dvvmath.berkeley.edu

Professor, Department of Mathematics, University of California, Berkeley, USA.

Operator algebras, Free Probability.

“Noncommutative probability theory views elements of a noncommutative von Neumann algebra as analogs of classical random variables. One of Voiculescu’s discoveries is that in this more general noncommutative framework there is room for a new notion of independence, called freeness. Freeness bears the same relationship to free products of algebras as ordinary independence does to Cartesian products of probability spaces. Amazingly, as Voiculescu showed, many theorems and concepts in classical probability have very nice free probability analogs; this list includes the central limit theorem, notions of convolution, infinitely divisible laws, and so on. Remarkably, there is also a free probability analog of the classical information-theoretic notion of entropy. Free probability theory has now grown into a rich field, with connections and applications to many other areas of mathematics. For example, Voiculescu’s discovery that certain random matrices are asymptotically free as their sizes go to infinity makes possible computations of expected asymptotic spectral density of their eigenvalues. On the other hand, his and his followers’ work in free probability has led to a number of revolutionary results in von Neumann algebra theory, especially for von Neumann algebras associated to free groups.” (Dimitri Shlyakhtenko, of the University of California, Los Angeles).

2004 NAS Award in Mathematics from the National Academy of Sciences(NAS)for “the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.”

Voiculescu, Dan (2006). Symmetries arising from free probability theory. In Frontiers in number theory, physics, and geometry. I 231-243 Springer Berlin.

Voiculescu, Dan (2005). Free probability and the von Neumann algebras of free groups. Rep. Math. Phys. 55 No.1, 127-133.

Voiculescu, Dan (2005). Aspects of free probability. In XIVth International Congress on Mathematical Physics 145-157 World Sci. Publ., Hackensack, NJ.

Voiculescu, Dan (2004). Free analysis questions. I. Duality transform for the coalgebra of partial_X: B. Int. Math. Res. Not. No.16, 793-822.

Voiculescu, Dan (2002). The topological version of free entropy. Lett. Math. Phys. 62 No.1, 71-82.

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