Subfactors, planar algebras, knots and graphs

子因子、平面代数、结和图

• 批准号: 1501116
• 负责人: Emily Peters
• 金额: $ 15.59万
• 依托单位: Loyola University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501116&HistoricalAwards=false
• 关键词: Subfactors; planar; algebras; knots; graphs

So-called von Neumann algebras have their origins in the attempt by Murray and von Neumann to put the study of quantum mechanics on firm mathematical footing. The two generalized ideas from linear algebra, about symmetries of three-, four-, or higher-dimensional space, to an infinite-dimensional setting that arises naturally in quantum physics. When trying to understand von Neumann algebras and their underlying structure, one is led naturally to consider "subfactors." These are pairs of minimal von Neumann algebras, one of which contains the other. The study of subfactors has resulted in some surprising connections with other areas of mathematics. One of the earliest is the famous Jones polynomial, which arrises in the mathematical study of knots. This polynomial has led to a deeper understand of the connections between subfactors and topology (the qualitative study of shape--as opposed to geometry, which is the quantitative study of shape). More recently, subfactors have been connected, through the seemingly abstract algebraic field of "category theory," to the topic of topological quantum computing. Quantum computers would be capable of efficiently computing a large class of problems that are widely believed to be intractable on traditional computers, and topology may be the answer to the difficult question of how to stabilize quantum systems enough to make computers from them. The principal research goals of this project deal with questions in subfactor theory: the study of inclusions of von Neumann algebras with trivial center. In this approach, questions about subfactors are translated into questions about finite invariants of subfactors, namely, planar algebras and principal graphs. Planar algebras have many advantages: they allow one to calculate in finite-dimensional spaces, exploit the connection between subfactors and topology, and illuminate the underlying symmetry of the subfactors. The principal investigator plans to extend her previous research on classification and construction of small index subfactors and also to apply ideas from this classification in new places. This project is guided by the following big questions that the principal investigator does not expect to answer completely but that lead one to ask smaller and more approachable questions: (1) Which graphs can occur as principal graphs? (2) What is the highest supertransitivity a subfactor can have? (3) Can subfactors with integer index have objects whose dimensions are not square roots of integers? (4) Are exotic subfactors common, or rare, at higher indices? (5) Are there symmetry principals relating graphs and subfactor planar algebras embedded in their graph planar algebras?

所谓的冯·诺依曼代数起源于穆雷和冯·诺依曼试图将量子力学的研究建立在坚实的数学基础上的尝试。 从线性代数（关于三维、四维或更高维空间的对称性）到量子物理学中自然出现的无限维设置的两个广义概念。 当试图理解冯·诺依曼代数及其基本结构时，人们自然会考虑“子因子”。这些是最小冯诺依曼代数对，其中一个包含另一个。 对子因子的研究与数学的其他领域产生了一些令人惊讶的联系。 最早的多项式之一是著名的琼斯多项式，它出现在纽结的数学研究中。 这个多项式使人们对子因素和拓扑之间的联系有了更深入的了解（形状的定性研究——与几何形状相反，几何是形状的定量研究）。最近，子因素已经通过“范畴论”看似抽象的代数领域与拓扑量子计算的主题联系起来。 量子计算机将能够有效地计算一大类被广泛认为在传统计算机上难以解决的问题，而拓扑学可能是如何稳定量子系统以制造计算机这一难题的答案。 该项目的主要研究目标涉及子因子理论中的问题：具有平凡中心的冯·诺依曼代数包含的研究。在这种方法中，关于子因子的问题被转化为关于子因子的有限不变量的问题，即平面代数和主图。平面代数有很多优点：它们允许人们在有限维空间中进行计算，利用子因子和拓扑之间的联系，并阐明子因子的基本对称性。首席研究员计划扩展她之前对小指数子因素的分类和构建的研究，并将这种分类的想法应用到新的地方。 该项目以以下大问题为指导，主要研究者并不期望完全回答这些问题，但会导致提出更小且更容易理解的问题：（1）哪些图可以作为主图出现？ (2) 子因子可以具有的最高超及物性是多少？ (3) 具有整数索引的子因子是否可以具有维度不是整数平方根的对象？ (4) 在较高指数下，外来子因素是常见还是罕见？ (5) 是否存在与图和嵌入图平面代数中的子因子平面代数相关的对称原理？