Collaborative Research: FRG: Hyperbolic Geometry and Jones Polynomials

合作研究：FRG：双曲几何和琼斯多项式

• 批准号: 0456275
• 负责人: Oliver Dasbach
• 金额: --
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456275&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Hyperbolic; Geometry

The works of Thurston and Jones revolutionized low dimensionaltopology. Thurston established the ubiquity of hyperbolic structure inlow dimensions. Jones' work, via such physical notions as quantumgroups and path integrals, led to vast families of topologicalinvariants associated with diagrammatic descriptions of topologicalobjects. The investigators have striking experimental evidence for adirect link between these disparate approaches. Establishing such alink is of fundamental importance to low dimensional topology. Thisproposal aims to establish this link, with particular focus on thegeneralized volume conjecture, which relates the most importantgeometric and quantum invariants: the hyperbolic volume andChern-Simons invariant, and the colored Jones polynomials of aknot. The theory of L2-invariants provides a combinatorial frameworkto study hyperbolic volume. Deforming this construction along thecurve of representations given by the A-polynomial involves thetwisted Alexander polynomial and the colored Jones polynomials. Treeentropy of graphs provides the bridge between L2-torsion and coloredJones polynomials.The volume conjecture relates classical geometric invariants ofthree-dimensional spaces with topological invariants motivated byideas from quantum physics. This conjecture originated in the theoryof quantum gravity, which cannot yet be verified experimentally. Themathematically rigorous verification sought by this project of thisand related conjectures will support the internal consistency ofquantum gravity. Unifying quantum and geometric invariants is also ofintrinsic mathematical importance, which will yield important newinsights in other fields. Computer programs to study geometricinvariants and tabulation of knots and their invariants areessential tools for this research. Undergraduate and graduate studentsinvolved in this project will be exposed to sophisticated mathematicsand computer tools.

瑟斯顿和琼斯的工作彻底改变了低维拓扑学。瑟斯顿确立了双曲结构在低维中的普遍存在。琼斯的工作通过量子群和路径积分等物理概念，导致了与拓扑学对象的图解描述相关的拓扑不变量的大家族。研究人员有明显的实验证据证明这些不同的方法之间存在直接联系。建立这样的链接对于低维拓扑是非常重要的。这个提议旨在建立这种联系，特别关注广义体积猜想，它联系着最重要的几何和量子不变量：双曲体积和Chern-Simons不变量，以及纽结的有色琼斯多项式。L2不变量理论为研究双曲体提供了一个组合框架。沿着A多项式所给出的表示曲线变形这一结构涉及到扭曲的Alexander多项式和有色的Jones多项式。图的树熵在L2-挠度和有色Jones多项式之间架起了桥梁，体积猜想将三维空间的经典几何不变量与量子物理中的拓扑不变量联系起来。这一猜想起源于量子引力理论，目前还无法得到实验验证。这个项目和相关猜想所寻求的严格的数学验证将支持量子引力的内部一致性。统一量子和几何不变量也具有内在的数学重要性，这将在其他领域产生重要的新观点。研究纽结及其不变量的几何不变量和列表的计算机程序是这项研究的基本工具。参与该项目的本科生和研究生将接触到复杂的数学和计算机工具。