Abstract:

"Quantum" mathematics postulates that like classical mechanics is a "shadow" of quantum mechanics, obtained from it by passing to a certain limit, many classical mathematical objects --- measure spaces,topological spaces, manifolds --- are but shadows of more general "quantum" objects (typically described by operator algebras). Thus probability spaces and random variables are particular cases of more general "non-commutative" (or "quantum") probability spaces and random variables. In the early 80s, Voiculescu realized that in addition to the "usual" notion of independence for (classical or more generally quantum) random variables, it is possible to define an entirely new notion of free independence --- a notion of independence that is specific to "quantum" probability spaces and does not make sense for ordinary commuting random variables. Amazingly, this at first sight very esoteric theory has found many applications to much more classical problems, for example, to the theory of random matrices. We'll give a brief introduction to free probability theory and its applications.