The geometry of convex bodies in high dimensions and correspondingly of normed vectors spaces is a beautiful subject of study that has fascinated mathematicians for over a hundred years. Questions that have driven the field include: how does measure concentrate in high dimensions? how close is every convex body to an ellipsoid? how much can one improve upon on the triangle inequality? To what extent does the parallelogram law hold? To gain a deep understanding of these questions, still today the subject of very active research, powerful techniques were developed by leveraging surprising connections between geometry, analysis and probability. These have now found numerous applications across mathematics and computer science. In particular, a rich interplay of continuous and discrete mathematics has developed in recent years.
| Number | Topic | Resources |
|---|
| 0 | Review of Convexity and Normed Spaces |
- Notes.
- Related content: Vershynin's Chapter 1 link.
|
| 1 | Geometry and Distances of l_p^n spaces |
- Notes.
- Related content: Vershynin's Chapter 1 link.
|
| 2 | The Lowner-John Ellipsoid |
|
| 3 | Prekopa-Leinder and Brunn-Minkowski Inequality |
- Notes: Pages 7-10 in Assaf Naor's course (link).
|
| 4 | Discrepancy Theory and Spencer's Theorem |
|
| 5 | Grothendieck's Inequality |
- Notes.
- Exercises: see lecture notes.
|
| 6 | Quasi-Random Graphs |
- Notes.
- Exercises: see lecture notes.
|
| 7 | Covering Numbers and Gaussian Inequalities |
- Related content:
Sudakov Inequalities, Giannopoulos and Milman link, Theorem 6.1.1 and 6.1.2.
Eps-Nets and Covering Numbers, Vershynin link, Lemma 3.5 and Section 4.1.
|
| 8 | Gaussian Concentration and Dvorestky's Theorem |
- Notes.
- Homework 1. (due Tuesday May 7th)
- Related content:
Chapter 3 (sections 1-3) in Vershynin link.
|
| 9 | Type, Cotype and Kwapien's Theorem |
|
| 10 | Type Failure and l_1 Containment |
|
| 11 | Non-Commutative Khinchine Inequality and the Alon-Roichman
Theorem |
|
| 12 | Graph Sparsification by Random Sampling |
- Notes: Daniel Spielman's lecture notes Notes.
Golden Thompson Inequality: Terry Tao's Blog Post link
|
Remark: we have chosen to arrange the lectures above by content instead of the
exact dates during which they were presented.
