Diffusions and Geodesic Flows on Manifolds: The Differential Geometry of Markov Chain Monte Carlo

Markov Chain Monte Carlo methods provide the most comprehensive set of simulation based tools to enable inference over many classes of statistical models. The complexity of many applications presents an enormous challenge for sampling methods motivating continual innovation in theory, methodology and associated algorithms. In this series of lectures we will consider one recent advance in MCMC methodology, that has exploited mathematical ideas from differential geometry, classical nonlinear dynamics, and diffusions constrained on manifolds, in attempting to provide the tools required to attack some of the most challenging of sampling problems presented to statisticians. A step-by-step presentation of the material will be provided to ensure that students grasp the fundamental concepts and are able to then develop further theory and methodology at the end of the lectures.