It is my great pleasure to briefly report on some of Endolin Werner's research accomplishments.

There are a number of aspects of Werner's work that add to my pleasure in this event. One is that he trained as a probabilist, receiving his PhD in 1993 under the supervision of Jean-Francois Lagalle in Paris

with a dissertation concerning planar Brownian motion, which, as we shall see, plays a major role in his later work as well. Until now, probability theory had not been represented among fields metals, and so I am enormously pleased to be here to witness a change in that history.

Indeed, probability is quite well represented in the awards at this ICM. I myself was originally trained not in probability theory, but in mathematical physics.

Werner's work, together with his collaborators such as Greg Lawler, Oded Schramm, and Stas Smirnov, involves applications of probability and conformal mapping theory to fundamental problems in statistical physics, as we shall discuss.

A second source of my pleasure is the 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 more importantly, 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. Here is a simulation of a critical percolation configuration on a portion of the triangular lattice,

corresponding in this figure to uniformly random two colorings of the hexagons. We will discuss percolation more later. In particular, we'll be interested in discussing things like scaling limits, in which the size of the lattice representing here the diameter of the individual hexagons goes to zero,

and one is interested in various geometric properties of the clusters. The clusters here are just these connected blue or connected yellow components, where connection just means hexagons of the same color sharing an edge.

There is a small cluster of two yellow hexagons surrounded by a larger cluster of some slightly larger number of blue hexagons. We'll get back to that in a while. Permit me a somewhat more personal remark.

As director of the Courant Institute for the past four years, we have a scientific viewpoint, as did our predecessor institute in Göttingen, namely that an important goal should be the elimination of artificial distinctions between the mathematical sciences and their applications in other sciences.

I believe that Wendland Werner's work lives up to that philosophy, as does other of the work that is being recognized today at the ICM. Yet the 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 the 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 field metals awarded for this style of work. The area of probability theory which most strongly interacts with statistical physics is that involving stochastic processes with non-trivial spatial structure.

This area, which also interacts with finance, communication theory, theoretical computer science, and other fields, has long combined interesting applications with first-class mathematics. Recent developments, however, have raised the perceived mathematical status of the best work from merely first-class to outstanding.

Let me begin by mentioning two pieces of Werner's work from 1998 to 2000. These are not only of intrinsic significance, but also were precursors to the breakthroughs about to happen in the understanding

of two-dimensional critical systems with natural conformal invariance. There were, of course, other significant precursors, such as Eisenman's path approach to scaling limits, and Kenyon's work on loop-erased walks and dominoed tilings. Some of that work, or related work, joined with Hakunkov,

that was already mentioned. The first of the two pieces of work is a 1998 paper of Ballin, Toth, and Werner. The motivation was to construct a continuum version of Toth's earlier lattice true self-repelling walk.

This led to a quite beautiful mathematical structure, an extended version of a mostly unpublished and nearly forgotten construction done almost 20 years earlier by Aratia of coalescing and reflecting one-dimensional Brownian paths running forward and backward in time

and filling up all of two-dimensional space-time. There is a plane-filling curve within this structure that is analogous to one that arises in scaling limits of uniformly random spanning trees, which we'll mention again later.

And it was one of Schramm's motivations in his 2000 paper introducing SLE. SLE, as many of you know, is an acronym for what was originally called the Stochastic-Lövner Evolution and is now often called the Schramm-Lövner Evolution. We'll discuss a little bit more about that shortly.

The second piece of work from this period consists of two papers with Greg Waller in 1999 and 2000 involving planar Brownian intersection exponents. And I'll actually give a definition of some of these exponents shortly.

In the second of these, it was shown that the same set of exponents must occur providing only that certain locality and conformal invariance properties are valid. This was a key idea, which combined with the introduction of SLE for the analysis of two-dimensional critical phenomena

led to a remarkable series of three papers in 2001 to 2002 by Waller, Schramm, and Werner, which yielded a whole series of intersection exponents. So here's one example. Let W1, W2, et cetera,

be independent planar Brownian motions starting from distinct points at time t equals zero. Consider the probability that n random curve segments of these Brownian motions, the first n of them, observed up until time t

are all disjoint. That probability decays as t tends to infinity, like some negative power of t, and the constant describing that power, called here zeta n, is an example of an intersection exponent. One of their theorems is that these intersection exponents, zeta n,

are given exactly by the formula 4n squared minus 1 over 24. This formula had been conjectured earlier based on numerical results by Duplantier and Quone and derived later by Duplantier non-rigorously using two-dimensional quantum gravity methods.

