: What you’re describing is what happens at the archimedean places, but what happens at the places corresponding to prime numbers?

C.S.: There are two ways to answer this question. The first is to take the Riemann–Roch theorem over $\mathbb{Z}$ with a completion at infinity: that’s what Henri and I did. The other approach, which is even better, is to complete at every place. This is what is done in arithmetic: one considers the adeles, that is, the product of the p-adic and real components.

There should therefore be a theory over $\mathbb{Q}_p$ analogous to Bismut’s theory of analytic torsion at infinity. Such a theory has not yet been developed. The adelic theory that ought to follow Arakelov theory is still in its infancy. What we currently have is a theory of adelic intersection, due to Zhang and Yuan. Even for the theory of Chern classes, much remains to be done. We understand $c_1$ for line bundles, but not Chern classes in the p-adic setting.

Nevertheless, we have a vague indication. If we take a graph, we have the determinant of the Laplacian on that graph, and we might therefore wonder whether the p-adic theory could be a generalization of this. But we are still far from understanding it. There are serious difficulties: Chambert-Loir and Ducros are currently working extensively on these questions, but the theory is not yet coming together.

P.P.: Could you explain why some people speak of Arakelov theory and others of Arakelov geometry?

C.S.: Szpiro spoke of Arakelov theory because he was interested in surfaces with significant arithmetic consequences. Gillet and I spoke of geometry because generalizing the theory to higher dimensions was a purely formal undertaking. We weren’t thinking about applying the theory. That was the direction Gillet and I had taken: we developed the theory for its own sake, as a kind of geometry.

P.P.: And has your geometry, in turn, led to arithmetic applications?

C.S.: Not yet, but I certainly hope it will! I’m thinking in particular of the conjectures of Vojta and Lang on arithmetic surfaces, which extend to all dimensions and are related to the ABC conjecture.

P.P.: How did you come to present Voevodsky’s work at the ICM in Seoul?

C.S.: Although Voevodsky’s results imply Lichtenbaum’s conjectures, on which I had worked extensively, his work is not directly related to mine. In fact, I presented Voevodsky’s results at the 2002 ICM because Suslin had declined. It was, of course, a great honor for me, and I explained what I understood of his work without examining the proofs in detail. I learned about Voevodsky’s theorems in Karoubi and Loday’s K-theory seminar.

P.P.: Let’s talk about another Fields Medalist whom you supervised, Laurent Lafforgue.

C.S.: Lafforgue completed his master’s degree under my supervision while he was in his third year at the École normale supérieure. I had asked him how Arakelov theory could be used to prove Dyson’s lemma. He wrote to Bombieri, who replied that he saw no point in using such elaborate machinery to prove a theorem that was already well understood.

For my part, I wondered whether Arakelov theory could improve the bounds in Dyson’s lemma, and I saw the project mainly as a warm-up for a thesis. After Bombieri’s reply, Lafforgue stopped making progress. I scolded him a little, and within three weeks he had solved the problem.

He then had the choice, as the top student in his class, of joining the CNRS or entering the École nationale d’administration. I called the director of the École normale supérieure mathematics department and told him that he absolutely must not steer Lafforgue toward the CNRS. Lafforgue himself was uncertain about his choice. I told him: “If you want to bang your head against the wall as hard as you like, choose research!” A week later, he had chosen the CNRS. I had appealed to his emotions!

He had told me that my idea involving Dyson’s lemma wasn’t leading anywhere particularly interesting, so I let him choose his own research topic, following the same approach that had been used with me at the time. Since he had job security, that seemed justified to me. He explored several topics, two or three in succession, and each time he had a good idea, but it didn’t work out. Eventually, he began working with Laumon, who gave him a magnificent problem that led to his Fields Medal.

P.P.: How did you get to IHES?

C.S.: My first memory is of giving a talk about my thesis with Deligne in the audience. I also remember Deligne’s Tuesday afternoon lectures. He chose a different subject every year. It began with a course on his proof of the Weil conjectures. There was also a seminar at Orsay on the same day.

Later, I complained to Kuiper that, at the university, we never knew who was at IHES. He then invited me to come and work at the Institute. His aim was to attract people from the university to IHES. He also wanted to bring in Laumon, for example. At the time, IHES was trying to broaden its audience.

