Abstract

The free probability theory is a probability theory of noncommutative random variables, whereusual independence isreplaced by free independence. lt was initially designed to studylongstanding problems about von Neumann algebras of freegroups. lt turns out to be an extremelypowerful framework to study the universality laws in random matrix theory due tothegroundbreaking work of Voiculescu. These limiting laws are encoded in abstract operators.called free random variables. Brown measure is a sort of spectral measure for free randomvariables. not necessarily normal, l will report some recentprogress on the Brown measure of thesum X+Y of two free random variables Xand Y, where Y has certain symmetryor explicit R-transform. The procedure relies on Hermitian reduction and subordination functions. The Brownmeasure resultscan predict the limit eigenvalue distribution of various full rank deformed randommatrix models. The talk is based on mywork on Brown measure of elliptic operators and ioint workswith Hari Bercovici. Serban Belinschi and Zhi Yin