Studies in free probability in operator algebras and a free analogue of the Riemann sphere

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Most of the PI's research will be motivated by the development of free probability theory and of its applications, and will deal especially with acquiring new analytic tools and with extensions of the framework of free probability. One main theme will be the noncommutative analogue of the Riemann sphere constructed by the PI and the fully matricial analytic functions on it. Work on this analytic machinery will aim at preparing it for use in operator-valued free probability and its applications to operator algebras, random multimatrix systems and some of the problems arising in connection with the analogue of entropy. Another theme will be the continuing phenomenon of new extensions of free probability parallling classical situations, like free extreme values and connections to free quantum groups.Free probability theory is a noncommutative probability framework, where independence is modelled on free products instead of tensor products, which makes it suitable for dealing with variables with the highest degree of noncommutativity. The free analogue of the Riemann sphere in the proposal aims at giving to a basic complex analysis object the same highest degree of noncommutativity to be adapted to uses in free probability. In free probability theory the role of the Gauss law is held by the Wigner semicircle law and the framework is particularly well suited for dealing with the asymptotics of systems of random matrices as their size increases. Via random matrices free probability has applications to multi-user telecommunication systems in engineering and to certain mathematical models in physics.