Mathematical Sciences: Probability: Symmetries and Local Structure

研究者打算在与他以前的工作密切相关的概率论的各个领域继续他的研究。特别是，他计划研究多元可交换性理论中的某些表征问题，其中表示可以表示为适当Hilbert空间张量积上的多个Wiener-Ito积分的和。他将进一步扩展他早期对可交换随机集、它们的Palm测度和局部性质的研究。在这两个一般领域中，还将讨论许多具有更基本性质的相关问题。研究者将继续研究概率论中的一些基本结构问题。这项研究将为其他研究人员在更多应用领域（如统计学和统计力学）的工作提供必要的一般概念框架和基本理论，这反过来又对技术和其他科学领域有用。

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.