Invariants of Links and 3-manifolds

DMS-0204158 Thang Le Thang Le计划继续他的研究量子和有限型不变量的链接和3-流形。特别是，他想研究的问题所产生的体积猜想连接量子不变量的经典对象，如fundamentalgroups，torsions，和卷。其他问题涉及量子不变量的积分性质（广义）及其背后的拓扑结构，以及它们的应用。场与几何学、组合学、数论和物理学相互作用。纽结和三维流形理论是数学的一个古老的分支，在发现琼斯多项式及其与理论物理学（量子场论、高能物理学）的关系之后，又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。数学家们用“不变量”来分类纽结和三维流形。本研究计画研究纽结与三维流形的新不变量类及其与经典不变量类的关系。新的不变量在区分纽结和三维流形方面是非常强大的。

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.