Free probability: combinatorial and analytic aspects, and interactions with other notions of independence

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The proposal is in the area of free probability theory - a line of research which parallels aspects of classical probability, in a context where tensor products are replaced by free products, and independent random variables are replaced by freely independent noncommutative random variables. Free probability originated in the 1980s, in connection to problems about free products of operator algebras. Since then, it has emerged as a subject in its own right, with connections to other parts of mathematics such as classical probability, algebraic combinatorics, and the theory of random matrices. Through its connection to random matrices, free probability has found applications in quantum information theory and in the area of wireless communications in electrical engineering.***The objective of the proposed research is to make progress in a direction which is quite active in the present stage of free probability, and concerns the interaction between free independence and other types of independence for random variables (either classical independence, or other types of noncommutative independence). The proposed research also has the feature that it combines in an essential way analytic and combinatorial methods, in the treatment of the problems that are being considered.