Extremes of stochastic processes and random fields: new directions

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This aim of this proposal is to investigate ``the shape of the extreme exceedance sets'' of very general infinitely divisible random fields. This is a very broad class of models, containing Gaussian, alpha-stable, Poisson, Gamma and many other important random fields as special cases. The notion of``shape'' includes the number of connected components of the exceedance sets, how many ``holes'' do these components have, and of what kind, etc. The infinitely divisible random fields provide an excellent class of models because of their rich mathematical structure and because of the richness of the possible ``shapes'' of their exceedance sets. The ideas and the tools that will be employed to investigate the exceedance sets will come from probability theory, algebraic topology, ergodic theory, geometry and data mining. This proposal aims to further our understanding of the extremes by using novel ideas coming from diverse areas of mathematics. Understanding extremal phenomena and the risks associated with these phenomena is highly beneficial for the society: analysis, forecasting and evaluating the impact of events like Hurricane Katrina of 2005 is hard to overestimate. The same is true for financial risks and security risks. Another impact of this project would be in medicine: it will give doctors new statistical tools to detect departures from the norm in scans by studying the shape the extremes tend to take under pure randomness, when no disease is present.