Free Probability and Cohomology in von Neumann Algebra Theory.

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

A fundamental theme in mathematics is that, in many cases, certain aspects of complicated finite-dimensional structures become simpler as the dimension of such structures goes to infinity. This very paradoxical behavior underlies the theory of thermodynamics in physics, in which complicated systems of particles admit a simpler global description (involving just a few variables, such as pressure and temperature) when the number of particles becomes large. A mathematical theory of free probability, initiated by Voiculescu in the 1980s, has shown that von Neumann algebras can be viewed as systems of 'thermodynamic variables' for asymptotic behavior of random matrices. A distinctive feature of the theory is that the variables no longer commute: the order of their multiplication matters. This results in an extremely rich theory that leads to free probability generalizations of classical objects such as partial differential equations, Brownian motion, and so on. The present project concerns questions of better understanding of asymptotic behavior of random matrices and of the resulting asymptotic limit through construction of invariants, which are both algebraic and analytic in nature. The project's aim is to continue the study of cohomological invariants associated to subfactors and more general inclusions of algebras, in particular L^2 invariants and their analogs. The study of some of these invariants turns out to involve many analytical tools supplied by free probability theory, such as free probability analogs of Brownian motion and of the associated Laplace-like operators. The study of these operators - arising in the asymptotic limit of random matrices - is of significant interest in random matrix theory as well. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.