Subfactors, bimodules and planar algebras

这个关于子因子和平面代数的项目旨在开发广泛的技术来更好地理解子因子的结构并发现新的例子。其中一个主要目标是了解在其相关的平面代数中组合子因子的技术。这将导致构造子因子和平面代数的新方法，特别是具有无限主图的新例子，这对结构理论的进展至关重要。此外，每个这样的子因子将提供一个潜在的非常有趣的双模融合类别，它可以显式计算。作为第一步，这些组合将被研究作为两个Temperley-Lieb平面代数张量积的平面子代数。对于更一般的组合，必须发展子因子和平面代数的上同调理论，以取代群上同调。从这些调查中很可能产生一个非常丰富的理论。特别是对于琼斯指数为黄金比例平方两倍的子因子，预计将得到新的分类结果。子因子可以看作是捕获数学或量子物理系统的量子对称性的数学对象。这些对称性在理解这些复杂系统的行为中起着关键作用，子因子理论为操纵和研究它们提供了有效的工具。这一理论在数学和物理的许多领域，如共形场论、统计力学、低维拓扑学和组合学中有许多深刻而令人惊讶的应用。子因子在理解自然发生的结构方面做出了重要贡献，这些自然发生的结构是先天的，在数学和物理的这些截然不同的领域。预计该项目将对其中一些基础科学领域作出重要贡献。平面代数及其相关的融合类别在固态物理和拓扑量子计算中的激动人心的应用是可能的。此外，该项目将涉及研究生和博士后研究人员，并有助于他们作为数学研究人员的培训。

This project on subfactors and planar algebras aims at developing a wide range of techniques to better understand the structure of subfactors and discover new examples. One of the main goals is to understand the technique of composing subfactors in terms of their associated planar algebras. This will lead to new methods of constructing subfactors and planar algebras, in particular new examples with infinite principal graphs which are essential for progress in the structure theory. Moreover, every such subfactor will provide a potentially very interesting fusion category of bimodules, which can be computed explicitly. As a first step, those compositions will be studied that arise as planar subalgebras of the tensor product of two Temperley-Lieb planar algebras. For more general compositions, a cohomology theory for subfactors and planar algebras has to be developed which will replace group cohomology. A very rich theory is likely to emerge from these investigations. In particular, it is expected that new classification results for subfactors with Jones index two times the golden ratio squared will be obtained.A subfactor can be viewed as a mathematical object which captures quantum symmetries of a mathematical or quantum physical system. These symmetries play a key role in understanding the behavior of these complex systems, and the theory of subfactors provides effective tools to manipulate and study them. This theory has had many profound and surprising applications to numerous areas of mathematics and physics, such as conformal field theory, statistical mechanics, low dimensional topology and combinatorics. Subfactors have contributed in an important way to the understanding of naturally occurring structures in these a priori quite distinct areas of mathematics and physics. It is expected that the project will make important contributions to some of these areas of basic science. Exciting applications of planar algebras and their associated fusion categories to solid state physics and to topological quantum computing are possible. Furthermore, the project will involve graduate students and postdoctoral researchers and contribute to their training as researchers in mathematics.