I prepared for the entrance examination to the Écoles Normales Supérieures (ENS) in Rennes which is not a famous place to prepare for such examinations so, although I was the best student in the class, I wasn't really sure I could compete. But finally I was admitted and I was very happy. I had to work a lot of course during these two years but for me it was not too much - I could stand it. I know that some people find it extremely hard but that was not the case or me. I wasn't admitted with a very good rank - I was 18th, I think.

I did not have good teachers of physics at school so I did not like physics at all. It was clear to me that I wanted to do mathematics. I didn't like physics experiments but above all I didn't like the lack of rigour in physics. I think my teachers were responsible for that for later I studied physics and finally I like some aspects of it.

When I entered the Écoles Normales Supérieures, I still didn't know I wanted to make mathematics my profession. I heard there were positions in CNRS for doing research in mathematics. Of course I knew that there were very few such positions, but this was very attractive for me. Before that I didn't really know what I should do to become a mathematician.

After two years, I passed the agrégation de mathématicque. It was a difficult examination and I had to learn a lot of mathematics - all branches of mathematics - to pass this examination. It was very useful for my research work after that.

I passed the agrégation which is one examination which gives a ranking of all students, not only from the ENS but all mathematics students in France. I was ranked first and I was very happy with that. It gave me some confidence to continue to do mathematics.

Of course, I didn't know probability theory at that point so there was no mathematical reason to choose that. After I began, very quickly I liked it and I wanted to continue.

Yor was very active - he always accepted new people. You could go and see him and even if he was very busy he always said, "Well you can come and work with me - we can discuss things". At that moment it was very convenient for I could come along each week to discuss with him. Even if I had nothing interesting to say, it was important for me to go and discuss with him.

... played an irreplaceable role in welcoming the best mathematics students interested in probability and engaging them in research, advising over 30 theses during his career as university professor. A large number of these students went on to be researchers at the CNRS or professors in France and other countries. Without him, the recent successes of the French probability school, most notably the Fields Medal of his 'grandstudent' Wendelin Werner in 2006, would most likely never have been achieved.
...
Two words which best describe Marc Yor's scientific personality are, without doubt, enthusiasm and generosity. Enthusiasm, because he knew so well how to communicate his taste for research and to share the joy of discovery of new theorems and formulas. Generosity, because he helped so many young researchers, publishing with them numerous research articles which everyone knew he had essentially written himself, but for which he was always happy to share the credit.

It was Marc Yor who, through his infectious enthusiasm, was able to convince me to take an interest in the fascinating mathematical object that is Brownian motion.

... mathematical Brownian motion is a curve depending on chance, accounting for the phenomenon studied by great physicists like Albert Einstein and Jean Perrin. To give an intuitive idea of what Brownian motion is, we imagine a walker moving in a very large park and taking a step every second in one of the four possible directions, North, South, East or West, chosen randomly every time. If we observe the movement of the walker over a long period of time, say a few hours, and on a suitable scale, we will see a random curve which is close to that of Brownian motion in dimension two. Brownian motion is thus a sort of ideal prototype of a purely random curve. My first research work dealt with various properties of the Brownian motion curve, notably concerning the intersections of this curve with itself: for example, we show that the Brownian motion in the plane passes an infinite number of times at certain points, and my work has enabled us to better understand this phenomenon.

I think I did some non-trivial work there with some simulations. I was not really interested because there were constraints - I had not chosen to work on that. I worked with someone who is still at the establishment and he published a paper including my results. Apparently my name is on the paper but I have never seen it! It is not in MathSciNet since its an internal CAE report. It is more related to numerical analysis. It was not a bad experience for I think I learnt certain things. I didn't have much money but for that year I had a very small apartment in Paris.

... greatly impressed me with the elegance and originality of his work. The few discussions I had with him had a lasting influence on my mathematical career, particularly in the field of random trees.

We give a presentation of several studies concerning Brownian motion, which can be grouped around two main themes. The first is the study of intersections of Brownian trajectories, and is linked to certain problems which have recently aroused the interest of physicists, particularly in polymer theory. A key role in this study is played by the notion of local intersection time, which allows, in a certain sense, to measure the number of intersections of a Brownian trajectory. The second theme concerns the study of the winding number of Brownian motion and some related questions.

It's a little team, where it is easy to talk with mathematicians in other areas. I joined several people there that I appreciate, like my former student Wendelin Werner. There was the desire not to become fossilised, but also to evolve in my research.

I am a specialist in probability theory. My first research work focused on mathematical Brownian motion. Simplifying a little, Brownian motion represents the movement of a particle which at each instant will change direction in a completely random manner, and we are interested in the geometric properties of the trajectory thus obtained. Secondly, I worked a lot on models describing a cloud of particles subject to a double phenomenon of Brownian displacement and random reproduction: for example we can imagine that at random times each Brownian particle either dies or splits into two new particles. My motivation for studying these models came in part from deep connections with another area of mathematics, the theory of partial differential equations. At the same time, I was interested in "continuous random trees" which describe the genealogy of large populations, and I constructed these models by connecting them to other classic objects of probability, Lévy processes. In the last twelve years, my work has focused on a new branch of probability, random geometry. This involves understanding the geometric properties of large graphs drawn randomly in the plane. This leads to the introduction of new fascinating mathematical objects, in particular the "Brownian map" which is a random metric space limit of many discrete models and whose existence and uniqueness I contributed to showing. This random geometry has close ties to other parts of mathematics, notably combinatorics, as well as to the physical theory called quantum gravity in two dimensions.

... 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.

... 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.

... for the originality, quality and importance of his work, recognised both nationally and internationally.

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

Personally, I consider each of my articles a bit like an object made by a craftsman who would strive for a long time to make it more beautiful and more elegant before offering it to his clients (for me, submitting it to a journal). I know that this perfectionist side has been very beneficial for my career.

We present some recent developments concerning the genealogy of branching processes, and their applications to superprocesses. We also discuss connections with partial differential equations, statistical mechanics and interacting particle systems.

We discuss scaling limits of random planar maps chosen uniformly at random in a certain class. This leads to a universal limiting space called the Brownian map, which is viewed as a random compact metric space. The Brownian map can be obtained as a quotient of the continuous random tree called the CRT, for an equivalence relation which is defined in terms of Brownian labels assigned to the vertices of the CRT. We discuss the known properties of the Brownian map. In particular, we give a complete description of the geodesics starting from the distinguished point called the root. We also discuss applications to various properties of large random planar maps.

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size n, for instance the class of all triangulations of the sphere with n faces. We equip the vertex set of the graph with the usual graph distance rescaled by the factor $n^{-1/4}$. We then prove that the resulting random metric space converges in distribution as $n \rightarrow ∞$, in the Gromov-Hausdorff sense, toward a limiting random compact metric space called the Brownian map, which is universal in the sense that it does not depend on the class of graphs chosen initially. The Brownian map is homeomorphic to the sphere, but its Hausdorff dimension is equal to 4. We obtain detailed information about the structure of geodesics in the Brownian map. We also present the infinite-volume variant of the Brownian map called the Brownian plane, which arises as the scaling limit of the uniform infinite planar quadrangulation. Finally, we discuss certain open problems. This study is motivated in part by the use of random geometry in the physical theory of two-dimensional quantum gravity.

I would like to express my deepest gratitude to Jean-François Le Gall for offering me an exciting thesis subject opening the way to numerous directions of research, for his availability, for the always fruitful mathematical discussions, for his always wise advice, for all the time devoted to the careful reading (and re[re]reading!) of the articles and finally for the total freedom granted during this thesis.