Subfactors and Planar Algebras

该项目涉及子因子和平面代数的结构。琼斯的平面缠结算子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对子因子理论基础上的代数组合结构的更深层次的理解。这些技术将用于研究与无限深度子因子相关的平面代数。将研究平面代数的一般组成，并分析这种组成的障碍。由Bisch和Jones在之前的工作中发现的平面代数的自由积将在这里发挥关键作用。该项目旨在更好地理解“平面关系”的概念在构成的背景下的坦波利-利布平面代数。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。将探讨平面代数与随机矩阵理论之间的联系。长期以来，算子代数和非交换几何的思想在量子物理学、统计力学和最近的弗里德曼量子计算方法中发挥了重要作用。琼斯的子因子理论是基于约翰·冯·诺伊曼在20世纪30年代提出的抽象数学对象，在数学和物理的几个领域有深远的应用，包括结理论、表示理论、统计力学和共形场论。子因子是一种数学对象，它允许人们捕捉构造它的数学或物理情况的非常普遍的对称性。平面代数提供了一个数学框架，它似乎是为描述固态物理现象而量身定做的。该项目的重点是研究这些新物体的结构及其在小规模物理问题上的潜在应用。

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.