### Tuesday, October 6, 2015 (3:00 p.m. in Yost 306)

**Title:** K-convexity (Part I)

**Speaker:** Stanislaw Szarek (Professor, Department of Mathematics, Applied Mathematics, & Statistics, Case Western Reserve University)

**Abstract:** Let (g_{i}) be a sequence of independent standard Gaussian random variables on some probability space, and let P be the orthogonal projection on the subspace generated by the g_{i}‘s. Then P can be written as Pf = sum_{i} E(f g_{i}) g_{i} , where E denotes the expected value (i.e., the integral); this is just the formula from MATH 424. Now comes a twist: instead of scalar valued functions, we allow f to take values in a normed space X and define P = P_{X} by the same formula. If this gives us a well-defined and bounded operator on L_{2}(X), we say that X is K-convex. This concept was introduced in 1970’s and turned out to be extremely useful.

The plan is to have two educational talks introducing and investigating this concept, hopefully culminating with a theorem showing that if dim X = n, then the norm of P_{X} as an operator on L_{2}(X) is at most of the order log n. Prerequisites will be kept to minimum.