The Rollo Davidson Trust was founded in 1975 in memory of Rollo Davidson, an accomplished mathematician of remarkable potential, and Fellow-elect of Churchill College, Cambridge, who died on the Piz Bernina in 1970. Initial funding from the Trust came from the royalties of two collections of papers Stochastic Analysis and Stochastic Geometry, published in 1973 and 1974 respectively as memorial tributes to Rollo Davidson by his friends and colleagues. Since 1976 the has Trust awarded an annual Prize for young probabilists and has benefited from the continuing association with the Davidson family. At a meeting on 10 March 1986 the Trustees awarded the Rollo Davidson Prize jointly to Peter Gavin Hall (1951-2016) and Jean-François Le Gall. It was awarded to:-

... Jean-Francois Le Gall of the Université Pierre et Marie Curie, Paris, for his use of local-time arguments to obtain uniqueness theorems for stochastic differential equations, and for his work on the multiple points of Brownian motion.

The Peccot course is a one-semester mathematics course at the Collège de France. Each course is given by a mathematician under the age of thirty who has distinguished themselves by their promising first work. The course consists of a series of lectures which allow the winner to present their recent research. Being a lecturer on a Peccot course is a distinction that often foreshadows an exceptional scientific career; among them are future holders of the Fields medal, the Abel prize, academicians and professors at the Collège de France. The first to deliver the Cours Peccot were Émile Borel (1899-1902), Henri Lebesgue (1902-03) and René Baire (1903-04). Jean-François Le Gall delivered the Cours Peccot at the Collège de France in 1989-90. His course was titled Quelques équations cinétiques et leurs limites fluides.

The 1997 Line and Michel Loève International Prize in Probability has been awarded to Jean-François Le Gall, professor at the École Normale Supérieure, Paris. The prize carries a monetary award of about $30,000.

3.1. Biographical Sketch.

Jean-François Le Gall was born on November 15, 1959, in Morlaix, France. He was a student at the École Normale Supérieure de Paris (1978-82), where he received his Agrégation de Mathématiques (rank 1) (1980). He finished his Ph.D. in 1982, under the direction of Marc Yor, and five years later his Thèse d'État de Mathématiques, both at Université Paris VI. In 1982 Le Gall became Chargé de Recherches of the Centre National de la Recherche Scientifique at Paris VI. He was a professor at Paris VI from 1988 to 1997, when he took his present position as professor at ENS. He received the Rollo Davidson Prize (1986), the Cours Peccot Prize of the Collège de France (1989), and was a junior member of the Institut Universitaire de France (1992-97). In 1992 he was an invited speaker at the first European Congress of Mathematics in Paris. He has been selected as an invited speaker at the International Congress of Mathematicians in Berlin in August 1998.

3.2. The Work of Le Gall.

Le Gall's early work was mainly concerned with fine properties of Brownian motion, particularly those that relate to cone points and multiple points for the Brownian path, intersection local times, and the Wiener sausage. It is only possible to describe a small selection of Le Gall's results in this area.

A cone point is a point $t$ in time for a planar Brownian motion $B$ such that $B(s) - B(t) \in C_{\alpha}$ for $0 ≤ s ≤ t$, where $C_{\alpha}$ is a fixed cone with angle $\alpha$. Le Gall showed that if $\alpha > \large\frac{\pi}{2}\normalsize$, then the set of cone points is a regenerative set that is the range of a stable subordinator with index $1 - \large\frac{\pi }{2}\normalsize \alpha$. This is an extension of a celebrated result of Spitzer.

A multiple point for the planar Brownian path is a point $z$ in the plane such that the set $\{t \in [0, 1] : B(t) = z\}$ has two or more elements. A classical theorem of Dvoretsky, Erdös, and Kakutani states that there exist points $z$ such that the set $\{t \in [0, 1] : B(t) = z\}$ has the same cardinality as [0, 1]. Le Gall established a far-reaching generalisation of this result by showing that given any totally disconnected compact subset $K \subset \mathbb{R}$, there exists with probability one a point $z$ such that $\{t \in [0,1] : B(t) = z\}$ has the same order type as $K$.

