New directions in free probability

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Free probability theory originated from questions about the structure of algebras of operators, but has evolved into a field with links to many other quite unrelated fields: most notably random matrices, but also combinatorics, representation theory of large groups, mathematical physics, or as applied fields as financial correlations and wireless communications. The proposer's approach to free probability theory has played a major role in many of those investigations. This proposal will concentrate on three new directions in free probability: (i) Quantum Symmetries in Free Probability, (ii) Free stochastic calculus and free Maliavan Calculus, and (iii) Operator-valued free probability theory.