Probabilistic symmetries, extreme values and random topology

具有一定分布对称性的模型是概率论中的重要课题，在物理学、天文学、气候学和海洋学等多个学科中都有应用。常用的对称性包括平稳性(平移下的分布不变性)、自相似性(标度下的分布不变性)、各向同性(旋转下的分布不变性)等。文献表明，一些对称性可以导致过程/随机场的极值的特定行为。另一方面，极值与随机拓扑学的新兴研究领域密切相关。它研究随机集的拓扑性质，如游程集(随机场的值大于阈值的区域)，与传统方法不同的是，它关注的是在变形下具有健壮性的特征。这项研究的目标是发展一种揭示概率对称性、极值和随机拓扑之间关系的理论。这项研究将作为两个密切相关和相互作用的流派进行。第一个是研究定义在各种空间上的具有概率对称性的过程/随机场，特别是它们的极值。我打算考虑紧空间(S^1，S^n)、离散空间(Z)或赋范不同的空间(p-进制数)。第二个流将研究具有概率对称性的随机场的游程集的拓扑。这是由大脑成像的一些研究强烈推动的，其中漂移集对应于大脑中的活跃区域。已有的结果要么是渐近的，要么是预期的欧拉特征。我计划导出其他一些重要拓扑量的界，例如有限阈值的游程集的连通分支的个数。通过提供一个统一的框架来研究概率对称下的极值和随机拓扑，所提出的研究方案将在概率和统计学方面做出重大贡献。这将导致更好地理解定义在不同空间上的过程，并反过来，使用随机位置来刻画空间。此外，这将是第一次得到游程集的某些拓扑量的界的工作。当拓扑特征用于数据分析时，例如在气候学和海洋学中，结果特别有用。最后，来自第二流的结果可以直接应用于脑成像，例如，检测某些脑部疾病，这样患者就可以得到更早和更准确的诊断。因此，这些作品对加拿大公众至关重要的几个部门具有潜在价值。参与的学生将获得概率及相关领域的知识，以及解决问题和沟通的技能，为未来的职业发展做好准备。

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.