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.

该提案的目的是研究数学的四个领域之间的联系：冯诺依曼代数;子因子理论;自由概率论和随机矩阵理论。 拟议的研究的主要工具来自这四个主题的想法的综合。 这涉及到自由随机分析和理论的平方可积上同调的自由概率论;波帕的变形刚性理论在冯诺依曼代数;组合结构的随机矩阵相关的随机多矩阵模型的潜力;和平面代数方法的子因子理论，作为开创性的琼斯。 上面提到的四个领域在数学上都是惊人的丰富。 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和其他领域（包括DNA结构的研究）都有应用。 随机多矩阵模型的研究具有工程应用价值，如手机设计。 本研究的重点是这四个领域之间的相互作用，旨在开发可能影响所有相关领域的工具和技术。