Geometric structures and invariants of links and 3-manifolds

链接和 3 流形的几何结构和不变量

• 批准号: 1404754
• 负责人: Efstratia Kalfagianni
• 金额: $ 22.44万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404754&HistoricalAwards=false
• 关键词: Geometric; structures; invariants; links; manifolds; Geometric; structures; invariants; links; manifolds

The research of this project falls in the area of 3-dimensional topology. The central objects of study in this area are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of low dimensional topology. The main theme of this project is to establish such connections and explore their ramifications and applications to other areas of mathematics.This project will establish relationships between geometry and combinatorial descriptions, properties, and quantum invariants of links and 3-manifolds. The PI has developed a setting for establishing new unexpected relations between the colored Jones link polynomials, the topology and geometry of essential surfaces in link complements, and hyperbolic geometry. One part of the project will continue developing this theory and exploring its applications. Another part, will combine several new techniques, to develop methods for recognizing geometric structures on 3-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data. A third part will study skein link theory in 3-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds. A fourth part will explore the applicability of quantum knot invariants to classical questions in knot theory and search for a classification of crossing changes that do not alter the topology of the underlying knots. The project also involves the research of graduate students currently working with PI.

该项目的研究属于三维拓扑领域。在该领域的研究中心对象是称为3个manifolds的空间。 3个manifold是一个本地看起来像普通的3维空间，但其整体结构可能很复杂的对象。三维拓扑结构的重要组成部分是对结的研究（嵌入了3个manifolds中的某种纠结的循环）及其分类。 Thurston的几何化猜想的解决方案已经确定，3个manifolds（以及其中的结的补充）分解为散布明确的几何形状的碎片，并且双曲线几何形状是更频繁的。但是，实际上，通常根据组合拓扑描述给出3个manifolds，寻求从这些描述中推断几何信息的方法是自然而重要的。拓扑师一直在研究3歧管的研究之一是通过使用不变性。在过去的几十年中，源于物理学的想法导致数学家发现了各种不变的结和3个manifolds。了解拓扑和组合数量以及与几何形状的连接是低维拓扑的核心和重要目标。该项目的主题是建立这种联系并探索其对数学其他领域的后果和应用。该项目将在几何形状与组合描述，属性和链接和3个manifolds的量子不变性之间建立关系。 PI开发了一个设置，以建立彩色琼斯链接多项式之间的新意外关系，链接补充中必需表面的拓扑和几何形状以及双曲线几何形状。该项目的一部分将继续发展该理论并探索其应用。另一部分将结合几种新技术，以开发从纯粹组合输入的3个manifolds上识别几何结构的方法，并从拓扑数据中得出对几何数量的估计。第三部分将研究3个manifolds，其不变性的链条链接理论，以及与3个manifolds几何分解的相互作用。 第四部分将探索量子结不变性在结理论中的经典问题上的适用性，并寻找不会改变基础结拓扑的交叉变化的分类。该项目还涉及目前与PI合作的研究生的研究。