Math 790:Stochastic Processes on Manifolds, Varieties, and Sheaves, Spring 2022

Description

Stochastic Processes on Manifolds, Varieties, and Sheaves We will discuss how to place stochastic processes on geometric, algebraic, and topological objects. We will mainly consider Brownian motion and Gaussian processes as example processes. We first go through the manifold case where we introduce the Eells-Elworthy-Malliavin formulation of lifting Brownian motion on manifolds and then discuss a SPDE formulation of Gaussian processes on manifolds that ensure the covariance function is positive (semi) definite. We then consider the case of algebraic varieties and adapt manifold models to the setting where there are singularities induced by algebraic structures. There is very little on stochastic processes on sheaves. We will consider some recent work where we consider probability spaces as Grothendieck sites on which Brownian motions are defined via sheaves. I will try to make this short course as self-contained as possible. I am hoping knowledge of Markov chains and basic differential geometry is enough for a base requirement. >

Course stuff

1. Main text:
• Stochastic Processes on Manifolds by Elton Hsu



2. Useful material:
• Slides by Sayan Mukherjee (Lecture 1 may be useful)
• Lecture notes by Steve Lalley on Gaussian processes
• Lecture notes by Steve Lalley on Brownian Motion
• Lecture notes by Steve Lalley on Martingales
• Lecture notes by Steve Lalley on Ito calculus
• Lecture notes by Steve Lalley on SPDEs
• Lecture notes by Pavel Tumarkin on Riemannian Geometry. see Handouts: relevant sections are Smooth manifolds -> Geodesics
• Lecture notes by Shreya Arya on basic geometry
• Lecture notees by Sebastian Roch on multivariate Brownian motion
• Horizontal diffusion maps by Tingran Gao
• Lecture notes on SDEs on euclidean spaces and manifolds by Sayan Mukherjee
• Lecture notes on stochastic horizontal lift by Sayan Mukherjee
• Slides on scaling limits and random simplicial complexes by Sayan Mukherjee
• Slides on generalizations of the giant component by Sayan Mukherjee