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.
|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|