Free Probability, Transport, and Applications

Optimal transport is a study of ways of moving large numbers of objects from one place to another while minimizing cost. For example, in 1781 French mathematician Monge considered the problem of moving piles of dirt to fill holes in an optimal way. This area has since developed into a thriving mathematical field with connections to probability theory, stochastic analysis, differential equations, and many other fields.

The current project aims to develop optimal transportation techniques in the context of Voiculescu’s free probability theory. 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.

This award supports the proposer's research in this area and contributes to US workforce development through the training of graduate students.

This project deals with the emergent use of PDE methods to produce results in operator algebras and free probability theory. These methods allow us to construct free transport maps and prove isomorphism and decomposition results for operator algebras; they give new insights into Voiculescu's theory of free entropy and free stochastic calculus; and strengthen the connection between random matrix theory and free probability theory.

We propose a mixture of problems, some coming from existing research directions and some exploring new lines of inquiry.

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.