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

该提案的目的是使用自由概率论的工具来研究三个数学领域：随机矩阵理论、子因子理论和von Neumann代数理论。这些工具涉及一种微积分，沃库列斯库早些时候在他的自由信息理论的背景下介绍了这种微积分。从这个“自由演算”研究微分算子的性质，将使研究者能够形成偏微分方程组和随机微分方程组技术的非对易类比。这些又将在von Neumann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von Neumann代数理论--在数学上都有惊人的丰富。例如，琼斯的子因子理论导致了一种新的纽结不变量的发现，它在数学的各个领域和其他领域都有用处(包括对DNA结构的研究)。随机多矩阵模型的研究在手机设计等领域具有工程应用价值。本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.