Mathematisches Forschungsinstitut Oberwolfach Report No . 15 / 2005 Free Probability Theory Organised

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. The theory grew out of attempts to solve some longstanding problems about von Neumann algebras of free groups. In the almost 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, topology. Free probability also has connections with applied mathematics (wireless communication) and some mathematical models in theoretical physics. The Oberwolfach workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. Mathematics Subject Classification (2000): 46L54. Introduction by the Organisers The workshop Free Probability Theory, organised by Philippe Biane (Paris), Roland Speicher (Kingston), and Dan Voiculescu (Berkeley) was held March 27th–April 2nd, 2005. This meeting was well attended with over 50 participants with broad geographic representation from Austria, Canada, Denmark, France, Germany, Hungary, Japan, Netherlands, Poland, USA. This workshop was sponsored by a project of the European Union which allowed us to invite in addition to established researchers also a couple of young people who were interested in learning about free probability. 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 828 Oberwolfach Report 15/2005 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 almost 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, topology. Free probability also has connections with applied mathematics (wireless communication) and some mathematical models in theoretical physics. The Oberwolfach workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. The diversity of the particants and the ample free time left in the programme stimulated a lot of fruitful discussions, laying the seed for many new collaborations. The programme consisted of 13 lectures of 50 minutes, supplemented by 13 shorter contributions of 30 minutes. Because of the various backgrounds of the participants much emphasis was put on making the lectures accessible to a broad audience; most of them provided a survey on the background as well as highlighting some recent developments in connection with free probability. Instead of trying to summarize all these developments we will let the following abstracts speak for themselves. Free Probability Theory 829 Workshop: Free Probability Theory