Free Probability Studies in von Neumann Algebras

PI将研究自由概率并用它来研究冯·诺伊曼代数。由香农微分熵的自由概率模拟引起的问题，将研究涉及自由差商推导的分析问题，非交换变量的自然推导。将自由概率平行于经典概率的扩展，如探索极值理论和沃瑟斯坦距离的自由类似物，将继续进行。算子代数方面的重点将放在自由群（如自同构群）的von Neumann代数的结构特征上。自由概率是一种适用于非交换性最高程度的量的概率框架。这与从高斯定律（其自由概率对应物是半圆定律）到信息理论熵的大面积经典概率相似。本文的目的是开发自由概率论中针对算子代数和随机矩阵问题的新工具。自由概率在大随机矩阵渐近性中的应用，在某些物理模型和多用户电信系统模型中的随机矩阵方面也引起了人们的兴趣。

The PI will work on free probability and use it to study von Neumann algebras. Motivated by problems arising from the free probability analogue of Shannon's differential entropy, analysis questions involving free difference quotient derivations,the natural derivations for noncommutative variables, will be studied. Extensions of the free probability parallel to classical probability, such as exploring free analogues of extreme value theory and of the Wasserstein distance will be continued. The focus on the operator algebra side will be on structural features of von Neumann algebras of free groups such as the automorphism group.Free probability is a probabilistic framework adapted to quantities with the highest degree of noncommutativity. This parallels a large area of classical probability from the Gauss law ( whose free probability counterpart is the semicircle law) to information theoretic entropy. The proposal is to develop new tools in free probability theory aiming at problems on operator algebras and random matrices. The applications of free probability to the asymptotics of large random matrices have also been of interest in connection with the random matrices in certain physics models and in models of multiuser telecommunication systems.