... for his works on the intersection exponents of Brownian motion and their impact in theoretical physics.

Lawler, Schramm and Werner received the prize for their groundbreaking work on the development and application of stochastic Loewner evolution (SLE). Of particular note is the rigorous establishment of the existence and conformal invariance of critical scaling limits of a number of 2D lattice models arising in statistical physics.

... for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.

Take scissors and cut completely at random a shape in a piece of paper. What can you say about this shape? Part of the question is to make sense of notion "completely at random" because there are infinitely many possibilities. One motivation to study this type of question comes from physics: If you consider a physical system and raise its temperature, then at certain values of the temperature, there occurs sudden change of its macroscopic behaviour: Liquid becomes vapour, iron looses its spontaneous magnetisation etc. It has been observed empirically that when a system is exactly at such a "critical" temperature, it can exhibit random macroscopic features. For instance, if the system is planar, then the two phases may coexist and the lines separating the regions corresponding to each of the phases are then random loops, just as those cut out by scissors.

Understanding the behaviour of certain natural very long random curves in the plane is a seemingly simple question that has turned out to raise deep questions, some of which remain unsolved. For instance, theoretical physicists have predicted (and this is still an open problem) that the number $a(N)$ of self-avoiding curves of length $N$ on the square lattice $\mathbb{Z} \times \mathbb{Z}$ grows asymptotically like $C^{N} N^{11/32}$ for some constant C. More generally, theoretical physicists ... have made predictions concerning the existence and values of critical exponents for various two-dimensional systems in statistical physics (such as self-avoiding walks, critical percolation, intersections of simple random walk) using considerations related to several branches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras).

Probability theory and statistical mechanics often focus on the behaviour of a functional of a large random system, with microscopic random inputs that sometimes interact with each other. In many cases, the output is close to being deterministic when the system is very large. In "critical cases," the outcome can however be random at any scale. The microscopic randomness then gives rise to a macroscopic continuous randomness and it is natural to try to understand it mathematically. This turns out to be rather difficult in general, but in some cases, it is possible thanks to an additional mathematical structure, and relates these random structures to other parts of mathematics. An example is given by the scaling limit of critical two-dimensional particle systems, where particles interact locally. The understanding of these phenomena is related to complex analysis and to representation theory, as predicted by theoretical physicists. Mathematical progress has been made towards a better rigorous understanding of these models during the last years.

There are a number of aspects of Werner's work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion ... Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history. ... Werner's work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Mapping Theory to fundamental issues in Statistical Physics ... A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are providing quite new conceptual understanding of basic phenomena - in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation ... Yet a third source of pleasure concerns the collaborative nature of much of Werner's work. Beautiful and productive mathematics can be the result of many different personal work styles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work.