Diffusions and jump processes on groups and manifolds

对科学实验或人类活动进行建模往往涉及随机性。洗牌程序提供了一个熟悉的，但复杂的和数学上有趣的例子，作为许多混合现象的模型。随机性被用来理解图像恢复和识别、通信和社交网络、金融市场的行为，以及总体上对大型数据集的分析。它是研究高效计算和科学模拟的重要工具。在所有这些应用中，与问题背后的复杂组合或几何结构相关的强结构约束决定了行为。这个项目涉及基本随机过程的基本性质，以及这些过程的行为如何与其不同环境的全局几何结构相关。作为该项目的一部分，博士后助理、研究生和本科生将接受指导和培训。被资助的研究集中于由相关的几何或代数结构(例如，黎曼流形和群)定义的随机过程。这些进程的全局行为由该底层结构决定。在某些情况下，这些行为可以提供有关底层空间及其结构的信息。这些探索是在分析、几何和概率的交界处进行的，其中群体结构的概念起着关键作用。偏微分方程组和位势理论，即研究调和函数和热方程的解，也是中心。黎曼流形上的布朗运动和有限生成群的Cayley图上的随机游动提供了关键的例子。这个奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Modeling scientific experiments or human activities often involves randomness. Card shuffling procedures provide a familiar, yet complex and mathematically interesting example that serves as a model for many mixing phenomena. Randomness is used to understand image restoration and recognition, communication and social networks, the behavior of financial markets, as well as in the analysis of large data sets in general. It is an important tool in the study of efficient computations and scientific simulations. In all these applications, strong structural constraints associated with the complex combinatorial or geometric structure underlying the problem determine the behavior. This project is concerned with the fundamental properties of basic stochastic processes and how the behavior of these processes relates to the global geometric structure of their different environments. Postdoctoral associates, graduate students, and undergraduate students will be mentored and trained as part of this project. The funded research focusses on random processes that are defined by a related geometric or algebraic structure (e.g., Riemannian manifolds and groups). The global behaviors of these processes are determined by this underlying structure. In some cases, these behaviors can provide information on the underlying space and its structure. These explorations are at the interface between analysis, geometry, and probability, with the notion of group structure playing a key part. Partial differential equations and potential theory, i.e. the study of harmonic functions and solutions of the heat equation, are also central. Brownian motion on Riemannian Manifolds and random walks on Cayley graphs of finitely generated groups provide key examples. The notion of stable-like processes on nilpotent groups is also studied.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.