Random walks are interesting stochastic processes in their own right but they also have closed connections to many different areas of mathematics including analysis, geometry and algebra. The goal of the project is to explore various aspects of discrete potential theory and heat kernel analysis and how they may be applied to finite/infinite graphs beyond the classical setting of Euclidean lattice.
To get a flavour of the project, see resources below.
Warning: supervisions will be remote and online throughout the entire Michaelmas term. Do not take this project if you are not ready for highly independent studies.
The project lies at the intersection of analysis and probability.
Prerequisites: 2H Markov Chain II and Probability II are essential.
Co-requisites: 3H Stochastic Processess III is essential; Analysis III is highly recommended.
The following two books will be our primary references for this project's background reading:
Random Walks and Heat Kernels on Graphs by Martin Barlow. (See Ch.1-2 for preliminaries)
Introduction to Analysis on Graphs by Alexander Grigor’yan.