# Overview

This workshop is devoted to random fields in geometry and physics that model the intuitive notion of a `random surface'. The focus is on the geometric properties or morphology of random surfaces. The simplest random surfaces are defined by graphs of Gaussian random functions of various kinds, which have been used in physics to model water waves, speckle patterns in laser light, the mass distribution of the early universe or vacua of string theory. The study of their geometry is sometimes called statistical topography. The interesting geometric properties range from connectivity properties and curvatures of contour lines, to the distribution and correlation of peak points to the structure of excursion sets. The `roughtness' of the random surfaces correspond to the Sobolev class of the random functions, and range from spherical harmonics of fixed but large degree to the GFF (Gaussian free field), which is a random distribution rather than a function.

Quantum gravity involves a more difficult notion of random surface, namely a random Riemannian metric on a fixed surface or manifold. To define a measure `e`^{- S(g)} ` D g` on the space `M` of all Riemannian metrics (modulo diffeomorphisms) often involves defining a Riemannian metric on `M` itself. Polyakov used the volume form `Dg` of the natural de Witt-Ebin metric on `M` together with the Liouville action to define 2 D quantum gravity coupled with matter. The geometry of such Liouville random surfaces has been studied at the physics level of rigor for several decades by Polyakov, Knizhik, Zamolodchikov, Friedan, David, Distler, Duplantier and many others. Recently a rigorous definition of the Polyakov measure on a conformal class of metrics on the disc has been given by Duplantier-Sheffield, based on exponentiating the GFF. They proved a scaling relation between areas of discs relative to random metrics and a fixed metric known as the KPZ relation.

Exponentials of smoother Gaussian fields have been studied by Y. Canzani, D. Jakobson and I. Wigman, who have proved rigorous results on the curvature of such random metrics. A different notion of random metric in a conformal class (or Kahler class) has been recently proposed by Ferrari-Klevtsov-Zelditch, based on the Mabuchi-Semmes-Donaldson metric on `M` and its approximation by finite dimensional symmetric spaces of Bergman metrics. The mathematical study of random Riemannian metrics is in its infancy and the workshop aims to review both the conjectural properties of random surfaces and the kinds of geometric features which can be rigorously studied. The infinite dimensional geometry of the Riemannian manifold $M$ of metrics (in particular, its own geodesics) is intimately involved in the integration over the space of metrics and will also be surveyed in the workshop.

Further notions of random surface arise in discrete settings such as height functions of random dimer tilings or random triangulations of a surface. They may be viewed as discretizations of the continuum notions of random surfaces above. The continuum limit shapes of such discrete random surfaces have been studied by Kenyon, Okounkov, Sheffield and many others. Although the workshop focusses on continuum approaches to random surfaces, it also aims to overview such related work on discrete random surfaces.

Random fields have been used extensively to model the behavior of high energy eigenfunctions in quantizations of chaotic systems. We hope to discuss some recent approaches to averaging over spaces of operators (e.g. the work of Toth and Eswarathasan), and possible relations to Random Wave conjectures in Quantum Chaos.

Finally, we intend to discuss applications of probabilistic methods to improve existence and regularity results for solutions of PDEs with random initial conditions.