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.

该项目将涉及列举组合和代数的几何拓扑和几何方法中的组合方法的研究。 这项工作的大部分将在量子组，量子拓扑不变的和3个manifolds的有限型不变性的背景下。 琼斯多项式区分解开的猜想是激发这项工作的问题的一个例子。 另一个例子是，猜想没有不变的瓦西利夫将结与反向区分开。这项研究涉及研究拓扑，枚举和量子力学之间的联系。 拓扑是对结和表面等物体如何与自身相连的研究。尽管它是一个世纪前最初牢固地确定为数学领域，但由于许多数学和物理学领域的贡献，在过去30年中，它在过去30年中取得了重大进展。 枚举是计算事物的艺术，例如，有多种方式与Dominos的正方形托盘瓷砖；它与材料的热特性和物理学中的其他问题有关。 量子力学是一个著名的主题，但鲜为人知的是它对数学产生了巨大影响。它部分重新定义了我们对几何和数学空间的概念。 这三个领域在1980年代看到了汇合处，最终是授予沃恩·琼斯，弗拉基米尔·德林菲尔德和爱德华·维滕的田野奖章。 希望扩大这些和其他数学家和物理学家的工作。