Invariants of Links and 3-Manifolds, Their Properties and Topology

题目：“结点和3-流形的不变量，它们的性质和拓扑”摘要：本文研究了3-流形的量子不变量和有限型不变量及其与经典拓扑不变量和几何不变量的关系。研究者将继续研究他与Murakami和T. Ohtsuki共同开发的同调三球的普遍有限型不变量。特别是，他计划进一步发展这一理论，以包括其他3流形，并研究与Le-Murakmi-Ohtsuki不变量相关的拓扑量子场论，这与通常的量子不变量相关的理论不同。他将尝试理解这些新不变量的拓扑结构，并找到它们的应用。其中一个项目是研究双曲的，或者更一般的，简单的，结点体积和它们的量子不变量之间的关系（证明kashaev - murakami - murakami猜想）。结和3流形理论是一个古老的数学分支，在发现琼斯多项式及其与理论物理（量子场论、高能物理）的关系之后，它重新引起了数学家和物理学家的兴趣。事实上，它现在是数学中最活跃的领域之一。结理论的许多结果也可以在分子生物学中找到应用。数学家使用“不变量”来对结和3流形进行分类。本课题研究了结点和3流形的一类新的不变量及其与经典不变量的关系。新不变量在区分结点和3流形方面非常有效。

Proposal: DMS-9971350PI: Thang LeTitle: "Invariants of knots and 3-manifolds, their properties and topology"Abstract: This research studies quantum and finite type invariants of3-manifolds and their relationships with classical topological andgeometrical invariants. The investigator will continue to studythe universal finite type invariant of homology 3-spheres which hedeveloped in joint work with J. Murakami and T. Ohtsuki. Inparticular, he plans to develop the theory further to includeother 3-manifolds and to study the topological quantum fieldtheory associated with the Le-Murakmi-Ohtsuki invariant which isdifferent from those associated with usual quantum invariants. Hewill try to understand the topology of these new invariants andfind their applications. One of the projects is to study relationsbetween the hyperbolic, or more general, simplicial, volume ofknots and their quantum invariants (to prove theKashaev-Murakami-Murakami conjecture).The theory of knots and 3-manifolds is an old branch ofmathematics which has gained renewed interest among mathematiciansand physicists after the discovery of the Jones polynomial and itsrelation to theoretical physics (quantum field theory, high energyphysics). In fact, it is now one of the most active domains inmathematics. Many results of knot theory may also findapplications in molecular biology. To classify knots and3-manifolds, mathematicians use "invariants". This researchproject studies new classes of invariants of knots and 3-manifoldsand their relationships with the classical ones. The newinvariants are very powerful in distinguishing knots and3-manifolds.