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: 0844485
• 负责人: Abhijit Champanerkar
• 金额: --
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0844485&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.

瑟斯顿和琼斯的作品彻底改变了低维度学。 Thurston确立了双曲结构嵌入尺寸的无处不在。琼斯通过诸如量子组和路径积分等物理概念的工作导致了与拓扑视图的图解描述相关的巨大拓扑成像家族。研究人员在这些不同的方法之间有了偏见的联系的惊人实验证据。建立这样的Alink对于低维拓扑至关重要。此Proposal旨在建立此联系，特别关注将总体构想的猜想，它与最重要的地面和量子不变性相关联：双曲线体积和Chern-Simons不变性，以及Aknot的彩色Jones多项式。 L2不变的理论为研究双曲线量提供了组合框架。 A-Polynomial给出的表示形式沿着该构建的形式变形，涉及亚历山大多项式和有色琼斯的多项式。图形的树木概念提供了L2-Torsion和ColoredJones多项式之间的桥梁。体积的猜想将三维空间的经典几何不变性与拓扑不变的空间与量子物理学动机的拓扑不变性。该猜想起源于量子重力理论，该理论尚无法通过实验验证。 该项目对此进行了严格的严格验证和相关的猜想将支持Quantum Gravity的内部一致性。统一的量子和几何不变也是信息的数学重要性，它将在其他领域产生重要的新吸引力。用于研究几何变异和结的计算机程序及其不变式的制表本质上是本研究的基本工具。该项目的本科生和研究生将接触精致的数学和计算机工具。