Random walks, diffusions, semigroups, and associated geometries

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Random processes play an important role in many aspects of science and other human activities. The study of card shuffling procedures provides an entertaining yet complex and mathematically interesting example. It also serves as a model for the many important mixing phenomena. We use randomness to understand complex phenomena, from the behavior of polymer molecules and DNA analysis, to image restoration and recognition, communication and social networks, and the behavior of financial markets. Random processes are also used as important tools for efficient computations. In all these cases, there are strong structural constraints underlying the behavior of the relevant random process. These constraints are expressed in terms of the environment of the process which often has a complex combinatorial or geometric structure. This proposal focuses on the study of the fundamental properties of such stochastic processes and on how they relate to the global structure of their environment.Many basic Markov processes evolve on a state space carrying a related geometric structure. Brownian motion on a Riemannian manifold, random walks on Cayley graphs of finitely generated groups and finite Markov chains on complex combinatorial structures such as trees or matchings are significant examples. This proposal focuses on the relationships between the behavior of such processes and the properties of the underlying geometric structure. It involves problems at the interface between analysis, geometry and probability with a major role played by groups and their actions. Potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation, is also at the center of many of these considerations.