Free Probability Studies in von Neumann Algebras

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The PI will work on free probability and use it to study von Neumann algebras. Motivated by problems arising from the free probability analogue of Shannon's differential entropy, analysis questions involving free difference quotient derivations,the natural derivations for noncommutative variables, will be studied. Extensions of the free probability parallel to classical probability, such as exploring free analogues of extreme value theory and of the Wasserstein distance will be continued. The focus on the operator algebra side will be on structural features of von Neumann algebras of free groups such as the automorphism group.Free probability is a probabilistic framework adapted to quantities with the highest degree of noncommutativity. This parallels a large area of classical probability from the Gauss law ( whose free probability counterpart is the semicircle law) to information theoretic entropy. The proposal is to develop new tools in free probability theory aiming at problems on operator algebras and random matrices. The applications of free probability to the asymptotics of large random matrices have also been of interest in connection with the random matrices in certain physics models and in models of multiuser telecommunication systems.