So I arrived at IHES as a visitor in 1982, and I stayed. I asked Serre to support my application, and he advised me to apply every year. IHES did not yet have a CNRS laboratory, so I could not be permanently assigned there. One day, I was summoned by Verdier, director of the mathematics department at Paris 7, who asked me to choose, and I chose IHES.

I remained a visitor at IHES until 2007 and have held a permanent position at the Institute ever since. I have benefited enormously from the exceptional environment at IHES, which has allowed me to discuss mathematics with people such as Quillen, Friedlander, and Mazur.

P.P.: We would now like to discuss your connections with the various scientists who have spent time at IHES. Let’s begin with Dennis Sullivan. How did you come to interact with him?

C.S.: In a note published in the Comptes Rendus de l’Académie des Sciences, Sullivan had proved that the Euler class of $SL_n(\mathbb{Z})$ is annihilated by the denominator of the nth Bernoulli number. This result interested me greatly because of its connections with my own work at the time, so I went to see him and asked whether he would be willing to supervise my thesis.

We talked, and at the end he said to me: “You and I do not have the same conception of mathematics.” I was flattered that he credited me with having a “conception of mathematics,” but it also meant the end of our discussions. I think what he meant was that we were interested in different aspects of the problem we were studying. What interested me was the arithmetic aspect of the statement, whereas he was interested in the fact that the Euler class was torsion, as part of his broader investigations into Euler classes of discrete groups.

P.P.: Could you also tell us about your relationship with computers?

C.S.: Jean-François Mestre, an old friend from the École Normale who later became a colleague at Paris 7, introduced me to computers. I had a complicated calculation to perform involving a polynomial depending on n, and he helped me carry it out using a computer. This marked the beginning of my work on computing the arithmetic Todd genus.

I told Mestre that I wasn’t sure about the signs of the polynomial’s coefficients, and he said that with a computer, we could “try out” the different possible signs. So we got to work and tested the various possibilities. Each time, the computer produced a horribly complicated result from which nothing useful could be extracted.

Finally, I said to him, never mind, let’s try one last possibility and then go for a drink. He programmed the final choice and, miraculously, we obtained a beautiful result. We could immediately see that there was a theorem to be derived from it, even before we understood what was going on. It was extraordinary!

Since then, I have continued using computers to help formulate conjectures. At the time, I used Maple and Maxima. This must have been in 1987 or 1988.

That same year, I refined my ideas and, with the help of Maxima, conjectured that two numbers I had defined for every integer n were equal. At the time, I was in Bonn for an Arbeitstagung, and I asked Don Zagier to look into the question. I had written out the definitions of these numbers on two pages and left my notes on Zagier’s desk in Bonn. Five days later, he returned six handwritten pages explaining why the two numbers were equal, using Stirling numbers.

Zagier’s calculation (which was published as an appendix to Analytic Torsion and the Arithmetic Todd Genus) produced the famous series involving the derivative of the zeta function and enabled Gillet and me to prove the arithmetic Riemann–Roch conjecture, which introduces a new, explicitly defined Todd genus.

In general, I think computers are extremely useful in mathematics. They make it possible to tackle difficult calculations or explore areas where one has ideas but is uncertain about one’s intuition. They are particularly useful for finding counterexamples. In my own work, I have used computers extensively to test ideas. If the computer produced nothing useful, I moved on to something else. For this, I didn’t need extraordinary computing power; I worked mainly with Maple on a desktop computer.

Later, with Herbert Gangl and Philippe Elbaz-Vincent, we used computers to calculate the cohomology of $SL_n(\mathbb{Z})$ for $3<n<8$. For that work, we used the most powerful machines available at the time.

P.P.: Could you also tell us about Voevodsky’s relationship with computers and his work on the foundations of proof assistants?

C.S.: As everyone knows, Voevodsky was deeply affected by a gap in one of his proofs. He therefore wanted to develop tools to prevent this kind of mishap, which in his case was fortunately possible to remedy.

His approach to proof assistants was very abstract and, to my knowledge, has not been pursued further. However, I saw that the Collège de France organized a conference on this subject in 2025.

P.P.: Could you tell us about your exchanges with Ofer Gabber?

C.S.: Gabber is truly an extraordinary person. You can ask him any question, and he will answer it. The only requirement is that the question be precise. He doesn’t answer questions in which certain terms are left vague.

In fact, simply asking him a question is extremely useful, because it forces you to formulate the question precisely for yourself. Once you’ve done that, you’re never disappointed by his answer!

