Invariants of Links and 3-manifolds

thang leang Le计划继续他对连杆和3流形的量子和有限类型不变量的研究。特别是，他想研究体积猜想引起的问题，体积猜想将量子不变量与经典对象（如基群、扭转和体积）联系起来。其他问题涉及量子不变量的完整性（广义上）和它们背后的拓扑结构，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。结和3流形理论是数学的一个古老分支，在琼斯多项式的发现及其与理论物理（量子场论、高能物理）的关系之后，数学家和物理学家对它重新产生了兴趣。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”对结和3流形进行分类。本课题研究结点和3流形的不变量的新类别及其与经典不变量的关系。新的不变量在区分结点和3流形方面非常有效。

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.