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 |