Subfactors, planar algebras, knots and graphs

所谓的冯诺依曼代数起源于默里和冯诺依曼试图把量子力学的研究建立在坚实的数学基础上。 从线性代数，关于三维、四维或更高维空间的对称性，到在量子物理学中自然产生的无限维环境的两个广义概念。 当试图理解冯诺依曼代数及其底层结构时，人们自然会考虑“子因子”。这些是成对的极小冯诺依曼代数，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。 其中最早的是著名的琼斯多项式，它出现在对纽结的数学研究中。 这个多项式使我们更深入地理解子因子和拓扑之间的联系（对形状的定性研究，而不是对形状的定量研究）。最近，子因子已经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起来。 量子计算机将能够有效地计算被广泛认为在传统计算机上难以解决的一大类问题，而拓扑学可能是如何稳定量子系统以使其成为计算机的难题的答案。 本项目的主要研究目标是子因子理论中的一些问题：研究具有平凡中心的冯诺依曼代数的包含问题。在这种方法中，问题的子因子被转化为问题的子因子，即，平面代数和主图的有限不变量。平面代数有许多优点：它们允许在有限维空间中计算，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以前对小指数子因子的分类和构建的研究，并将这种分类的想法应用于新的地方。 这个项目是由以下大问题，主要研究者不希望完全回答，但导致一个问更小，更平易近人的问题：（1）哪些图可以出现作为主图？ (2)一个子因子所能具有的最高超传递性是什么？ (3)具有整数索引的子因子可以有维数不是整数平方根的对象吗？ (4)在较高的指数下，外来子因子是常见还是罕见？(5)图与子因子平面代数之间是否存在嵌入其图平面代数的对称性原理？

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?