Random Surfaces: growth, fluctuations and universality

The goal of this project is the mathematical study of random surfaces, which are involved in many real- world phenomena. One example is provided by surface growth phenomena. This can be observed with a simple experiment: if some coffee is dropped on a sheet of paper, you will observe the colored stain growing in an approximately round shape. A closer observation, however, reveals that the shape of the coffee stain is not perfectly round and its boundary or surface looks more and more wiggly and random as the stain grows larger. A similar behavior is observed experimentally in the growth of combustion fronts, of crystals, of bacterial colonies, etc. While in the example of the coffee stain the growth phenomenon occurs in a two-dimensional environment (the sheet of paper), many other growth phenomena occur in the usual three-dimensional space (think, for instance, of the snow layer growing in your garden during a snowfall). Based on experimental observations and theoretical studies, physicists have predicted random surfaces arising in extremely different physical systems to show the same pattern of fluctuations on large scales. This phenomenon is called universality. It is an exciting challenge for mathematicians to understand this phenomenon rigorously, using the tools of probability theory. In order to study random surfaces, mathematicians describe them with the help of models or equations that are somewhat simplified but retain the essential features of the real world phenomena. These simplified models are still extremely challenging for mathematicians and their study requires sophisticated tools of probability theory, such as the theory of stochastic differential equations, and the combinatorics of discrete surface models. Two-dimensional growth phenomena have been enormously studied by mathematicians in the last few decades, revealing many unexpected and fascinating facets. In contrast, three-dimensional ones, and more generally random surfaces in high spatial dimension, are largely unexplored and it is one of the main goals of this project to make substantial progress in this direction. Another line of research we will pursue is the mathematical study of the running time of probabilistic algorithms that are used to sample the configurations of random surfaces. This topic is part of a very active branch of mathematics, at the interface between probability theory, computer science and combinatorics.