Jean-François Le Gall is a Professor at Université Paris-Sud. He is laureate, together with Gregory Lawler, Professor at Chicago University, of the 2019 Wolf Prize in Mathematics.
Would you tell us about your research work?
I am a specialist of probability theory. My first research contributions pertained to Brownian motion. To put it simply, Brownian motion models the movement of a particle which constantly changes direction in a totally random way, and one is interested in the geometric properties of the trajectory thus obtained. Later, I studied models describing the evolution of a population of particles subject to both Brownian motion and a branching phenomenon: imagine for instance that at random times, each Brownian particle either dies or splits into two new particles. A motivation for my interest in such models came from the existence of deep links with another domain of mathematics, the theory of partial differential equations. At the same time, I became interested in "continuous random trees", which describe the genealogy of large populations, and I constructed such models from the other classical objects of probability theory called Lévy processes. During the last twelve years, I have concentrated on a new branch of probability theory: random geometry. The goal is to understand the geometric properties of large graphs randomly drawn in the plane. This study has led to the introduction of fascinating new mathematical objects, in particular the Brownian map, which is a random metric space arising in the continuous limit of large discrete random graphs. Much of my work has been about proving the existence and uniqueness of the Brownian map. Random geometry has close connections with other fields of mathematics, especially combinatorics, as well as with the physical theory called 2D quantum gravity.
Would you tell us about mathematicians who impacted your research, or whom you particularly admire?
Among the mathematicians who impacted my work, let me first mention my PhD supervisor Marc Yor, who conveyed his enthusiasm for mathematics and his passion for Brownian motion to me. I was also influenced by Jacques Neveu, who was the head of my laboratory when I started my career at the CNRS and who impressed me by the elegance and the originality of his work. I had only a few discussions with him, but they certainly had a strong impact on my future research, particularly in the domain of random trees. Lastly, and perhaps above all, I have been in regular contact for some twenty years with Eugene Dynkin, a great Russian mathematician who worked at Cornell University. His articles opened up new research horizons to me and he always welcomed my own contributions with interest and kind attention. Along with them, there are many mathematicians whom I admire, but it would be difficult to draw a complete list of them. Let me just quote Laurent Schwartz, who helped me a lot at the beginning of my mathematical career.
What does it mean to you to be a mathematician?
Let me give you a personal definition: it means diving into the huge field of mathematical knowledge and trying to extend its limits, whether it be by a tiny step. Mathematics are a great collective adventure (as opposed to other sciences, mathematics are instantly open to everyone via free preprint repository servers), to which every researcher contributes. The preceding definition
of a mathematician is probably too restrictive: people who are able to find new applications of well-established
mathematical theories to real life problems surely also deserve to be called mathematicians.
Even when mathematics are called applied mathematics, they still are fundamental. What does this mean to you?
Mathematics play a fundamental part in other sciences, in the sense that they provide, particularly in physics, both the models and a language in which scientific theories are phrased. A famous article by the physicist Wigner is entitled « The unreasonable effectiveness of mathematics in natural sciences ». If you think of « fundamental » as opposed to « applied », it is true that
many mathematicians do not care so much about the applications that their results may have beyond mathematics, and that they are more concerned by aesthetic issues: a « beautiful » result will be more valued than a « useful » one. However, one should not forget that mathematical results which seemed abstract and disconnected from any possible application later proved very useful, for example in physics (one could also mention the applications of number theory to cryptography). I think that mathematicians who are exclusively motivated by the fundamental character and the "mathematical beauty" of the concepts they study may also lay out theories which will play an important role in future applications.
You are rewarded for your "deep and elegant contributions". What would you say is mathematical elegance?
Mathematical elegance is something very important to me, yet very difficult to define at the same time, although any mathematician can spot an elegant argument. I could mention the simplicity and conciseness of the proofs, the power of the new concepts produced, the clarity of the developments leading to the desired result, ... I personally consider each one of my
articles somehow like an object shaped by a craftsman who would strive to make it more beautiful and more elegant before offering it to his clients (submitting it to a journal in my case). For sure, that perfectness issue proved very beneficial to my career.
The award announcement mentions the applications of your works to the mathematical understanding of 2D quantum gravity. Quantum gravity is also a theory in physics. It is supposed to be a theory that two other physical theories, quantum mechanics and general gravity, could approximate. Does it matter to you that your works in fundamental physics find applications in physics?
I will certainly answer yes, but let me make one point precise. I cannot say that my works have yet found applications in physics, even though I know they have been quoted in physics conferences. However, the mathematical models I study are essentially the same as the ones that many theoretical physicists are interested in (I think in particular of members of
the Institut de Physique Théorique de Saclay), and I very much enjoyed the interactions I have had with these researchers, even if the points of view, the methods and the goals differ when you are a mathematician or a physicist.
You started your career at the CNRS, and then, five years later, you moved on to be a Professor at University. What impact would you say the CNRS had on your career?
I owe a lot to the CNRS. When I was done with my years as a student at the Ecole Normale Supérieure at the beginning of the 1980s, academic opportunities were almost nonexistent, and getting a position as a research fellow (« attaché de recherche ») at the CNRS was practically the only way for a junior mathematician to take his first steps in research. I only stayed five years at the CNRS but those years, where I benefitted from a remarkable freedom of work, were very productive and determining for the rest of my career.
Jean-François Le Gall is a Professor at Université Paris-Sud. He is a member of the Laboratoire de Mathématiques d'Orsay and a member of the French Académie des Sciences.