Mathematical Sciences: Probability: Symmetries and Local Structure

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The investigator intends to continue his research in various areas of probability theory that are closely related to his previous work. In particular, he is planning to study certain characterization problems in multivariate exchangeability theory, where the representations may be expressed as sums of multiple Wiener-Ito integrals on tensor products of suitable Hilbert spaces. He will further extend his earlier study of exchangeable random sets, their Palm measures and local properties. In both of these general areas, there are many related problems of a more foundational nature that will also be addressed. The investigator will continue his research on certain basic structural problems in probability theory. This research will provide the general conceptual framework and basic theory necessary for work of other researchers in more applied fields, such as statistics and statistical mechanics, which in turn are useful for technology and other areas of science.