Random Fields and Stochastic Geometry (09w5040)

Objectives

The primary aim of this workshop is to bring together researchers in the areas of random fields (stochastic processes over general parameter spaces) and stochastic geometry, to discuss and develop a new class of results at the interface of the two subjects.

The secondary aim is to involve researchers in areas that have traditionally applied results of these kinds, with the complementary purposes of aquainting them with the new theory, much of which was generated by specific issues arising in applications, and of identifying new areas of further theoretical development.

The motivation for holding this workshop now is a recent explosion of interest in the study of the sample path properties of smooth random fields, arising from a collection of important results that have been proven over the past few years. In some ways, their roots are from the early 1970's, when it was realised that the study of two basic properties of Gaussian random processes, boundedness and continuity, was of an essentially Banach space geometric nature rather than the purely probabilistic problem that it had been considered until then. At the time, this realisation revolutionised the study of Gaussian processes, and has shaped its structure ever since.

Over the last 6-7 years a further revolution, has been taking place, this time in the way the sample path properties of smooth multi-parameter stochastic processes, or random fields, have been studied. Although the setting is still primarily Gaussian, this time it also extends out into the non-Gaussian world, but once again it is primarily geometric. However, whereas the 1970's
saw a replacement of general parameter spaces by a canonical Hilbert space with an inner product induced by the random field, in the new set of problems all parameter spaces become Riemannian manifolds with Riemannian metrics induced by the fields. These results involve an intruiging blend of probability and geometry, and although the revolution has, primarily, been taking place at the level of theoretical mathematics, it has already had an impact on applications of random field theory in applied settings as wide apart as astrophysics and medical imaging.

The interface between random field theory and stochastic geometry has been at the centre of this activity, which is why most of the participants in the workshop will also come from the area of this interface. However, it turns out that this theory also has had a major impact on the problem of obtaining practical approximations for the tail of the distribution of the suprema of many random fields, an area of extremal theory that is rich in applications to many areas of applied probability and statistics. This area has also seen many independent developments of its own over the past few years, after a decade or two of relative stagnation. Consequently another group of participants in the workshop will be comprised of mathematicians from extreme value theory, the aim being to try to formulate a common framework for the extremal theory of Gaussian and related random fields that combines many of the recent approaches.

Yet another group of participants will come from the area of theoretical physics. In that area Gaussian random field geometry has become a central component in topics as widely dispersed as the study of nodal domains in optical waves and quantum chaos, and in the study of the geometry of liquid crystals and superfluid helium.

The meeting itself will be structured around three types of activities: A small number of 45-60 minute expository talks, a larger (but still small) number of standard 25-30 minute talks, a poster session and round table discussions to highlight the multi-disciplinary nature of the topics of the workshop.