Spring School in Geometry and Quantum Topology

几何与量子拓扑春季学校

• 批准号: 1106739
• 负责人: Stavros Garoufalidis
• 金额: $ 2.5万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-03-01 至 2012-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106739&HistoricalAwards=false
• 关键词: Spring; Geometry; Quantum; Topology

Modern Knot Theory started with the seminal discovery of the Jones polynomial by V. Jones in the mid eighties. Shortly after Jones's discovery, a plethora of polynomial invariants were constructed using the theory of Quantum Groups, and the later developed TQFT. Among the giants in the field are Fields Medalists V. Drienfeld, M. Kontsevich, V. Jones and E. Witten. Although the Jones polynomial is a combinatorially defined object, certain limits of it contain important information about the algebraic topology and the differential geometry of the knot complement.Three such limits are the focus of the school: namely the Volume Conjecture, the AJ Conjecture and Hyperbolic Geometry. In addition, the asymptotics of the Jones polynomial is an important example of a solvable topological quantum field theory, which illustrates some of the latest ideas in string theory and M-theory. The geometric limits also contain nontrivial new arithmetic of 3-dimensional objects, which is also one of the themes of the school.

现代纽结理论始于80年代中期V.Jones对Jones多项式的开创性发现。在琼斯的发现之后不久，大量的多项式不变量使用量子群理论和后来发展的TQFT来构造。该领域的巨人包括菲尔兹奖获得者V。Kontsevich，V. Jones和E.维滕。虽然琼斯多项式是一个组合定义的对象，但它的某些极限包含了纽结补的代数拓扑和微分几何的重要信息，其中三个极限是该学派的焦点：体积猜想，AJ猜想和双曲几何。此外，琼斯多项式的渐近性是可解拓扑量子场论的一个重要例子，它说明了弦理论和M理论中的一些最新思想。几何极限还包含了三维物体的非平凡的新算术，这也是该学派的主题之一。