Topology, Geometry and Algebraic Combinatorics

本项目将涉及几何拓扑学中的组合方法和枚举组合学和代数中的几何方法的研究。大部分工作将在量子群、量子拓扑不变量和3流形的有限型不变量的背景下进行。琼斯多项式区分解结的猜想是激发这项工作的一个例子。另一个例子是没有瓦西里耶夫不变量来区分一个结和它的反面的猜想。这项研究涉及对拓扑学、枚举和量子力学之间联系的调查。拓扑学研究的是像结点和表面这样的物体是如何相互连接的；虽然它在一个世纪前就被确立为数学的一个领域，但由于数学和物理的许多领域的贡献，它在过去的30年里取得了重大进展。枚举是一种计数事物的艺术，比如用多米诺骨牌铺正方形托盘有多少种方法；它与材料的热性质和物理学中的其他问题有关。量子力学是一门著名的学科，但它对数学的巨大影响却鲜为人知；它在一定程度上重新定义了我们对几何和数学空间的概念。这三个领域在20世纪80年代融合在一起，在沃恩·琼斯、弗拉基米尔·德林菲尔德和爱德华·威滕获得菲尔兹奖时达到了顶峰。希望是扩展这些和其他数学家和物理学家的工作。

This project will involve the study of combinatorial methods in geometric topology and geometric methods in enumerative combinatorics and algebra. Most of this work will be in the context of quantum groups, quantum topological invariants, and finite-type invariants of 3-manifolds. The conjecture that the Jones polynomial distinguishes the unknot is one example of a problem that motivates this work. Another example is the conjecture that no Vassiliev invariant distinguishes a knot from its reverse. This research involves the investigation of connections between topology, enumeration, and quantum mechanics. Topology is the study of how objects such as knots and surfaces are connected to themselves; although it was first established firmly as a field of mathematics a century ago, it has seen major advances in the past 30 years due to contributions from many areas of mathematics and physics. Enumeration is the art of counting things, such as how many way that there are to tile a square tray with dominos; it is related to the thermal properties of materials and other questions in physics. Quantum mechanics is a famous subject, but it is less well known that it has had an enormous influence in mathematics; it has partly redefined our conception of geometry and mathematical spaces. These three areas saw a confluence in the 1980's, culminating in the Fields medals awarded to Vaughan Jones, Vladimir Drinfeld, and Edward Witten. The hope is to extend the work of these and other mathematicians and physicists.