Subfactors and Planar Algebras

该项目研究子因子和平面代数的结构，平面缠结的琼斯运算自然地作用于子因子的标准不变量。由此产生的平面代数技术，导致了更深入的理解的代数组合结构的理论的子因子。这些技巧将用于研究与无限深度子因子相关的平面代数，研究平面代数的一般合成，并分析这种合成的障碍。自由产品的平面代数所发现的Bisch和琼斯在以前的工作将发挥关键作用在这里。该项目旨在更好地理解“平面关系”的概念，在Temperley-Lieb平面代数的组成的背景下。子因子和平面代数理论在固态物理和拓扑量子计算中的潜在应用将被研究。平面代数和随机矩阵理论之间的联系将被探讨。从算子代数和非交换几何的想法已经发挥了很长一段时间的重要作用，在量子物理学，统计力学和最近，在弗里德曼的方法来量子计算。琼斯的子因子理论是基于约翰·冯·诺依曼在20世纪30年代提出的抽象数学对象，它在数学和物理学的几个领域都有着深远的应用，包括纽结理论、表示论、统计力学和共形场论。子因子是一个数学对象，它允许人们捕捉它所构造的数学或物理情况的非常一般的对称性。平面代数提供了一个数学框架，似乎是为描述固态物理现象而量身定制的。该项目侧重于研究这些新对象的结构及其在小尺度物理问题中的潜在应用。

The project deals with the structure of subfactors and planar algebras.Jones' operad of planar tangles acts naturally on the standard invariant of a subfactor. The resulting planar algebra techniques have led to a deeper understanding of the algebraic-combinatorial structures underlying the theory of subfactors. These techniques will be used to investigate planar algebras associated to infinited depth subfactors.General compositions of planar algebras will be studied, and obstructions for such compositions will be analyzed. The free product of planar algebras discovered by Bisch and Jones in prior work will play a key role here. The project seeks a better understanding of the notion of "planar relations" in the context of composition of Temperley-Lieb planar algebras. Potential applications of the theory of subfactors and planar algebras to solid state physics and topological quantum computation will be investigated. Connections between planar algebras and random matrix theory will be explored.Ideas from operator algebras and noncommutative geometry have played for a long time an important role in quantum physics, statistical mechanics and more recently, in Freedman's approach to quantum computing. Jones' theory of subfactors, which is based on abstract mathematical objects introduced by John von Neumann in the 1930's, has had profound applications to several areas of mathematics and physics, including knot theory, representation theory, statistical mechanics and conformal field theory. A subfactor is a mathematical object which allows one to capture very general symmetries of a mathematical or physical situation from which it was constructed. Planar algebras provide a mathematical framework which seems tailor-made to describe phenomena in solid state physics.The project focuses on investigating the structure of these new objects and their potential applications to problems in small scale physics.