Just two days ago, I asked him another question, and I’m eagerly awaiting his reply. I don’t remember when it started, but I ask him a great many questions. Gabber is an exceptionally generous person and an invaluable member of the mathematical community.

P.P.: Could you tell us about your work on motives and the Beilinson–Soulé conjecture?

C.S.: From my perspective, this is a continuation of my thesis work, in which étale Chern classes appeared. What interested me was computing the K-theory of $\mathbb{Z}$. For a given K-theory group of $\mathbb{Z}$, unlike topological Chern classes, there is not just one étale Chern class but several. In fact, each of these Chern classes detects a part of the K-theory group. What I conjectured was that the K-theory of $\mathbb{Z}$ contained no more parts than those detected by the étale Chern classes.

During a trip to Moscow, I met Alexander Beilinson, who told me that he had given these parts of K-theory a name: “motivic cohomology.” This was the first time an adjective had been used instead of the noun “motive,” a profound concept that intimidated the mathematical community because establishing motives in Grothendieck’s sense would require proving the standard conjectures.

From Beilinson’s point of view, motivic cohomology could not exist in negative degrees, and this corresponded exactly to the number of groups that appeared in the K-theory of $\mathbb{Z}$ from my perspective. This correspondence is what is called the Beilinson–Soulé conjecture. So I arrived at it not through motives but through K-theory.

That must have been in 1982 or 1983. I had traveled to Moscow with a small group of mathematicians, including Hélène Esnault, Eckart Viehweg, Jean-Luc Brylinski, and Christian Kassel. We had left West Berlin for East Berlin, where we caught a flight to Moscow.

We had been invited by our Moscow colleagues to give talks at their seminar. The trip gave me the opportunity to meet Yuri Manin, Alexander Beilinson, and Vadim Schechtman, whose work I already knew.

The Beilinson–Soulé conjecture remains open to this day and is still a great mystery to me.

P.P.: You have also worked on the field with one element. Could you tell us more about that?

C.S.: The field with one element is an idea introduced by Jacques Tits in a 1957 paper on Chevalley groups. In that paper, he observed that everything behaves as though there were an underlying field with one element associated with these groups. For example, if we take $GL_n(k)$ and let $k$ be the field with one element, then $GL_n(k)=\mathfrak{S}_n$.

It had been known since Springer that the characters of $GL_n(k)$, where $k$ is a field with p elements, recover the characters of the symmetric group when one sets $p=1$. This is what motivated Tits to define the field with one element.

There was little follow-up to Tits’s paper until the Russian mathematician Alexander Smirnov revived the idea after noticing a connection with the zeta function and the Riemann hypothesis. If there were a field with one element and $\operatorname{Spec}(\mathbb{Z})$ were a curve over that field, one could try to adapt Weil’s proof of the Riemann hypothesis for curves over finite fields, and this would yield the Riemann hypothesis.

In 1993, Manin wrote a programmatic paper calling for the development of algebraic geometry over the field with one element. I had this in mind when I began trying to define varieties over the field with one element. I found a rather complicated definition and wrote a paper on the subject, in which I made it clear that this was only a preliminary attempt and that other definitions were undoubtedly possible.

In the years that followed, other mathematicians proposed alternative definitions, but unfortunately no comparisons were made between these definitions, which ultimately appear quite different.

Alain Connes and Caterina Consani devoted considerable effort to this question. They found a very elegant definition, and for Connes, this was indeed part of his thinking about the Riemann hypothesis. At present, work on the field with one element has once again reached a standstill, and I must admit that it is no longer among my concerns.

P.P.: What was the nature of your exchanges with Alain Connes?

C.S.: We have never really worked together, but we have had many discussions. The only time I took up his ideas was precisely in connection with the Riemann hypothesis. Connes described it in terms of an operator, and I tried to generalize his idea so that it would work for the Godement–Jacquet zeta functions.

P.P.: Was it also through Alain Connes that you began working with Jean-Benoît Bost?

C.S.: Yes, Bost was a student of Alain Connes. He began by working on $C*$-algebras and later became interested in Arakelov theory.

When he came to IHES, he did extraordinary work with the physicist Thomas Jolicoeur, observing that a statement in string theory could be understood through Arakelov theory. This was around 1986, at about the same time that Manin and a co-author made the same discovery.

We therefore organized a seminar on the subject. It began with a physicist writing a complicated formula on the blackboard, followed by a second talk in which I started from that physics formula and gradually transformed it into a mathematical formula, namely the Riemann–Roch formula in Arakelov theory.

