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

免费概率技术：冯诺依曼代数、随机矩阵和子因子。

• 批准号: 1161411
• 负责人: Dimitri Shlyakhtenko
• 金额: $ 34万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161411&HistoricalAwards=false
• 关键词: Free; probability; techniques; von; Neumann

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.

该提案的目的是使用自由概率理论的工具来研究数学的三个领域：随机矩阵理论，亚比例理论和von Neumann代数理论。这些工具涉及Voiculescu早期在其自由信息理论的背景下引入的一种差分演算。从该“自由演算”中研究差分运算符的性质将使研究者能够制定PDE和SDE技术的非交换类似物。反过来，这些将在von Neumann代数，随机基质和子因子理论中提供应用。四个领域 - 随机矩阵理论，亚比例理论和von Neumann代数的理论 - 数学上都非常丰富。例如，琼斯的子因子理论导致发现了一种新颖的结，该结构不变了，该结中已经用于数学及其他地区（包括DNA结构的研究）。随机多矩阵模型的研究具有工程应用，例如手机设计。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。