Invariants of Links and 3-manifolds

DMS-0204158 Tang LeTang Le计划继续他对环和3-流形的量子和有限型不变量的研究。特别是，他想研究围绕体积猜想产生的问题，体积猜想将量子不变量与基本群、挠率和体积等经典对象联系起来。其他问题涉及量子不变量的积分性质(广义)及其背后的拓扑学，以及它们的应用。该领域与几何学、组合学、数论和物理学相互作用。纽结和三维流形理论是一个古老的数学分支，自从琼斯多项式及其与理论物理(量子场论、高能物理)的关系被发现后，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。纽结理论的许多结果也可能在分子生物学中得到应用。为了对结和三维流形进行分类，数学家使用“不变量”。本研究项目研究纽结和3-流形的新的不变量及其与经典不变量的关系。新的不变量在区分纽结和3-流形方面非常强大。

DMS-0204158Thang LeThang Le plans to continue his study of quantum and finite typeinvariants of links and 3-manifolds. In particular, he would liketo study problems arising around the volume conjecture whichconnects quantum invariants to classical objects like fundamentalgroups, torsions, and volumes. Other problems involve integralityproperties (in broad sense) of quantum invariants and thetopology behind them, and their applications. The field hasinteractions with geometry, combinatorics, number theory, andphysics.The theory of knots and 3-manifolds is an old branch ofmathematics which has gained renewed interest amongmathematicians and physicists after the discovery of the Jonespolynomial and its relation to theoretical physics (quantum fieldtheory, high energy physics). In fact, it is now one of the mostactive domains in mathematics. Many results of knot theory mayalso find applications in molecular biology. To classify knotsand 3-manifolds, mathematicians use "invariants". This researchproject studies new classes of invariants of knots and3-manifolds and their relationships with the classical ones. Thenew invariants are very powerful in distinguishing knots and3-manifolds.