The seminar continued, and at the end Laurent Moret-Bailly gave a talk on his recent work. It was truly exciting!

The connection between mathematics and theoretical physics was very much in the air at the time, culminating in the ICM in Kyoto, where people joked that three “Quantum Fields Medals” had been awarded: to Drinfeld, Jones, and Witten.

Later, Bost, Gillet, and I also worked on heights on arithmetic varieties, following a paper by Gerd Faltings. We eventually produced a very systematic paper, characteristic of Jean-Benoît’s mathematical style: he likes to explore every aspect of a concept.

That is also what he has done very recently with François Charles, developing a theory of numbers in infinite dimensions. In their work, they examine all possible invariants on a lattice and extend these invariants to a lattice of infinite rank. A paper with arithmetic applications, which I imagine will once again be very comprehensive, is due to appear soon.

Jean-Benoît is an extremely rigorous mathematician whose work I greatly admire.

P.P.: We recently learned that Nicușor Dan, your former doctoral student, has been elected President of Romania. Could you tell us more about his career?

C.S.: Dan came to France after brilliant secondary studies in Bucharest, during which he notably achieved the maximum score at the International Mathematical Olympiad twice in succession.

He came through an exchange program between the École Normale and Romania and was admitted as an international student at the École normale supérieure. He completed a master’s degree in algebraic geometry at Orsay and, for his thesis, chose Arakelov theory and came to see me. This was at the time when Damian Rössler was also my student.

Dan wrote an excellent thesis, which was published in Mathematische Annalen in 2002 in an article entitled “Prolongement méromorphe des courants de Green.” He could certainly have pursued an academic career, but he preferred to become involved in public and political life.

Daniel Barsky was co-supervisor of his thesis because IHES could not award doctorates at the time, so I had to work alongside a university professor.

After his thesis, Dan returned to Romania, where he continued doing mathematics until 2011. At the same time, he founded an École Normale Supérieure in Bucharest, modeled on the one in Paris.

He then devoted himself to an association he had founded to combat uncontrolled urban development in Bucharest. This eventually led him to become Mayor of Bucharest in 2010. The rest is history.

I visited him in Bucharest when he founded the École normale supérieure there in the early 2000s. I am now trying to obtain his contact details so that I can congratulate him, as he seems to have changed his email address.

P.P.: Could you tell me about your work in mathematical biology?

C.S.: In fact, my interest in biology comes from my father, who told me at the time that there was a field destined to develop: the intersection of mathematics and biology. This was in 1967.

He said that the day would come when biologists would have so many results that they would need mathematics to make sense of them. His prediction proved entirely correct.

When I was a student at the École normale supérieure, I considered working on mathematics related to biology. I spoke to the director of studies, who told me that there was nothing interesting in that field and that, in any case, I would need a medical degree to pursue that direction if I wanted to be respected in the field.

That frightened me, and I ended up choosing pure mathematics, telling myself that I would see later whether that direction was still possible.

I also tried reading Thom’s work on structural stability and morphogenesis, but I didn’t get very far. At the time, it was one of the few active areas connecting mathematics and biology.

When I turned 30, I began learning a little biology by attending François Jacob’s lectures at the Collège de France. I was studying biology for pleasure until the day Misha Gromov organized a mathematics–biology seminar at IHES. It was then natural for me to take part.

Through this seminar, I came into contact with people in Évry and found a problem concerning positive circuits in gene networks that was genuinely mathematical. The conjecture stated that positive circuits must exist in gene networks if those networks admit multiple steady states.

This can be translated into a precise mathematical statement, which I proved in the early 2000s. Afterward, I continued discussing these questions with Nadya Morozova, but when she moved to the Institut Curie, the collaboration gradually came to an end.

Gromov, for his part, remains interested in mathematical biology and teaches courses on the subject at IHES.

I look back on this work with great pleasure, and I believe there is still much to be done in this direction, particularly by younger generations!

Reference

[1] N. Bourbaki, Groupes et Algèbres de Lie, Chapter 3, Exercises 6, 7 and 8, pp. 273–274.

[2] C. Soulé, Groupes arithmétiques et K-théorie des anneaux d’entiers de corps de nombres. Thèse d’État. 1978.

IHES wishes to thank Cécile Gourgues for typesetting Christophe Soulé’s thesis.