# Lecture Series

Venue: HIM lecture hall, Poppelsdorfer Allee 45

## Mark Jerrum, London: Estimating the mixing time of Markov chains

- Monday, September 10, 2007: 14:00 - 15:00
- Wednesday, September 12, 2007: 14:00 - 15:00
- Friday, September 14, 2007: 14:00 - 15:00
- Monday, September 17, 2007: 14:00 - 15:00
- Wednesday, September 19, 2007: 14:00 - 15:00
- Friday, September 21, 2007: 14:00 - 15:00

Abstract: A "Monte Carlo" algorithm works by generating random samples from a specified probability distribution, often one of a combinatorial or statistical-mechanical nature. Because this distribution is often complex, it may not be feasible to sample from it directly. However, it is often possible to set up a Markov chain whose stationary distribution is the one we wish to sample from. This is the idea behind the "Markov chain Monte Carlo" (MCMC) method. The MCMC approach is efficient only when the Markov chain in question is "rapidly mixing", i.e., converges rapidly to its stationary distribution. As a result, the mathematical analysis of the mixing time of Markov chains has become a highly developed branch of probability theory and theoretical computer science.

A wide variety of techniques have been developed for bounding the mixing time of Markov chains that arise in the study of MCMC algorithms. So much so that the lecture notes [1] provided only partial coverage of the area, even at the time of publication. In this short course, I’ll just have time to cover a couple of these techniques, one classical, one more recent.

[1] Mark Jerrum, Counting, sampling and integrating: algorithms and complexity. Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag, Basel, 2003.

## Eric Carlen, New Brunswick: From particles to continuous mechanics

- Monday, September 24, 2007: 13:00 - 15:00
- Wednesday, September 26, 2007: 13:00 - 15:00
- Friday, September 28, 2007: 13:00 - 15:00

Abstract: Physical systems consisting of many interacting particles generally have very different mathematical descriptions on different scales. A familiar case in point is a dilute gas of colliding particles. On the microscopic level, one has the direct particle description. On the mesoscopic scale, one has the kinetic theory description in terms of an evolving phase space probability distribution. Finally on the macroscopic scales, one has a fluid mechanical description, in terms of the Euler equations or Navier-Stokes equation, depending on the particular macroscopic scale.

In all such problems, in the passage from one scale to the next, certain dissipation inequalities govern the rate at which "extraneous" detail is washed out, leaving the simplified description on the larger, longer scale. The inequalities suggested by the physical problems often turn out to be very interesting from the point of view of geometric analysis, and the subject has been a fruitful source of conjectures and results in this area.

These lectures will focus on examples in which such dissipation inequalities can be proved, and will discuss a number of open problems as well. It is of interest to prove and apply such inequalities not only for the actual physical microscopic dynamics, but also for simpler microscopic models that lead to the same mesoscopic scaling limits, since doing so provides information on the behavior of solutions of these equations.

## Maxim Kontsevich, Bures-sur-Yvette: Analytic aspects of Quantum Field Theory

- Monday, October 15, 2007: 13:30 - 15:00
- Wednesday, October 17, 2007: 13:30 - 15:00
- Monday, October 22, 2007: 13:30 - 15:00
- Wednesday, October 24, 2007: 13:30 - 15:00

## Christian Borgs, Washington: Statistical Physics and Graph Theory: from the Ising Model to Chromatic Polynomials and Markov Chains

- Friday, November 2, 2007: 13:30 - 15:00
- Monday, November 5, 2007: 13:30 - 15:00
- Wednesday, November 7, 2007: 13:30 - 15:00
- Friday, November 9, 2007: 13:30 - 15:00

Abstract: In the past few years, a new community has started to form at the boundary of two previously largely disjoint groups of mathematicians: namely, mathematical statistical physicists and probabilists on one side, and combinatorialists and theoretical computer scientists on the other. Among the areas of shared interest between these groups are the theory of phase transitions in combinatorial models, the study of graph polynomials such as the chromatic polynomial, and the study of mixing times for Monte Carlo Markov chains.

In this lectures, I will review a set of techniques that were originally developed by mathematical physicists to study models of statistical physics and phase transitions in these models, and show how they can be applied to problems of interest in combinatorics and theoretical computer science.