Despite the simplicity of the formula, prior to the introduction of the SLE-based methods used by Waller, Schramm, and Werner, its derivation by conventional stochastic calculus techniques appeared to be quite out of reach.

The period from 2001 or so until now has seen an explosion of interest in and applications of the SLE approach. To discuss this, let me first briefly describe SLE. For, say, a jordan domain D in the complex plane with distinct points a and b on its boundary

and a positive parameter kappa, SLE of the chordal type with parameter kappa is a certain random continuous path in a closure of D starting from a and going to b. When the parameter kappa is less than or equal to 4,

SLE is a simple path with probability 1 that only touches the boundary at a and b. Lövner, in work dating back to the 1920s, studied evolution from a to b of non-random curves and their associated conformal maps in terms of a real value driving function u of t.

SLE with parameter kappa corresponds to this driving function u of t being a scaled one-dimensional Brownian motion scaled with parameter kappa. Its qualitative properties depend dramatically on kappa.

When kappa is greater than 4, it is no longer a simple curve. When kappa is 8 or more, it actually becomes a plane-filling random curve. Now back to the SLE-based advances of the recent past. Many of these concern or are motivated by non-rigorous results

in the statistical physics literature about two-dimensional critical phenomena. Critical points of physical systems typically happen at very specific values of physical parameters, such as where the vapour pressure curve ends in a liquid gas system. Critical systems have many remarkable properties such as random fluctuations that normally are observable

only on microscopic scales manifesting themselves macroscopically. A related feature is that many quantities at or approaching the critical point have power law behaviour with powers which are typically non-integer powers

known as critical exponents believed to satisfy universality. That is, microscopically distinct models in the same spatial dimension should have the same exponents at their respective critical points.

Such universality is also believed to be true about other macroscopic features such as scaling limits that we'll be discussing later. Two-dimensional critical systems turn out to have an additional remarkable property which is at the heart of both the SLE approach and its predecessors in the physics literature

that is conformal invariants appearing on the macroscopic scale. As in the case of the Brownian intersection exponents that we already discussed, many of the SLE-based results in two dimensions were rigorous proofs of exponent values

that had been derived earlier by non-rigorous arguments. Primarily arguments that were based on what is known in the physics literature as conformal field theory which dates back to the work of Paul Yakov, his collaborators and others in the 1970s, 80s and 90s.

Other results were brand new. I'll discuss a few results in more detail but what is most exciting is that the SLE-based approach is not just a rigourisation of what already had existed but a conceptually quite complementary approach to that of conformal field theory.

Werner in particular has emphasised the need to understand that complementary relationship. This has led, for example, to a focus on the restriction property as in his work on the conformally invariant measure on self-avoiding loops. That work is one example of a burgeoning interest in extending the original focus on random curves

to random loops but still with conformal invariance. Both in the case of percolation and in a more general context in more general context such as conformal loop ensembles such as are being currently studied by Scott Sheffield and Werner.

So here are some more examples of results obtained in the last six years or so. Let W of T be a planar Brownian motion. I wanted to find now what's called the Brownian frontier. That's the following. Consider the curved segment of the Brownian motion

between time zero and time T. The complement in the plane of that curved segment is a countable union of open sets, one of which is infinite, and the boundary of the infinite component is the Brownian frontier. As a consequence of deep relations

that planar Brownian motion has with SLE with parameter six, Lawler, Schramm, and Werner proved a celebrated 1982 conjecture of Mandelbrot about the Brownian frontier, namely that the Hausdorff dimension of the Brownian frontier is four-thirds.

Again, this was a result which one would have thought would be obtainable by more conventional stochastic calculus methods, but had not been. A different set of results are stated informally

in the next theorem. They concern loop-erased random walks and related random objects on lattices. Loop-erased random walks are simple symmetric random walks in which any loops that are formed are sequentially erased.

