体积猜想中的若干问题

Knots and links are very visual and rich objects, and hyperbolic 3-manifolds are very essential in studying low dimensional topology, and other areas of mathematical physics. Knots and hyperbolic structures are encountered in daily life from tying a knot to simple yard work and in the mathematical physics from Einstein's relativity to Witten's topological quantum field theory. Volume conjecture as one of fundamental problems is essentially unreachable in this challenging field full of rich interactions with many other fields of mathematics and beyond. This proposal attempts to unveil the geometric and topological properties around the volume conjecture. The novelty of the proposal is that infinite dimensional analysis and algebraic geometry techniques give new insights in knot theory and low dimensional topology theory. The problems addressed in this proposal are in the area of knot theory, 3-manifold and its geometric and topological ramifications. The proposed problems exhibit profound interactions among algebraic geometry, K-theory, complexified Chern-Simons theory, the L^2-geometry of the universal covering of knot complements, and mathematical physics. This research project is aimed at understanding fundamental questions in knot theory, building a rigorous mathematics for the volume conjecture, and finding a deeper relation between knot theory and algebraic geometry as well as mathematical physics. The proposed project will integrate current research with teaching at all levels from the calculations of matrix representations and programming the knot invariants (for undergraduate research experiences in representations of knot groups) to compute algebraic geometry quantities for knots (for post-doctoral and junior researchers) and to explore the L^2-invariants with other representations (for researchers). It is also expected that the techniques developed in this project will have an impact on the related mathematical subjects and mathematical physics.

扭结和扭结环与双曲3维流形在低维拓扑的研究中是非常直观丰富的，而且与其他数学领域以及数学物理有着非常紧密的联系。这些在一百多年前用于研究化学分子结构和爱因斯坦相对论。体积猜想是一个深刻刻划拓扑与几何之间相互关联的猜想，来源于数学物理的量子场论，牵扯到几何、分析、拓扑、代数几何以及低维拓扑等领域。本项目课题包含建立L^2-Jones不变量和L^2-同调群内的相交理论，以及深入理解复化Chern-Simons理论与代数几何之间关系，刻划高维的K群与Deligne上同调群与双曲扭结之间的内在结构。这些课题都是当前国内外低维拓扑研究中所关心的热门课题。项目组全体成员将努力刻苦地在项目课题的各个方向上进行钻研与探索，希望在项目支持期间得以在理论和方法上取得创新性成果以推进学科的深入研究和发展。

扭结和扭结环与双曲3维流形在低维拓扑的研究中是非常直观丰富的，而且与其他数学领域以及数学物理有着非常紧密的联系。这些在一百多年前用于研究化学分子结构和爱因斯坦相对论。体积猜想是一个深刻刻划拓扑与几何之间相互关联的猜想，来源于数学物理的量子场论，牵扯到几何、分析、拓扑、代数几何以及低维拓扑等领域。本项目课题包含建立L^2-Jones不变量和L^2-同调群内的相交理论，以及深入理解复化Chern-Simons理论与代数几何之间关系，刻划高维的K群与Deligne上同调群与双曲扭结之间的内在结构。这些课题都是当前国内外低维拓扑研究中所关心的热门课题。项目组全体成员将努力刻苦地在项目课题的各个方向上进行钻研与探索，在项目支持期间定义了L2 形式下的Jones不变量。一方面给出了同调表示的无穷维空间的L²-形式推广，同时该同调表示的方法也可以描述Burau表示，从而为建立L²-形式的twisted Alexander不变量和L2形式下的Jones不变量的关联提供了一个统一平台。这对低维拓扑的研究提供了一个新的研究领域。为解决体积猜想探索一条新的路径。 这方面的研究也会促进L2不变量领域的深入。