The University of Michigan Combinatorics Seminar
The probabilistic method has redefined functional analysis in high dimensions. Random spaces and operators are to analysis what random graphs are to combinatorics. They provide a wealth of examples that are otherwise hard to construct, suggest what situations we should view as typical, and they have far-reaching applications, most notably in convex geometry and computer science. With the increase of knowledge about random structures one begins to wonder about their universality. Is there a limiting picture as the dimention increases to infinity? Is this picture unique and independent of the distribution? What are the deterministic implications of probabilistic methods? This talk will survey progress on some of these problems, in particular a proof of a conjecture of Von Neumann and Goldstine on random operators and its connections to the Littlewood-Offord problem in extremal combinatorics.