Le Professeur Kurt Johansson (KTH Stockholm) fera une série d'exposés dans le cadre de la Chaire de la Vallée Poussin 2012 du 22 au 25 mai 2012
Titre : "Random matrices and related models"
PROGRAMME
Mardi 22 mai 16h - Leçon inaugurale
"Random matrix universality"
Mercredi 23 mai 15h
"Determinantal point processes"
Jeudi 24 mai 15h
"Szegö's theorem: history and recent developments"
Vendredi 25 mai 10h
"Dimer and random tiling models"
La leçon inaugurale sera consacrée au concept d'universalité en théorie des matrices aléatoires. Les matrices aléatoires sont une source naturelle de nouvelles lois de probabilité, qui apparaissent dans de nombreux contextes apparemment non reliés entre eux. Ceci est à rapprocher du fait que la loi Gaussienne décrit les fluctuations de nombreux phénomènes, ce que l'on peut appeler l'universalité de la distribution Gaussienne.
DETAILS
Lecture 1
Title: Random matrix universality
Abstract: In the last decades random matrix theory has had a very rapid development with applications to many areas e.g. in physics, statistics, telecommunications and other parts of mathematics. One central aspect is that of universality. Random matrices are a source of natural probability laws and stochastic processes which appear in apparently unrelated contexts. This is reminiscent of the fact that the Gaussian law describes the fluctuations of quantities in many areas of science, which we can call the universality of the Gaussian distribution. In this lecture I will give a non-technical overview of universality in random matrix theory.
Lecture 2
Title: Determinantal processes
Abstract: Random point processes are of interest in many contexts. The most well known example is the Poisson process which describes random, uncorrelated points. Determinantal point processes form a relatively new class of point processes which have been much studied recently since they have an interesting mathematical structure and appear naturally in many problems, e.g. in random matrix theory and in some two-dimensional statistical mechanical models. This lecture will outline basic properties and give several examples of determinantal processes.
Lecture 3
Title: Szegö´s theorem - history and recent developments
Abstract: The strong Szegö limit theorem for the asymptotics of Toeplitz determinants has several connections to problems coming from theoreticalphysics, e.g. the Ising model and random matrix theory, and there is a wealth of interesting mathematics around the theorem. In this lecture I
will describe some of the history of the theorem and its applications and also some more recent developments.
Lecture 4
Title: Dimer and random tiling models
Abstract: Random tiling models or more generally dimer models are two-dimensional statistical mechanical models which have an interesting mathematical structure and theory. They give examples of determinantal processes and also define certain models of random surfaces. In some of these models we have a boundary between a frozen and a liquid region and continuum scaling limits in these regions give rise to random matrix limit laws. In the lecture I will discuss some parts of the theory of dimer and random tiling models.