## Topics in Probability: Gaussian Analysis Math 7880-2, Fall 2018 University of Utah

Time & Place. Instructor. TH 10:50 a.m.-12:20 a.m. JWB 108 Davar Khoshnevisan JWB 102/JWB 237

Course Synopsis. Let $\mathbb{P}_n$ denote the canonical Gaussian measure - or the standard multivariate normal - on $\mathbb{R}^n$; that is, $\mathbb{P}_n(E) := \int_E \frac{\exp\left(-\frac12\|x\|^2\right)}{(2\pi)^{n/2}}\,{\rm d}x \hskip1in \text{for all Borel sets $E$ in $\mathbb{R}^n$}.$ This is an object that you have seen, say in the context of the classical central limit theorem. And some of you have studied many of the elementary properties of $\mathbb{P}_n$ in your graduate, and sometimes also undergraduate, courses. In this course we study some of the deeper structure of the "Gauss space" $(\mathbb{R}^n\,,\mathcal{B}(\mathbb{R}^n)\,,\mathbb{P}_n)$. En route we take a tour through a good deal of beautiful mathematics, and also encounter a few applications worthy of note.

Prerequisites. Basic measure-theoretic probability at the level of Math. 6040.

Text. The material of the course is based in part on a set of notes that Tom Alberts and I are in the process of writing. There are chapter links below; they will become live in due time, and will continuously be updated. Also, here is a link to the table of contents.

Topics. Here are some of the topics that we will likely cover.

1. The Finite-Dimensional Theory
2. The Infinite-Dimensional Theory