Free probability theory, group boundaries and von Neumann algebras

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Shlyakhtenko will study classes of operator algebras arising in connectionwith Voiculescu's free probability theory. The main emphasis of thisresearch is placed on the study of operator-valued free randomvariables and operator algebras that they generate. Analysis of thesevon Neumann algebras is intimately tied with the goal ofclassification of free Araki-Woods factors, which are free probabilityanalogs of ITPFI type III factors. Such analysis is also importantfor understanding of subfactors and automorphisms of amalgamated freeproduct algebras. One of the goals of the present research is todevelop free entropy-based techniques for dealing with operator-valuedrandom variables.Free probability theory is a highly non-commutative parallel to basicprobability theory. Matrix-valued random variables (such as randommatrices) naturally fit in the non-commutative probability framework;the asymptotic behavior of large random matrices is well-modeled bymatrix-valued free random variables. Applications are in mathematicsto the theory of operator algebras, subfactors, ergodic theory, aswell as the theory of random matrices, which have connections withcertain physical models.