Free probability, von Neumann algebras, subfactors and random matrices

自由概率、冯诺依曼代数、子因子和随机矩阵

• 批准号: 0900776
• 负责人: Dimitri Shlyakhtenko
• 金额: $ 67.04万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900776&HistoricalAwards=false
• 关键词: Free; probability; von; Neumann; algebras

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.

[摘要]该提案的目的是研究四个数学领域之间的联系：冯·诺依曼代数；次级因素理论;自由概率理论和随机矩阵理论。拟议研究的主要工具来自这四个学科的综合思想。这涉及到自由随机分析和自由概率论中的平方可积上同理论；von Neumann代数中的Popa变形-刚性理论具有势的随机多矩阵模型的随机矩阵组合结构以及琼斯开创的用平面代数方法来研究子因子理论。上面提到的四个领域在数学上都非常丰富。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。