The Geometry, Topology and Number Theory of the Jones Polynomial

纽结理论是通过从三维空间中去除环而获得的三维形状的研究。在固态物理、生物学(特别是DNA分子)和量子计算的各种模型中，大量出现了打结环的微观例子。宏观上的打结环以星系、黑洞和等离子体物理的形状出现。所提出的研究集中在纽结理论的理论方面，更准确地说，是关于经典纽结不变量和量子纽结不变量之间的关系。量子拓扑学提供了一种按各种复杂程度对纽结进行分类的方法，例如，使用纽结的琼斯多项式(及其平行关系)。理解这种过滤导致纽结理论与双曲几何、数论、复分析、代数几何和数学物理之间的深刻联系。研究生的培养和与资深科研人员的合作是本研究的一个重要方面。这项研究涉及困难的数值实验，并导致猜想和证明的意想不到的发现。除了数学和数学物理之外，量子纽结理论是一种强大而有效的工具，有可能影响拓扑量子计算和长分子形状的研究。经典的纽结不变量包括W.Haken的不可压缩曲面不变量和法向曲面不变量，代数几何和数学物理的特征变量，以及W.瑟斯顿开创的三维流形的双曲几何方法。量子拓扑学的诞生是在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，包括菲尔兹奖牌获得者V·琼斯、E·威滕、V·德恩菲尔德和M·康采维奇。有一些精确的、可用数值测试的猜想将经典不变量和量子不变量联系起来。这些猜想(包括Kashaev的体积猜想、Slope猜想和AJ猜想)的进展是拟议研究的重点。数值实验与证明的结合导致了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，并与不同领域的资深研究人员建立联系。

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.