# CBMS Conference: Probabilistic and Combinatorial Approach in Analysis

**Lecturer:** Professor **Mark Rudelson**, University oF Missouri-Columbia

**Organizers**

- Richard Aron
- Per Enflo
- Joe Diestel
- Victor Lomonosov
- Andrew Tonge
- Artem Zvavitch

Invited Talks August 6-10

Talks August 11-12

Abstracts

Participants

Printable Poster

PHOTOS

**LECTURES SLIDES:**

**Mark Rudelson:** 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 , Open problems.

**The tentative lectures titles/abstracts:**

**Entropy and the supremum of a random process.***We define the covering numbers of convex bodies and sketch the proofs of two classical inequalities (Dudley's and Sudakov's) relating covering numbers to the supremum of associated Gaussian processes.***Application of random processes to geometry and harmonic analysis.***We apply methods developed in the previous lecture to modeling a random vector distributed in the convex body. This allows us to give an alternative proof of a result of Bourgain. We also show how these methods lead to an alternative proof of the Uniform Uncertainty Principle of Candes, Romberg and Tao.***Combinatorial structure of subsets of a discrete cube.***We show that any large subset of a discrete cube possesses a certain combinatorial structure. Namely, it has a projection which coincides with a discrete cube of a smaller dimension, called the VC-dimension of the initial subset.***Approximation of a convex body by parallelepipeds.***We show how the combinatorial results of the previous lecture can be used to estimate the Banach-Mazur distance from a given convex body to a cube of the same dimension.***Combinatorial dimension and its connection to random processes.***We introduce combinatorial dimension, which is an extension of VC-dimension to the non-discrete setting. We discuss the relation between the combinatorial dimension of a set and the probabilistic properties of a random process indexed by this set.***Covering Conjecture and separating trees.***We show that large entropy of a convex body implies the existence of a nice combinatorial structure in it.***Dimension reduction.***We use random coordinate projections to show that the size of the combinatorial structure constructed in the previous lecture is independent of the dimension of the convex body.***Entropy versus combinatorial dimension.***We prove that under mild regularity assumptions the entropy of a set is equivalent to its combinatorial dimension. We also solve a long-standing problem of Talagrand concerning estimates of random processes in terms of the combinatorial dimension of the set of parameters.***Applications of combinatorial dimension to functional analysis and convexity.***We show how the combinatorial dimension approach can be applied to extend the Bourgain--Tzafriri restricted invertibility principle for linear operators and to derive a coordinate analog of the classical Dvoretzky theorem in convex geometry.***Convexity and the combinatorial dimension.***We establish the relation between the combinatorial dimension of a set and that of its convex hull. We finish with a series of open problems related to combinatorial dimension.*