Free Probability Theory

Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the twenty years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, algebraic combinatorics. Free probability also has connections with some mathematical models in theoretical physics. The BIRS workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. This continued a sequence of very successful 5-day workshops organized on these lines, like the ones at the Fields Institute in March 1995, at CIRM Luminy in January 1998, and at MSRI in January 2001. In this report we look in more detail at what are the current directions of development in free probability, with an emphasis on how they were represented in the BIRS workshop.