New directions in free probability

自由概率论起源于关于算子代数结构的问题，但已经发展成为一个与许多其他完全不相关的领域有联系的领域：最著名的是随机矩阵，还有组合学、大群的表示理论、数学物理，或者像金融关联和无线通信这样的应用领域。提出自由概率论的方法在许多这样的研究中发挥了重要作用。本提案将集中于自由概率论的三个新方向：(i)自由概率论中的量子对称性，（ii）自由随机演算和自由马里亚万演算，以及（iii）算子值自由概率论。

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.