Subfactors and Planar Algebras

该项目涉及子因子和平面代数的结构。琼斯的平面纠缠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使人们对子因子论背后的代数组合结构有了更深的理解。这些技巧将被用来研究与无限深子因子相关的平面代数。我们将研究平面代数的一般合成，并分析这种合成的障碍。Bisch和Jones在以前的工作中发现的平面代数的自由积在这里将起到关键作用。该项目试图更好地理解Temperley-Lieb平面代数合成中的“平面关系”概念。我们将探讨子因子论和平面代数理论在固体物理和量子拓扑计算中的潜在应用。我们将探讨平面代数和随机矩阵理论之间的联系。来自算子代数和非对易几何的思想长期以来在量子物理、统计力学以及最近弗里德曼的量子计算方法中扮演着重要的角色。琼斯的次因式理论建立在约翰·冯·诺伊曼于20世纪30年代的S提出的抽象数学对象的基础上，它在数学和物理的几个领域都有深刻的应用，包括纽结理论、表象理论、统计力学和保形场论。子因子是一种数学对象，它允许人们捕捉构成它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似乎是为描述固态物理中的现象量身定做的数学框架。该项目专注于研究这些新对象的结构及其在小规模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.