Free probability, von Neumann algebras, subfactors and random matrices

Shlyakhtenko该建议的目的是研究四个数学领域之间的联系：冯·诺伊曼代数；子因子论；自由概率论和随机矩阵理论。拟议研究的主要工具来自于这四个主题的想法的综合。这包括自由概率论中的自由随机分析和平方可积上同调理论；von Neumann代数中的Popa形变-刚性理论；与具有势的随机多矩阵模型相关的随机矩阵的组合结构；以及由Jones开创的子因子论的平面代数方法。上面提到的四个领域在数学上都令人惊讶地丰富。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。

AbstractShlyakhtenkoThe aim of the proposal is to study connections between four areas of mathematics: von Neumann algebras; subfactor theory; free probability theory and random matrix theory. The main tools for the proposed research come from a synthesis of ideas from these four subjects. This involves free stochastic analysis and the theory of square-integrable cohomology in free probability theory; Popa's deformation-rigidity theory in von Neumann algebras; combinatorial structure of random matrices related to random multi-matrix models with a potential; and the planar algebra approach to subfactor theory, as pioneered by Jones. All of the four areas mentioned above 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.