Low-Dimensional Topology from a "Super" View Point

“超”视点的低维拓扑

• 批准号: 0706725
• 负责人: Nathan Geer
• 金额: $ 11.82万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706725&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; Super; View

The project has three components, all concerning interactions between low-dimensional topology and quantum algebra. The first main theme of the proposed research is multivariable link invariants. The Principal Investigator and Patureau-Mirand have shown that Lie superalgebras naturally give rise to multivariable link invariants. The project continues this work with an aim of further understanding these invariants and how they relate to other link invariants. This initial work suggests that these multivariable link invariants are related to the Volume Conjecture. The proposed research will also focus on the categorification of a quantum invariant arising from a particular Lie superalgebra. This could give a topological categorification of the Alexander polynomial, in the sense of Bar-Natan's topological interpretation of Khovanov homology. A final theme in the proposed research is the construction of a 3-manifold invariant associated to a Lie superalgebra.Knots appear in many places ranging from everyday items such as tied shoe laces to string theory and protein folding. In mathematics, knots and more generally links are fundamental basic objects in topology. Knots have been studied since the 19th century. More recently, with the quantum field theory point of view moving into low-dimensional topology, the late 20th century saw the emergence of a revolution in the theory of knots and links. In particular, the theory of quantum groups associated to Lie algebras has been widely and productively used in low-dimensional topology. Not as well developed is the theory of quantum groups associated to Lie superalgebras and its applications. The theory of Lie superalgebras has particular properties that do not arise in the theory of Lie algebras. The focus of this proposal is to use these unique properties to gain new information about well known problems and objects in low-dimensional topology.

该项目有三个组成部分，都是关于低维拓扑和量子代数之间的相互作用。 建议的研究的第一个主题是多变量链接不变量。 主要研究者和Patureau-Mirand证明了李超代数自然地产生多变量链接不变量。 该项目将继续这项工作，目的是进一步理解这些不变量以及它们与其他链接不变量的关系。 这项初步工作表明，这些多变量链接不变量与体积猜想。 拟议的研究还将集中在一个特定的李超代数产生的量子不变量的分类。 这可以给出一个拓扑分类的亚历山大多项式，在这个意义上的巴尔-纳坦的拓扑解释霍瓦诺夫同调。 拟议研究的最后一个主题是构建与李超代数相关的3-流形不变量。结出现在许多地方，从日常用品，如系鞋带到弦理论和蛋白质折叠。 在数学中，结和更一般的链接是拓扑学中的基本对象。 自世纪以来，人们一直在研究绳结。 最近，随着量子场论的观点转向低维拓扑学，世纪末出现了一场关于节点和链接的理论革命。 特别是，与李代数相关的量子群理论在低维拓扑中得到了广泛而富有成效的应用。 与李超代数相关的量子群理论及其应用还没有得到很好的发展。 李超代数理论具有李代数理论中没有的特殊性质。 这个建议的重点是使用这些独特的属性，以获得新的信息，众所周知的问题和对象在低维拓扑。