Heat Kernels and Geometries in Discrete and Continuous Settings

人类活动的模型往往带有随机性。随机性被用来理解DNA、图像恢复和识别、通信和社交网络以及金融市场的行为。它是有效计算和科学模拟的重要工具。在所有这些应用中，模型的随机过程的行为受到所研究系统的组合或几何结构的约束。这个项目涉及这种随机过程的基本性质，以及它们如何与其环境的全局几何结构相关联，无论是离散的还是连续的。该项目为研究生提供了培训机会。该项目专注于由相关几何或代数结构定义的随机过程。这些进程的全局行为由这种全局结构决定。在某些情况下，这些行为对于获取有关底层空间的信息很有用。这项研究位于分析、几何和概率之间的界面上，群体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，也是许多这些考虑的中心。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。该奖项反映了NSF的法定使命，并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。

Models of human activities often involve randomness. Randomness is used to understand DNA, image restoration and recognition, communication and social networks, and the behavior of financial markets. It is an important tool for efficient computations and for scientific simulations. In all these applications, the behavior of the model's random process is constrained by the combinatorial or geometric structure underlying the system under study. This project is concerned with the fundamental properties of such stochastic processes and how they relate to the global geometric structure of their environment, be it discrete or continuous. The project provides training opportunities for graduate students. The project focuses on random processes that are defined by a related geometric or algebraic structure. The global behaviors of these processes are determined by this global structure. In some cases, these behaviors are useful to obtain information on the underlying space. This research lies at the interface between analysis, geometry, and probability, with the notion of group 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 at the center of many of these considerations. Brownian motion on a Riemannian manifold and random walks on Cayley graphs of finitely generated groups provide key examples.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.