THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Focus Program on Noncommutative Distributions in Free Probability Theory July 2013 Host by the Fields Institute Lead Organizers: Roland Speicher (Saarland U. and Queen's U.) Dan-Virgil Voiculescu (U.C. Berkeley) Organizing Committee: Serban Belinschi (Saskatchewan U.) Benoit Collins (U. of Ottawa), James Mingo (Queen's U.) Alexandru Nica (U. of Waterloo)

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Objectives

In the one-month program, the many connections with other fields will appear, but a focus will be provided by emphasizing the distributions of the noncommutative variables. Such variables, like the quantum mechanical observables are operators and the distributions are expectations associated with them. In the case of one variable the noncommutative distributions are expectations of spectral measures and are classical, i.e. probability measures, For several variables such distributions are expectation values of noncommutative monomials (there are many more of these than commutative ones). The program will emphasize in all parts of free probability the distributions aspect: explicit determinations of distributions if possible, methods of computing distributions, uses of facts about distributions in applications. This is a timely topic since recent developments in a variety of directions in free probability are also advances in the distributions aspect, ranging from new noncommutative analysis tools, new combinatorial and discrete mathematics aspects, the appearance of new quantum symmetries and new types of distributions.

Scientific Activity

Program activities will be around two one-week-workshops (at the beginning and at the end of July) and, in the 2 weeks between workshops, seminars and other activities, mostly especially adapted for the very young (including graduate students and postdocs).

The first workshop, July 2 - 6, 2013, will be more on the combinatorics side, but will also include random matrix aspects, while the second workshop, July 22 - 26, 2013, will be geared towards the analytic, stochastic and operator algebra aspects. In the intervening two weeks we expect to organize a special program aimed at bringing graduate students and postdoctoral fellows quickly to frontiers of the subject by providing them with a short series of lectures that covers the basic definitions and theorems.

Organizers for first workshop: R. Speicher, A. Belinschi, A. Guionnet, and A.Nica
Organizers for second workshop: D.V. Voiculescu, H. Bercovici, B. Collins, J. Mingo, and R.Speicher.

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