For each integer $p ≥ 2$, the set of multiple points $z$ for which the pre-image $\{t \in [0, 1] : B(t) = z\}$ has cardinality $p$ supports a random measure called a renormalised self-intersection local time. Le Gall established a remarkable asymptotic expansion as $\epsilon \downarrow 0$ of the area of the Wiener sausage $\{x : \exists 0 ≤ t ≤ 1, |x - B(t)| ≤ \epsilon\}$ in terms of these objects.

Le Gall's more recent work has focused on measure-valued diffusion processes, particularly the Dawson-Watanabe super Brownian motion. Super Brownian motion arises as the high density limit of a system of branching Brownian motions with critical, finite-variance branching mechanism. Le Gall has introduced a new representation of super Brownian motion with finite variance branching mechanism in terms of the Brownian snake, a process that takes values in the space of continuous paths in $\mathbb{R}^{d}$. The basic idea of the snake is deceptively simple: it is essentially an attempt to carry over to a continuous setting the notion of a depth-first search of a tree. However, the snake turns out to be a powerful tool that enables techniques from Markovian potential theory and excursion theory to be brought to bear on super Brownian motion.

Super Brownian motion is connected with the potential theory of the nonlinear operator $u \rightarrow \Delta u - u^{2}$ in a manner that is somewhat analogous to the connection between ordinary Brownian motion and the Laplace operator $\Delta$. Le Gall has shown in a series of papers how the snake can be used to explore this connection and obtain new analytic results using probabilistic tools. Also, Le Gall and Perkins used the snake to give an extremely delicate analysis of the exact Hausdorff measure properties of the support of planar super Brownian motion.

In recent work with Le Jan, Le Gall has found an analogue of the snake that is useful for studying super processes with arbitrary branching mechanisms. Moreover, this work establishes new connections between Lévy processes and continuous state branching processes that are deep extensions of the classical interrelationship between random walks, branching processes, and queues.

The Sophie Germain Prize is an annual mathematics prize awarded by the French Academy of Sciences to researchers who have carried out fundamental research in mathematics. The award has been conferred every year since 2003 and comes with a €8000 cash prize.

Jean-François Le Gall provided in-depth knowledge of the intersection properties of plane Brownian motion, thanks to the systematic use of local intersection times. He defined the Brownian snake, thanks to which he provided a probabilistic solution to the equation $\Delta u = u^{2}$. He also developed the studies of random trees. The quality of his work has earned him, over the past twenty years, first-rate international recognition.

The Fermat Prize for research in mathematics rewards the research work of mathematicians in fields where Pierre de Fermat's contributions were decisive: (i) statements of variational principles; (ii) foundations of the calculus of probabilities and of analytic geometry; and (iii) number theory. Within these fields, the spirit of the prize is to reward results which are accessible to the greatest number of professional mathematicians. Since 1989, The Fermat Prize is awarded every two years by the Institute of Mathematics of Toulouse, under the aegis of Université Paul Sabatier.

The 2005 Fermat Prize was awarded jointly to Pierre Colmez, Institut de Mathématiques de Jussieu, and Jean-François Le Gall, Université Paris VI and École Normale Supérieure. Le Gall was chosen for his contributions to the fine analysis of planar Brownian motion and his invention of the Brownian snake and its applications to the study of nonlinear partial differential equations.

The Centre national de la recherche scientifique (CNRS) awards the Silver Medal to outstanding researchers for the originality, quality and importance of their work, recognised both nationally and internationally. It was created in 1954 in order to highlight and better understand internal achievements. The medal is part of the "CNRS Talents" medals. The Gold Medal rewards the entirety of a scientific career, the Bronze Medal rewards young researchers, the Innovation Medal honours remarkable work on the technological or economic level, and the Crystal Medal rewards research support staff.

The silver medal was awarded to Jean-François Le Gall, professor at the University Paris-Sud 11 - Orsay Mathematics Laboratory (CNRS / University Paris-Sud 11), on 16 September 2009 at the CNRS in Gif-sur-Yvette [about 20 km from the centre of Paris] by Patrick Dehornoy, deputy scientific director of the Institute of Mathematical Sciences. Also present were Michèle Saumon, Ile-de-France South regional delegate of the CNRS, Guy Couarraze, president of the University of Paris-Sud 11, and Patrick Gérard, director of the Orsay Mathematics Laboratory. Le Gall's colleagues, friends and family were also at the presentation.

