Heat Kernels and Geometries in Discrete and Continuous Settings

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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.