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

这个提议是在自由概率论领域——在一个张量积被自由积取代，独立随机变量被自由独立的非交换随机变量取代的背景下，一条与经典概率方面相似的研究路线。自由概率起源于20世纪80年代，与算子代数的自由积问题有关。从那时起，它就成为了一门独立的学科，与数学的其他部分有联系，如经典概率论、代数组合学和随机矩阵理论。通过与随机矩阵的联系，自由概率在量子信息理论和电子工程中的无线通信领域中得到了应用。

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.