250067 VO Probabilistic Methods in Analysis (2021S)

The aim of the course is to explore four questions in analysis that share a rather surprising feature: their formulation has nothing to do with Probability Theory, but their solution is heavily based on probabilistic arguments.
I will use the four problems to illustrate the power of the probabilistic method: proving the existence of an object based on some event having a positive measure. Along the way, I will develop the necessary machinery, which is importnat in its own right: it is of constant use in many areas of modern mathematics, statistics and computer science (for example, it is of central importance in data science).   

To give a flavour of the questions, here is a somewhat inaccurate formulation of two of them:

Q1) If X is an infinite dimensional normed space, does X contain finite dimensional subspaces of arbitrarily high dimension that are arbitrarily close to being Euclidean? 

This question was asked by Grothendieck, and its solution - especially the breakthrough made by V. Milman in the 70's-, has led to the development of Asymptotic Geometric Analysis: the quantitative study of the geometry of high dimensional convex bodies. 

Q2) Let (X,d) be an arbitrary metric space consisting of n points. How well can X be embedded in a Hilbert space?

Here "well" means that the embedding does not distort distances by "too much". 

We will present Bourgain's result, which shows that any metric space of cardinality n can be embedded in a Hilbert space with distances distorted by a multiplicative factor of at most log(n). Moreover, a distortion of log(n)  is the best that one can hope for (actually Bourgain showed a slightly suboptimal lower bound - by a log log(n) factor).

Clearly, neither question has "probability" in its formulation, nor is it clear why probability should come into the game at all. As it happens, it is of crucial importance in the solutions.

Although I will try to make the course as self-contained as possible, it will require mathematical maturity. Knowledge of some Functional Analysis, measure and integration and probability theory is highly recommended.