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Professors Lauritzen and Werner will give keynote talks at the forthcoming World Congress.
Brownian Loop-soups, SLE and Conformal Field Theory: Wendelin Werner
Professor Werner will give the Lévy Lecture.
It has been observed in many occurrences that large systems from statistical physics can behave randomly on large scale if taken at their critical temperature (i.e., when a phase transition occurs). For two-dimensional systems, it has been pointed out by theoretical physicists that conformal invariance of the scaling limit of these systems should play an important role, and should help to describe their scaling limits.
This did for instance give rise to the development of Conformal Field Theory, at the crossroads of theoretical physics, representation theory and geometry. Some aspects of the mathematical understanding of these systems from statistical physics has however remained limited until recently, as the precise link between them and conformal field theory remained somewhat unclear. Schramm, combining basic ideas from complex analysis and probability constructed a class of two-dimensional random paths (the Schramm-Loewner Evolutions) that are the only possible interfaces for the scaling limits of two-dimensional models provided that they are conformally invariant. This led to various developments, including the proof of some of the physicists' predictions.
After reviewing some of these recent developments, I will describe a family of random "Sieprinsky carpet-type" conformally invariant fractal, based on a Poissonian cloud of Brownian loops (the fractal is the complement of a countable union of Brownian loops in a domain - the Brownian loop-soup that we introduced in a joint paper with Greg Lawler), and its remarkable properties. In particular, the boundaries of the "holes" in this fractal turn out to be SLE-type loops. It gives a way to tie more concrete links with conformal field theory.
Estimation of Structure in Graphical Models: Steffen L. Lauritzen
Professor Lauritzen will give the Laplace Lecture.
Graphical models have become increasingly used to describe complex relationships between several random variables. Such models are generally built up by a structural component, represented by a finite graph, which might contain directed and undirected links, and a specification of a joint probability distribution which satisfies conditional independence restrictions associated with the graph. Methods for identification of the structural component from available data have been developed, mainly within the computer science community. This lecture describes and discusses some of these methods, also from a general theoretical perspective.