Analytical and Combinatorial Aspects of Subfactors

摘要bischbisch将继续子因子理论的研究。具有有限琼斯指数的子因子具有令人惊讶的丰富的数学结构，并且分析，代数组合和拓扑技术的相互作用是该理论的内在特征。平面代数方法将用于研究子因子及其标准不变量的结构。在开创性的工作中，Bisch和Jones使用这些技术证明了子因子的一般结构定理，从而导致了小维单生成平面代数的分类。计划通过研究广义量子Yang-Baxter方程将这种分类进一步推进。新的代数塔有望在这一分析中出现。研究两个平面代数的自由积的概念，并研究平面代数上的各种自然梯度代数结构。该项目的其他方向包括子因子的刚性和Popa系统的子系统的研究。算子代数理论是子因子理论的一个分支，由冯·诺伊曼提出，为量子力学提供了一个充分的数学框架。例如，海森堡不确定性关系是冯·诺伊曼发明的抽象数学对象的性质的自然结果。在过去的几年里，越来越清楚的是，经典的数学结构，如群，并不总是足以捕捉在数学或物理情况下存在的所有相关对称性，特别是在量子场论的背景下。子因子可以被看作是一个数学对象，人们可以用它来捕捉正在研究的问题的量子对称性。然后，可以应用子因子技术以代数组合术语明确地描述这些对称性。例如，某些图形作为该数据的一部分出现。子因子理论在数学和理论物理的许多领域的惊人应用，例如结理论和共形场论，都是以这种方式被发现的。因此，子因子以一种重要的方式对理解这些先验的、截然不同的数学和物理领域中自然发生的结构做出了贡献。

AbstractBischBisch will continue his investigations in subfactor theory. Subfactors with finite Jones index have an amazingly rich mathematical structure and an interplay of analytical, algebraic-combinatorial and topological techniques is intrinsic to the theory. Planar algebra methods will be used to investigate the structure of subfactors and their standard invariants. In pioneering work, Bisch and Jones have used these techniques to prove a general structure theorem for subfactors that leads to a classification of singly generated planar algebras with small dimension. It is planned to take this classification one step further by studying a generalized quantum Yang-Baxter equation. New towers of algebras are expected to appear in this analysis. The notion of free product of two planar algebras will be investigated and various natural graded algebra structures on a planar algebra will be examined. Other directions of the project include work on rigidity of subfactors and the investigation of sub-systems of Popa systems.The theory of operator algebras, of which subfactor theory is a branch, has been introduced by John von Neumann to provide an adequate mathematical framework for quantum mechanics. For instance, the Heisenberg uncertainty relation appears as a natural consequence of the properties of the abstract mathematical objects that von Neumann invented. It has become increasingly clear in the last few years that classical mathematical structures, such as groups, are not always sufficient to capture all the relevant symmetries present in a mathematical or physical situation, especially in the context of quantum field theory. A subfactor can be viewed as a mathematical object with which one can capture what one might call the quantum symmetries of the problem under investigation. Subfactor techniques can then be applied to describe these symmetries explicitly in algebraic-combinatorial terms. For instance, certain graphs appear as part of this data. Many surprising applications of the theory of subfactors to several areas of mathematics and theoretical physics, for instance knot theory and conformal field theory, have been found in this way. Subfactors have thus contributed in an important way to the understanding of naturally occurring structures in these a priori quite distinct areas of mathematics and physics.