Subfactors, planar algebras, knots and graphs

所谓的冯·诺伊曼代数起源于默里和冯·诺伊曼试图将量子力学的研究建立在坚实的数学基础上。这两个广义的概念，从线性代数，关于三维、四维或更高维度空间的对称性，到量子物理学中自然出现的无限维设置。当试图理解冯·诺伊曼代数及其基本结构时，人们自然会考虑“子因子”。这是一对最小的冯·诺伊曼代数，其中一个包含另一个。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。最早的一个是著名的琼斯多项式，它出现在结的数学研究中。这个多项式使我们对子因子和拓扑（形状的定性研究——与几何相反，几何是形状的定量研究）之间的联系有了更深的理解。最近，子因子通过看似抽象的代数领域“范畴论”与拓扑量子计算的主题联系在一起。量子计算机将能够有效地计算大量被普遍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。本课题的主要研究目标是处理子因子理论中的问题：具有平凡中心的冯·诺伊曼代数包涵的研究。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。平面代数有许多优点：它们允许在有限维空间中进行计算，利用子因子和拓扑之间的联系，并阐明子因子的潜在对称性。首席研究员计划扩展她之前在小指数子因子分类和构建方面的研究，并将该分类的想法应用于新的领域。这个项目是由以下大问题指导的，主要研究者不希望完全回答这些问题，但这会导致人们提出更小、更容易接近的问题：(1)哪些图表可以作为主要图表？(2)子因子的最高超传递性是多少？(3)具有整数索引的子因子是否有维度不是整数平方根的对象？(4)在较高的指数中，外来子因子是常见的还是罕见的？(5)图平面代数中是否嵌入了与图和子因子平面代数相关的对称原理？

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?