7.1. Report from the Institute of Mathematical Statistics.

The Wolf Prize for Mathematics 2019 was awarded to Gregory Lawler from Chicago University:-

... for his comprehensive and pioneering research on erased loops and random walks

and to Jean-François Le Gall from Paris Sud Orsay University:-

... for his profound and elegant works on stochastic processes.

The work undertaken by these two mathematicians on loops and probability, which have been recognised by multiple prizes, became the stepping stone for many consequent breakthroughs.

Jean-François Le Gall has made several deep and elegant contributions to the theory of stochastic processes. His work on the fine properties of Brownian motions solved many difficult problems, such as the characterisation of sets visited multiple times and the behaviour of the volume of its neighbourhood - the Brownian sausage. Le Gall made ground-breaking advances in the theory of branching processes, which arise in many applications. In particular, his introduction of the Brownian snake and his studies of its properties revolutionised the theory of super-processes - generalisations of Markov processes to an evolving cloud of dying and splitting particles. He then used some of these tools for achieving a spectacular breakthrough in the mathematical understanding of 2D quantum gravity. Le Gall established the convergence of uniform planar maps to a canonical random metric object, the Brownian map, and showed that it almost surely has Hausdorff dimension 4 and is homeomorphic to the 2-sphere.

7.2. Extracts from the Citation.

The Wolf Prize in Mathematics, the third most prestigious distinction in mathematics after the Abel Prize and the Fields Medal, was awarded in 2019 jointly to Jean-François Le Gall, professor at the University of Paris-Sud:-

... for his profound and elegant work on stochastic processes

and Gregory Lawler, professor at Chicago University. It is the highest distinction of all the marks of academic recognition that Jean-François Le Gall has received: plenary speaker at the International Congress of Mathematicians in 2014, member of the Academy of Sciences since 2013, CNRS silver medal, Fermat prize, Loeve prize and Sophie Germain prize... Let us take this opportunity to briefly list some of his most important results in the field of probability theory in reverse chronological order.

Random planar maps and the Brownian map.

The Central Limit Theorem states that after proper centring and rescaling, the sum of n independent copies of a random variable with finite variance converges in law as $n \rightarrow ∞$ to a standard Gaussian distribution. Donsker's invariance principle provides a functional version of the latter; it asserts that the Wiener measure, that is the distribution of the Brownian motion, arises as the scaling limit of any random walk with finite variance. ...

Informally, Le Gall established in a series of works the counterpart of Donsker's invariance principle for surfaces instead of paths. More specifically, he solved the crucial problem raised by Oded Schramm of establishing the convergence in distribution of several discrete models of random planar geometries, such as uniform random quadrangulations [A quadrangulation of the sphere is a proper embedding of a finite connected graph in the sphere, without edge-crossings, and such that all faces are bounded by exactly 4 edges] with $n$ faces of $S^{2}$, to the Brownian map (the problem of the uniqueness of the Brownian map was solved independently and at the same time by Grégory Miermont).

Jean-François Le Gall has established a number of deep features of this new object. He proved that its Hausdorff dimension is 4 almost surely, that it enjoys a remarkable invariance property after uniform random re-rooting, described precisely its geodesics, etc. The Brownian map and its relatives form one of the most important and active fields of research on stochastic phenomena nowadays, and their connexions to some other fundamental objects including Conformal Loop Ensembles and Gaussian Free Field, confirm their central role in Probability Theory.

Branching processes.

The branching property lies at the core of many contributions of Jean-François Le Gall. Informally, it refers to a basic assumption that is very frequently made in random populations models, namely that different individuals evolve independently one from the other. Although this notion is easy to formalise when reproduction events do not accumulate, because then there is no difficulty for defining the genealogical structure in terms of a discrete tree, dealing with the continuous setting is often much more complex. Probably the most remarkable contribution of Jean-François Le Gall in this area is the introduction of the Brownian snake, a process with values in a space of trajectories, which is a tool of fundamental importance for solving a variety of problems.

