Probabilistic symmetries, extreme values and random topology

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Models exhibiting certain distributional symmetries are important topics in probability, and have applications in various disciplines such as physics, astronomy, climatology and oceanography. Commonly used symmetries include stationarity (distributional invariance under translation), self-similarity (distributional invariance under scaling), isotropy (distributional invariance under rotation), etc. Literature shows that some symmetries can lead to specific behavior of the extreme values of the process/random field. Extremes, on the other hand, are closely related to the emerging research area of random topology. It studies the topological properties of random sets such as the excursion set (the area on which the value of a random field is larger than a threshold), and differs from the traditional approaches by focusing on the features which are robust under deformation. The goal of the proposed research is to develop a theory that reveals the relation among probabilistic symmetries, extreme values, and random topology. The research will be pursued as two closely related and interacting streams. The first is to study processes/random fields having probability symmetries defined on various spaces, especially their extremes. I plan to consider the spaces which are compact (S^1, S^n), discrete (Z), or normed differently (p-adic numbers). The second stream will study the topology of the excursion sets for the random fields having probabilistic symmetries. It is strongly motivated by some research in brain imaging, where the excursion sets correspond to the active regions in the brain. The existing results are either asymptotic or for the expected Euler characteristic. I plan to derive bounds for some other important topological quantities, such as the number of the connected components of the excursion sets, for finite thresholds. The proposed research program will contribute significantly in probability and statistics by providing a unified framework to study the extreme values and random topology under probabilistic symmetries. It will lead to a better understanding of the processes defined on different spaces and conversely, the characterization of the spaces using the random locations. Moreover, this will be the first work where bounds are derived for certain topological quantities of the excursion sets. The results are especially useful when topological features are used for data analysis, such as in climatology and oceanography. Finally, the results from the second stream can be directly applied to brain imaging, for example, to detect certain brain illness, so that the patients can get earlier and more precise diagnosis. Thus, these works have potential value for several sectors that are critical to the Canadian public. The students involved will acquire knowledge in probability and related areas as well as skills in problem solving and communication, which prepare them for future career developments.