New Frontiers in Random Geometry (RaG)

The simplest experiment of probability theory is the toss of a fair coin. A sequence of coin tosses may be viewed as a one-dimensional process, and the ensuing theory is classical. When the randomness occurs in more general spaces, such as higher-dimensional euclidean spaces, the theory has wider importance and applicability, but also confronts difficulties of a very much greater order of magnitude. The basic challenge is to devise a calculus of probability that is well adapted to the problem under consideration and to the geometry of the encompassing space. Such problems may be static or dynamic in time.There have been many successes in recent years in areas including random walks, percolation and statistical physics, and models for aggregation and fragmentation. Two-dimensional systems are special for a variety of reasons, not least because of conformal structure and complex analysis.The current project will develop the frontiers of random geometry through a portfolio of linked themes including models for fragmentation and aggregation, percolation, random surfaces. The emphasis will be upon the development of new methodology, together with applications across a range of topics. We will pay special attention to three areas. The study of random fragmentations of a planar domain promises connections to processes similar to the so-called Gaussian free field. The study of surfaces with specified topological properties within percolation-type models makes connections to a multiplicity of random processes in three and more dimensions. The fractal nature of models for aggregation will be studied via conformality and other methods of stochastic geometry.In this six-year project, the three investigators will collaborate with research associates in mounting a concerted study of random geometry, with its special conjunction of stochastic processes inhabiting spaces of given geometry. Workshops will be organised on nominated topics of significance. Workers and students from the UK/EU and further afield will be invited to participate in the associated activity.

It is hard to overstate the degree to which probability theory is able to contribute to the advancement of the economy and public services, and to the enhancement of life. Fundamental advances in probability, devised originally for their theoretical value, are core to many aspects of modern life. We mention three examples: (i) the use of Brownian Motion in finance, (ii) methods of stochastic control in technological applications, (iii) Markov chain Monte Carlo as a key technique in computation. The translation of fundamental theory into applications rarely takes place overnight, and thus the chain of impact can be prolonged. The interface where probability meets geometry is one of the most active and promising areas where fundamental science leads to concrete outputs. For example, it has become clear that many applied problems of fundamental practical importance that are considered to be `hard' (in the sense of computational complexity) may be formulated as problems in geometry. The basic methodology that we will develop will come quickly to serve as a benchmark for a number of concrete questions directly relevant to the UK's well-being and economic growth. Our first example concerns the topic of `biodiversity'. Possible loss of genetic and phenotypic diversity is one of the major issues raised by global warming, and the scientific community needs adequate tools to monitor and interpret the levels of genetic variation. The Programme Grant will aid in the development of a theoretical foundation for a sound statistical analysis of genetic datasets. Our second example is the geometry of `random graphs and networks'. These are the most widely used models to describe a wide array of applied problems ranging from gene regulatory networks to telecommunication networks, and from traffic flow to worldwide-web dynamics. Properties of such models are captured by quantities depending on the geometry of the network. The scaling limits of the corresponding random geometric spaces allows for concrete computations. We expect these limiting objects to display a significant amount of fractal behaviour. Further down the line, there are many other examples of applications of the theory to be developed in RaG, and much scope remains for further investigation in the longer term. For instance, so-called k-SAT is one of the best-known problems in computer science, and is of central importance in various applied areas including hardware design and artificial intelligence. By developing sufficient intuition for the geometry of the random solution space, we believe that it will become possible to design better algorithms. Similar ideas may apply to many classical problems of combinatorial optimisation such as the travelling salesman problem, and decision problems such as scheduling under constraints. Such problems are faced by industry on a regular basis. Members of the team have extensive experience of engaging with researchers in these and other areas of application, including the British Antarctic Survey and the Wellcome Trust Sanger Institute within the cancer genome project. There are strong institutional links between the Cambridge Statistical Laboratory and local high-tech industry, including for instance the Microsoft Group on Systems and Networking. RaG will foster a culture of industrial collaboration, in which our research associates will be encouraged to participate. We make the following concrete proposal. Once a year, we will hold one-day brainstorming meetings with members of relevant applied and industrial groups in Cambridge and the UK. We anticipate strong synergy with the CCA (Cambridge Centre for Analysis) through its industrial connections and activities.