Free probability techniques: von Neumann algebras, random matrices and subfactors.

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The aim of the proposal is to use tools from free probability theory to study three areas of mathematics: random matrix theory, subfactor theory and the theory of von Neumann algebras. The tools involve a kind of differential calculus introduced earlier by Voiculescu in the context of his free information theory. Studying properties of differential operators from this "free calculus" will allow the investigator to formulate non-commutative analogs of PDE and SDE techniques. These in turn will give applications in von Neumann algebras, random matrix and subfactor theory. All of the four areas - random matrix theory, subfactor theory and the theory of von Neumann algebras - are amazingly rich mathematically. For example, Jones' subfactor theory has led to the discovery of a novel knot invariant, which has uses in diverse areas of mathematics and beyond (including the study of structure of DNA). Research in random multi-matrix models has engineering applications such as cell phone design. The focus of the present research is on the interplay between these four areas with the aim towards developing tools and techniques that are likely to impact all of the areas involved.