Gaussian Processes and Geometry – (Seminar in Probability and Stochastic Processes – 098435)

Instructor: Robert J. Adler

Time and place:
Tuesday, 08:30-10:30, Bloomfield Building (IE&M) 310.

Content: The course will effectively be a continuation of last semester’s course Gaussian Processes (098445) but this semester I plan to give all the proofs in reasonable detail, after spending considerabel time on the geometric background. If students wish to enrol in this course, without having taken last semester, I am prepared to give a two hour pep talk on where we shall be going, and why, in additional to the regular lectures.

Who might be interested in this course: Probabilists and differential and integral geometers.

Textbook: A book with Jonathan Taylor, entitled Random Fields and Geometry , and available as a free download. I strongly recommend bringing a copy to lectures.

Grade: The final grade will be based on student presentations during the semester.

Course Outline

Course Outline

Weeks Topic
1-2 Basic Reimannian geometry: Connections, curvature and integration.
3 Piecewise smooth manifolds: Whitney conditions, tameness and cone spaces.
4 Morse theory on stratified manifolds
5-6 Weyl’s tube formula in R^n and the n-sphere
7 Lipshitz-Killing and Minkowski curvatures
8 Gaussian fields on manifolds. The induced metric
9-10 Crofton’s formula: In Euclidean space and in (Gauss) function space
11 Kinematic fundamental formulae in R^n and the n-sphere
12 Poincare’s limit theorem and uniform processes on the n-sphere
13-14 A new kinematic fundamental formula in Gauss space