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

这项提议是在自由概率理论领域--这是一门与经典概率论平行的研究领域，在这种情况下，张量积被自由积取代，独立的随机变量被自由独立的非交换随机变量取代。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，它本身就成为了一门学科，与数学的其他部分，如经典概率、代数组合学和随机矩阵理论有联系。通过它与随机矩阵的联系，自由概率在量子信息理论和电子工程中的无线通信领域得到了应用。 这项研究的目的是在自由概率的现阶段相当活跃的方向上取得进展，并关注自由独立性与随机变量的其他类型的独立性(无论是经典独立性还是其他类型的非对易独立性)之间的相互作用。建议的研究还具有一个特点，即在处理正在考虑的问题时，它以一种本质的方式结合了分析方法和组合方法。

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.