Topology, Geometry and Algebraic Combinatorics

拓扑、几何和代数组合

• 批准号: 9704125
• 负责人: Greg Kuperberg
• 金额: $ 6.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704125&HistoricalAwards=false
• 关键词: Topology; Geometry; Algebraic; Combinatorics

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.

该项目将涉及几何拓扑中的组合方法以及枚举组合学和代数中的几何方法的研究。 这项工作的大部分内容将在量子群、量子拓扑不变量和 3 流形的有限型不变量的背景下进行。 琼斯多项式区分未结的猜想是激发这项工作的问题的一个例子。 另一个例子是这样的猜想：没有瓦西里耶夫不变量能够区分结和反结。这项研究涉及拓扑、枚举和量子力学之间的联系的调查。 拓扑学是研究结和面等物体如何相互连接的学科。尽管它在一个世纪前就作为数学领域首次确立，但由于数学和物理学许多领域的贡献，它在过去 30 年中取得了重大进展。 枚举是计数事物的艺术，例如将多米诺骨牌铺在方形托盘上有多少种方式；它与材料的热性能和其他物理问题有关。 量子力学是一门著名的学科，但它对数学产生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，最终授予沃恩·琼斯 (Vaughan Jones)、弗拉基米尔·德林菲尔德 (Vladimir Drinfeld) 和爱德华·威滕 (Edward Witten) 菲尔兹奖。 希望能够扩展这些以及其他数学家和物理学家的工作。