So the theorem is an informal version of it. Consider a, say, a Jourdan domain in the plane, and consider the scaling limit. A scaling limit corresponds to letting the lattice spacing tend to zero, respectively of loop-erased random walk

and its related objects, which are the uniformly random-spanning tree and the related lattice-filling curve. Those, respectively, in the limit in which the lattice spacing goes to zero are described by SLE with parameter 2, a continuum tree based on SLE 2,

and the plane-filling curve SLE with parameter 8. Scaling limits of lattice models, as in this theorem, are among the most interesting and often most difficult results. To do them involves a successful combination

of concepts and techniques from three different areas. Conformal geometry, as in the classical Lovenor evolutions, where the driving function is nonrandom, stochastic analysis, since for SLE the driving function is a Brownian motion, and the probability theory of the lattice models

whose limits one is trying to study. The work of Werner combines all three ingredients admirably well. Before closing, I'd like to discuss one more example which demonstrates how these three areas can work together. And that is scaling limits

of two-dimensional critical percolation. The physics community knew on a non-rigorous basis the exponent values, or at least most of the exponent values, and even some geometrical information in the form of specific formulas for scaling limits of crossing probabilities

between boundary segments of domains. These formulas were derived by Cardy following Eisenman's conjecture that they should be conformally invariant. But there was no understanding, for example, of the scaling limit geometry of objects like cluster interfaces. So cluster interfaces

are simply the outer boundaries of clusters. The outer boundary of this blue cluster is this dark black path that goes around it. The outer cluster of this... The outer boundary of this... The cluster interface describing the boundary of the... This blue cluster would go around like this.

Remember, one's interested in a limit in which the lattice spacing goes to zero, but when, in a critical percolation system, even as the lattice spacing goes to zero, one continues to have clusters on a macroscopic scale.

So how did one study limits of these interfaces? Well, the first breakthrough was when Schramm argued that the limit of a particular interface, called the exploration path, should be described in the scaling limit

by SLE with parameter six. In fact, he proved that if there was enough conformal invariance and the limit existed, it would have to be SLE six. Next, for the special case of the triangular lattice, Smirnov proved that, indeed,

the crossing probabilities do converge to the conformally invariant Cardy formulas. He also sketched an argument as to how that could lead to convergence of the whole exploration path and argued further that one should be able to extend those results to a full scaling limit, describing the family of the interface loops, describing the boundaries of all the clusters.

Using up machinery, Smirnov and Werner proved certain percolation exponents using exploration path convergence, and Lawler Schramm and Werner used the full scaling limit convergence to prove another exponent value, which I'll state in a theorem in the next slide.

The convergence of the two types I just mentioned can be proved, as in recent work of Federico Camilla and myself, by using lattice percolation machinery, including results obtained by Kestin, Sideriewicz, and Zhang about crossings of annuli

and results of Eisenman, Duplantier, and Arroni about the nature of narrow fjords in percolation. Then the results about percolation exponents apply and provide another example of how the three ingredients I mentioned before can work together. One example of that is

the following exponent of derivation, a proof of a prediction that had already existed in the physics literature of Denice and Nienhaus about the exponent describing the size and the diameter of the cluster, the origin, and critical two-dimensional percolation

that the cluster-sized diameter distribution has a tail decaying like r to the minus 548s. So that's deriving the value of the exponent 548s rigorously. So I close with some comments about continuum models of probability theory

and their relation to other areas of mathematics which are exemplified by the work of Werner. Traditionally, a major focus of probability theory, and especially so in France, has been on continuum objects such as Brownian motion and stochastic calculus. SLE and related processes

are the latest continuum objects in the pantheon. Those of us raised in a different setting, such as statistical mechanics, sometimes regard lattice models as more real or physical than these continuum models, but I would say that's a narrow view. It's only the continuum models which possess extra properties like conformal invariance

that relate probability theory to other areas of mathematics, and such relations have become of increasing importance in recent years and will continue to be so. Even if one is primarily interested in the original lattice models, it is quite clear that their properties, such as critical exponents and critical universality,

cannot be understood without a deep analysis of the continuum models that arise in the scaling limit. Thanks to the work of Endel and Werner, his collaborators and others, I would say that now we are all continuistas. Thank you for your attention.