Free probability techniques: von Neumann algebras, random matrices and subfactors.

该提案的目的是利用自由概率论的工具来研究数学的三个领域：随机矩阵理论、子因子理论和冯·诺伊曼代数理论。这些工具包括一种由Voiculescu在他的自由信息理论背景下引入的微分学。从这种“自由演算”中研究微分算子的性质将使研究者能够制定PDE和SDE技术的非交换类似物。这些又将在冯·诺伊曼代数、随机矩阵和子因子理论中得到应用。这四个领域——随机矩阵理论、子因子理论和冯·诺伊曼代数理论——都有着惊人的丰富的数学知识。例如，琼斯的子因子理论导致了一个新的结不变量的发现，它在数学的各个领域都有应用（包括DNA结构的研究）。随机多矩阵模型的研究具有工程应用，如手机设计。目前研究的重点是这四个领域之间的相互作用，目的是开发可能影响所有相关领域的工具和技术。

The aim of the proposal is to use tools from free probability theory to study three areas of mathematics: random matrix theory, subfactor theory and the theory of von Neumann algebras. The tools involve a kind of differential calculus introduced earlier by Voiculescu in the context of his free information theory. Studying properties of differential operators from this "free calculus" will allow the investigator to formulate non-commutative analogs of PDE and SDE techniques. These in turn will give applications in von Neumann algebras, random matrix and subfactor theory.All of the four areas - random matrix theory, subfactor theory and the theory of von Neumann algebras - are amazingly rich mathematically. For example, Jones' subfactor theory has led to the discovery of a novel knot invariant, which has uses in diverse areas of mathematics and beyond (including the study of structure of DNA). Research in random multi-matrix models has engineering applications such as cell phone design. The focus of the present research is on the interplay between these four areas with the aim towards developing tools and techniques that are likely to impact all of the areas involved.