Combinatorics, Dynamics, Probabilities

Stockholm, 3-7 October 2000

Abstracts of mini-courses

 

Prof. D. Anosov (Steklov Institute, Moscow)
Flows on surfaces and related geometric questions

Prof. O. Häggström (Chalmers University of Technology and Göteborg University)
Trends in percolation theory
Percolation theory deals with connectivity properties of random media. As a mathematical idealization, one usually takes an infinite graph (such as the Zd lattice), removes edges (or vertices) at random, and asks about the possible existence of infinite connected components in the remaining subgraph.
Spectacular progress was made in mathematical percolation theory in the 1980's. This was followed by a more silent period in the first half of the 1990's. In the last few years, things have started to look better again, and the subject has began to develop in a variety of new directions. In this talk, I intend to survey some of these developments. Topics that I plan to discuss include: conformal invariance, continuum percolation, dynamical percolation, entanglement and rigidity percolation, and percolation on nonamenable graphs.
References (PS file)

Prof. S. Janson (Uppsala University)
Brownian limits for combinatorial problems
Many problems on random (discrete) combinatorial structures have asymptotics that can be described using Brownian motion and processes derived from it, such as the Brownian bridge and Brownian excursion. I will discuss some such problems, involving random trees, forests and applications to hashing in computer science.

Doc. K. Johansson (KTH, Stockholm)
Random matrices, random growth and random tilings
Random matrices have a rich mathematical structure with connections to many different parts of mathematics. Recently, connections with certain problems in combinatorial probability have been the focus of much attention. In this minicourse I will give some background on random matrices and discuss how random matrix distributions and random matrix-like distributions appear in certain random growth models, related to the length of the longest increasing subsequence in a random permutation, and in some types of random tilings.

Prof. V. Kaimanovich (CNRS, France)
Ergodicity and conservativity: random walks, Brownian motion, geometric flows
Ergodicity and conservativity are the most basic and fundamental properties of a group of transformations (particular case: just a single invertible transformation) with a quasi-invariant measure. The aim of the minicourse is to discuss these properties in two closely related situations:

    1. for Markov processes on algebraic and geometric structures (random walks on groups, Brownian motion on Riemannian manifolds);
    2. for geometric flows on negatively curved manifolds (the geodesic and the horocycle flow)

In particular, it will be shown that ergodicity of the both geometric flows admits a natural interpretation in terms of the Brownian motion on the manifold (recurrence for the geodesic flow and absence of bounded harmonic functions for the horocycle flow).

Prof. A. Vershik (Steklov Institute, St.-Petersburg)
Ergodic theory of the polymorphisms
A polymorphism is a "multivalued map" in the category of measure spaces; also this is an analogue of Markov map for such spaces.
The dynamic theory of polymorphisms is a generalization of classical ergodic theory with tight links to the stationary Markov processes.
Ergodic theory of the polymorphisms includes

    1. Geometrical part: the set of polymorphisms is a convex semigroup compact in weak topology.
    2. Spectral theory and theory of dilations.
    3. Entropy theory.
    4. Connection to (Kolmogorov) Markov processes.
    5. Trajectory theory and connections to gruppoids and C*-algebras.

Some of those topics will be discussed in a series of lectures.

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