The Geometry, Topology and Number Theory of the Jones Polynomial

琼斯多项式的几何、拓扑和数论

• 批准号: 1406419
• 负责人: Stavros Garoufalidis
• 金额: $ 37.71万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406419&HistoricalAwards=false
• 关键词: Geometry; Topology; Number; Theory; Jones

Knot theory is the study of 3-dimensional shapes obtained by removing loops from the 3-dimensional space. Microscopic examples of knotted loops appear abundantly in solid state physics, biology (notably, the DNA molecule) and various models of quantum computing. Macroscopic knotted loops occur in the shapes of galaxies, black holes and plasma physics. The proposed research focuses on the theoretical aspects of knot theory and more precisely on the relation between classical and quantum knot invariants. Quantum topology offers a classification of knots by various measures of complexity, for example by using the Jones polynomial of a knot (and its parallels). Understanding this filtering leads to deep connections between knot theory and hyperbolic geometry, number theory, complex analysis, algebraic geometry and mathematical physics. An important aspect of this research is the training of graduate students and the collaboration with senior researchers. The research involves difficult numerical experiments and leads to unexpected discoveries of conjectures and proofs. Aside from mathematics and mathematical physics, quantum knot theory is a powerful and effective tool with the potential to impact topological quantum computation and the study of the shape of long molecules.Classical knot invariants include the incompressible and normal surface invariants by W. Haken, the character varieties of algebraic geometry and mathematical physics and the hyperbolic geometry approach to 3-dimensional manifolds initiated by W. Thurston. The birth of quantum topology was the Jones polynomial in the mid eighties. The discovery of the Jones polynomial attracted mathematicians and physicists of the highest caliber, including the Fields Medalists V. Jones, E. Witten, V. Drinfeld and M. Kontsevich. There are precise, numerically testable conjectures that relate classical and quantum invariants. Progress on these conjectures (which include the Volume Conjecture of Kashaev, the Slope Conjecture and the AJ Conjecture) is the focus of the proposed research. A combination of numerical experimentation with proofs leads to deep connections between low dimensional topology and hyperbolic geometry, number theory, complex analysis, algebraic geometry and mathematical physics. This combination helps to train young researchers, and to make connections with senior researchers in diverse fields.

纽结理论是研究通过从三维空间中移除环而获得的三维形状。打结环的微观例子在固态物理学、生物学（特别是DNA分子）和各种量子计算模型中大量出现。宏观打结环出现在星系、黑洞和等离子体物理学的形状中。拟议的研究集中在结理论的理论方面，更确切地说，在经典和量子结不变量之间的关系。量子拓扑学通过各种复杂性度量提供了一种对结的分类，例如使用结（及其平行物）的琼斯多项式。理解这种过滤导致纽结理论和双曲几何，数论，复分析，代数几何和数学物理之间的深刻联系。这项研究的一个重要方面是研究生的培养和与高级研究人员的合作。这项研究涉及困难的数值实验，并导致意想不到的发现和证明。除了数学和数学物理之外，量子纽结理论是一个强大而有效的工具，有可能影响拓扑量子计算和长分子形状的研究。Haken，代数几何和数学物理的特征簇以及由W。瑟斯顿量子拓扑学的诞生是80年代中期的琼斯多项式。琼斯多项式的发现吸引了最高水平的数学家和物理学家，包括菲尔兹奖获得者V。维滕、V.德林菲尔德和M.孔采维奇有精确的，数值上可检验的关系，经典和量子不变量。这些猜想（包括Kashaev体积猜想、斜率猜想和AJ猜想）的进展是本文研究的重点。数值实验与证明的结合导致低维拓扑与双曲几何、数论、复分析、代数几何和数学物理之间的深刻联系。这种结合有助于培养年轻的研究人员，并与不同领域的高级研究人员建立联系。