The Brownian snake yields notably a most useful construction of a class of measure-valued Markov processes called super-processes, from which a number of features of the latter can be derived. In turn, as it was initially stressed by E B Dynkin, super-processes have deep connexions to certain non-linear PDE's (typically $\Delta u = u^{2}$), and the Brownian snake can be used to establish fine properties about the singularities of the solutions, their asymptotic behaviours close to the boundary in situations when the solution explodes, etc. In a somewhat different direction, together with Yves Le Jan, Jean-François Le Gall constructed continuous state branching processes with a general (i.e. non-binary) branching mechanism from Lévy processes (i.e. processes with independent and stationary increments) without negative jumps. This enables to extend the construction of the Brownian snake to Lévy snakes, and to generalize many results which have been established previously for the binary branching mechanism to general ones. Furthermore, the genealogy of a Lévy snake is described by a continuous random tree, called a Lévy tree, which is a close relative to the celebrated (Brownian) Continuum Random Tree of Aldous, and has appeared since in many limit theorems for random structures. Last but not least, it should be stressed that the Brownian snake plays also a key role in the construction and the study of the Brownian map.

Intersection of planar Brownian paths.

Some of the earliest contributions of Jean-François Le Gall dealt with multiple points of the planar Brownian motion or of the simple random walk. ...

By building upon the concept of intersection local time, which had just been introduced by Jay Rosen, Jean-François Le Gall obtained a number of remarkably fine results. In particular, he has been able to determine the exact Hausdorff function for the set of points with a given multiplicity. He also enlightened the role of the self intersection local times and its renormalization in the asymptotic study of the area of the so-called Wiener Sausage (the trace left by some compact set translated along a Brownian trajectory), considerably refining an earlier result due to Kesten, Spitzer and Whitman in the 60's.

Jean-François Le Gall is one of the brightest and most influential probabilists of his generation. This brief description of some of his major contributions is of course far from being complete, and his mathematical works covers many more interesting topics.

The BBVA Foundation belongs to financial group Banco Bilbao Vizcaya Argentaria which is partnered in the scheme by the Spanish National Research Council. The awards were established in 2008, with the first set of winners receiving their prizes in 2009. Basic Sciences is one of eight award categories and previous winners in this category include Ingrid Daubechies (2011), David Mumford (2011), and Stephen Hawking (2014).

The BBVA Foundation Frontiers of Knowledge Award in Basic Sciences 2021 was presented to professors Charles Fefferman of Princeton University and Jean-François Le Gall of Université Paris-Saclay in recognition of their fundamental contributions in two areas of mathematics that have had multiple ramifications, with applications extending across a broad array of scientific and technology fields.

8.1. Jean-François Le Gall's biography.

Jean-François Le Gall (Morlaix, France, 1959) earned a PhD in Mathematics from the Université Pierre et Marie Curie (Paris 6) in 1982 and read his "Doctorat d'État" thesis in Mathematics at the same centre in 1987. From 1983 to 2007 he held professorial appointments at Pierre et Marie Curie and the École Normale Supérieure in Paris, where he was also Director of Mathematical Studies and of the Master's in Mathematics and Computer Science. Since 2006, he has worked in the Orsay Mathematics Laboratory (LMO) at Université Paris-Saclay, where he co-led the Master's in Probability and Statistics as well as heading the Probability and Statistics Group. During this period, he was elected a senior member of the Institut Universitaire de France. Le Gall is also Principal Investigator on the GeoBrown project funded with an ERC Advanced Grant, Vice President of Research in the Orsay Mathematics Department (DMO) and Vice President of the Comité National Français des Mathématiques. Author of over 130 articles in scientific journals, he serves on the editorial boards for the series Grundlehren der mathematischen Wissenschaften (Principles of mathematical sciences) and Annales de la Faculté des Sciences de Toulouse.

