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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
Recent Developments in Random Walks
Monday 2nd July - Thursday 12th July 2007


Random walks have been objects of intensive study by mathematicians over many decades with a well developed classical theory now in place. In recent years there have been a number of advances in this area which have spurred on research and have seen exciting progress on some novel problems. This workshop will cover recent work on random walk across a broad spectrum of activity but will have some major themes of focus. The key areas that we wish to focus on are specifically random walks in random media, random walks on groups and random walks with self-interaction. Our aim is to ensure that all are aware of new techniques and the challenges that are arising for our areas of focus. By bringing together a broad range of random walk experts we have the potential to provide significant impact on the development of these rich areas of research.

Central Themes
Our aim is to have speakers from a wide range of topics within the area of random walk. However we will provide more focus on three topics in which there has been exciting progress in the last few years.

Random walks in random environment:
The field of random media has been the object of intensive mathematical research over the last thirty years. It collects together a variety of models, mainly originating from physical sciences such as condensed matter physics, physical chemistry, and geology, where one is interested in materials which have defects or inhomogeneities. Many of these features can be modelled by taking a medium given by a suitable random process. The study of `transport phenomena' (heat conduction, fluid flow) in random media leads to study of the properties of the Laplacian and also of random motion or diffusion in such media. The study of random media is hard, and quite frequently what was initially thought of as a simple toy model has ended up as a major mathematical challenge.

Self-interacting random walk:
Self-interacting random walks (SIRW) are random processes evolving in an environment constantly modified by their own behaviour. The processes can be self-repelling or self-attracting, in other words more likely to stay away from or to come back to the places already visited before. These non-Markovian random walks ``learn'' from their past behaviour, either localizing on particular subsets or scattering on the graph, as a consequence of the interaction feature.

Random walks on graphs and groups:
Aside from these two areas of focus we will consider more classical problems of relevance such as the behaviour of random walks on groups and graphs. As a result we have invited experts in analysis and probability for random walks in this setting.

Organising Committee:
Ben Hambly (Oxford University) Laurent Saloff-Coste (Cornell University) Pierre Tarrès (Oxford University)