Asymptotic Geometric Analysis [MA5024]
Sommersemester 2015
News
- Next exercise session will be Wed. 8.7., 14:15 in 03.12.020A.
- The examination will be oral and in the week from July 13th to 17th.
Content
Mixing concepts from geometry and analysis we will study high dimensional linear structures with a strong focus on asymptotic (in the dimension of the space) and probabilistic results. The course will cover:
- Banach-Mazur distance and John's theorem
- Brunn-Minkowski and concentration of measure on the sphere
- Johnson-Lindenstrauss results and dimensionality reduction
- Bourgain's minimal distortion embedding
- Dvoretzky type theorems
- Applications in complexity theory and data analysis
Lecture notes
| File |
Date |
Content |
Comments/further reading |
| Lecture 1 |
16.04.15 |
Introduction, normed spaces and convex bodies, duality and polars |
|
| Lecture 2 |
30.04.15 |
Dual norm, nuclear norm, Lewis' theorem |
|
| Lecture 3 |
07.05.15 |
Extremal volume ellipsoids, John's theorem, Banach-Mazur distance |
|
| Lecture 4 |
13.05.15 |
Banach-Mazur metric, Brunn-Minkowski inequalities, isoperimetric inequality |
Survey  on Brunn-Minkowski |
| Lecture 5 |
20.05.15 |
Concentration of measure on the sphere, Levy's Lemma for median and expectation value |
Book on measure concentration |
| Lecture 6 |
27.05.15 |
Haar measures on compact metric spaces, Johnson-Lindenstrauss flattening Lemma |
remarks added (04.06.) |
| Lecture 7 |
03.06.15 |
Enflo's Hamming cube bound, Frechet embedding, Bourgain's minimal distortion embedding |
|
| Lecture 8 |
11.06.15 |
Bourgain embedding into l_p, epsilon nets, Dvoretzky criterion for existence of almost Euclidean subspaces |
|
| Lecture 9 |
18.06.15 |
Spherical sections of ellipsoids, Dvoretzky-Rogers Lemma, Dvoretzky's thm. part I |
Notes on Dvoretzky's thm. |
| Lecture 10 |
25.06.15 |
Dvoretzky's thm. part II, Dvoretzky dimension and duality |
comments added (02.07.) |
| Lecture 11 |
02.07.15 |
Dvoretzky's thm. with projections, global version of Dvoretzky's thm. |
|
Exercises
Literature
The following books have overlap with the lecture but also go considerably beyond it. More literature will be provided during the course.
- G. Pisier, "The volume of convex bodies and Banach space geometry", Cambridge University Press 1989. (Chap. 1-4)
- V.D. Milman, G. Schechtmann, "Asymptotic Theory of Finite Dimensional Normed Spaces", Springer 2001.
- M. Ledoux, "The Concentration of Measure Phenomenon", American Mathematical Society 2005.
- J. Matousek, "Lectures on discrete geometry", Springer 2002 (Chap. 13-15).