8.2. Jean-François Le Gall's achievements.

The geometry of random movements

Jean-François Le Gall has "profoundly transformed probability theory," writes Emmanuel Royer, Scientific Director of the National Institute for Mathematical Sciences and their Interactions - INSMI (CNRS, France), which put his name forward for the award.

For Marta Sanz Solé, Professor of Mathematics at the University of Barcelona, who also researches on probability and is a close follower of Le Gall's work, his contributions are "truly pivotal, in the sense of spurring new research around his results, and strengthening the connections with mathematical physics."

Many of the problems Le Gall works on come from physics, although he describes himself in the interview granted after hearing of the award as "a theoretical mathematician who works on mathematical objects of inherent interest, without thinking of the applications." Advances in mathematics, he insists, derive overwhelmingly from an "aesthetic motivation."

His first object of study was mathematical Brownian motion. This field traces its ideas back to Albert Einstein, who was able to explain the random movement of pollen grains floating in water as the result of the vibration of molecules in the fluid, and thus prove that atoms and molecules really exist. Le Gall has explored the geometry arising from the trajectories of particles in Brownian motion: "I have made an extensive study of this kind of motion, which describes the random movement of a particle that is constantly changing direction, and have introduced several key objects related to Brownian motion."

In the last fifteen years, his research has birthed a new branch within probability theory based on the study of "Brownian spheres." These are not in fact spheres but irregularly surfaced "mathematical objects" - the awardee explains - that appear when tens of thousands of minute triangles stick randomly to one another. "Physicists invented these spheres as a model for the theory of quantum gravity," he explains. "My contribution was to make this model rigorous." The field is now a hive of mathematical activity and "has opened new perspectives in research."

A result that Le Gall names among his favourites dates from nine years back and refers precisely to these Brownian sphere, specifically to proving their "uniqueness" in the mathematical sense: "That was a major issue, a problem that had been open for eight years," he relates. "Because if you aren't able to prove the uniqueness of your model, you can't tell if it really works."

Le Gall talks not only about the essential role of mathematics in the technology we use in our daily lives, "like GPS, which is based on advanced mathematical analysis," but also about its indispensable contribution to advancing knowledge across all domains: "Mathematics is the language of science, so it is important to stress that physicists, like chemists or biologists, use mathematics to understand nature. Quantum mechanics, for instance, or relativity rely on deep mathematics. It is essential for science to start from sound mathematical models."

8.3. Le Gall's acceptance speech.

Let me first say that I am extremely honoured to receive this award from the BBVA Foundation. I am also glad to share this award with Professor Fefferman, who is one of the great mathematicians of our times.

Let me try to explain the mathematical contributions for which I have received this prestigious award. I have worked a lot on Brownian motion, which is a mathematical model for a purely random curve. In the last 15 years, however, my area of research has been the definition and study of random geometry in two dimensions. This line of research is motivated by the physical theory known as quantum gravity, which aims at unifying general relativity and quantum mechanics. It would be too long to describe the connection between random geometry and quantum gravity in detail, but I can try to explain how random geometry is constructed. To this end, imagine a world consisting of a large number of cities located on a sphere like the earth. Some of these cities are connected by roads that do not cross except at cities, and the distance between two cities A and B is measured by the minimal number of cities one has to cross when going from A to B by following the roads. The point is now to choose the configuration of cities and roads completely at random: this can be done in a precise mathematical way, and one can then prove, for instance, that the typical distance between two cities is roughly of the same size as the total number of cities raised to the power $\large\frac{1}{4}\normalsize$. In the limit where the number of cities and the number of roads tend to infinity, one arrives at a model called the Brownian sphere, which is a new fundamental mathematical object. Most of my recent research contributions have dealt with the construction of this model and of several variants, and the study of their properties.

The Frontiers of Knowledge Award has a major significance for me as it means the recognition of the importance of the new field of research which I have developed together with colleagues all around the world. It is also a strong encouragement for me to push forward this line of research.

In conclusion, let me say that I have had fruitful contacts with Spanish colleagues for a long time. In particular, my very last PhD student, who came from Barcelona to France for his doctoral studies, is an extremely talented young Spanish mathematician, who has made deep contributions to the Brownian sphere. For these reasons, I am also very happy to receive this award here in Spain.