## Anton Bovier, Berlin: Long term dynamics of disordered systems: from metastability to ageing

- Monday, November 19, 2007: 14:00 - 15:00
- Wednesday, November 21, 2007: 14:00 - 15:00
- Friday, November 23, 2007: 14:00 - 15:00
- Monday, November 26, 2007: 14:00 - 15:00
- Wednesday, November 28, 2007: 14:00 - 15:00
- Friday, November 30, 2007: 14:00 - 15:00

Abstract: In this series of lectures I will give an introduction into work done over the last years concerning the dynamics of spin glasses from a rigorous point of view. The main paradigm characterizing the particular properties of these (and other) systems is that of , and I will begin my lectures by introducing this concept and by reviewing the main models one is studying. This will include a brief review on the equilibrium theory of mean-field spin glasses, with emphasis on the simplest example, the Random Energy Model (REM) and its connection to extreme value theory and Poisson processes.

This will lead us to the simplest "trap-model" caricature of the dynamics of such models, the REM-like trap model of Bouchaud. I will review three methods of solving this model: renewal theory, spectral theory, and the analysis of the clock process. The latter will lead us to the key dynamical mechanism that will be seen to play a dominant role in many ageing systems, the -stable L'evy-subordinator.

The remainder of the lectures are devoted to show that the trap model correctly describes the asymptotics of more complicated models. I will first give a simple proof of this in the Glauber dynamics of the REM, on all time-scales before equilibration, and then show that the same is true for -spin SK (Sherrington-Kirkpatrick) models on sufficiently short, but still exponentially large, time scale. I will discuss the connection of these dynamical results with recent work on the universality of local energy statistics in disordered systems. (There will be lecture notes available before the beginning of the course on my web page).

## Alice Guionnet, Lyon: Lecture on Random matrices

- Monday, December 3, 2007: 13:00 - 15:00
- Wednesday, December 5, 2007: 13:00 - 15:00
- Friday, December 7, 2007: 13:00 - 15:00

Abstract: The theory of random matrices has known a great interest in the last twenty years from different mathematical communities, from the study of the local fluctuations of the spectrum of random matrices and integrable systems to the study of macroscopic quantities such as the empirical distribution of their eigenvalues and free probability.

These lectures will provide an introduction to the study of random matrices, at a macroscopic level. After discussing the celebrated Wigner theorem, we shall consider interacting matrices and their asymptotic distribution. We shall discuss applications in combinatorics, physics and operator algebras.

## Martin Barlow, Toronto: Random walks and percolation

- Wednesday, December 12, 2007: 13:30 - 15:00
- Friday, December 14, 2007: 13:30 - 15:00
- Monday, December 17, 2007: 13:30 - 15:00
- Wednesday, December 19, 2007: 12:30 - 15:00

## Yuri Kondratiev, Bielefeld: Interacting particle systems in continuum

- Monday, January 21, 2008: 14:00 - 15:00
- Wednesday, January 23, 2008: 14:00 - 15:00
- Friday, January 25, 2008: 14:00 - 15:00
- Monday, January 28, 2008: 14:00 - 15:00
- Wednesday, January 30, 2008: 14:00 - 15:00
- Friday, February 1, 2008: 14:00 - 15:00

Abstract: The lectures will be devoted to a review of recent studies in the theory of interacting particle systems in continuum. Main subjects of the considerations will be Markov processes in configuration spaces of continuous systems. These processes appear, in particular, in a relation with an analysis of stochastic evolutions of complex adaptive systems and have several motivations coming from problems of statistical physics, ecology, genetics, economics etc.

Between others, we suppose to discuss the following topics:

- birth-and-death processes in configuration spaces;
- contact models in continuum;
- individual based models in spatial ecology: Bolker-Pacala and Dickmann-Low systems;
- stochastic models of spatial economic systems.

In the case of the Glauber type dynamics in continuum, spectral gap properties for corresponding Markov generators and a construction of non-equilibrium processes will be described.

## Tomas Björk, Stockholm: Topics in interest rate theory

- Monday, February 11, 2008: 13:00 - 15:00
- Wednesday, February 12, 2008: 13:00 - 15:00
- Friday, February 15, 2008: 13:00 - 15:00

Abstract: The lecture series provides an introduction to arbitrage free interest rate theory. This will include standard results like short rate models, affine term structures, inversion of the yield curve, and forward rate (HJM) models. Time permitting we will also go into more recent developments, like market models, geometric term structure theory and the potential approach to positive interest rates.