Knot Theory: 3 and 4-dimensional manifolds

纽结理论：3 维和 4 维流形

• 批准号: 1309081
• 负责人: Shelly Harvey
• 金额: $ 31.87万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309081&HistoricalAwards=false
• 关键词: Knot; Theory; dimensional; manifolds

This project will study knot and link concordance and its implications for 3- and 4-manifolds. We will study the smooth concordance classes of topologically slice knots via the recently defined bipolar filtration, using techniques of gauge theory, Heegaard Floer homology and von Neumann signatures. We will attempt to exhibit a primary decomposition of the knot concordance group using noncommutative localization techniques. We will also study the set of concordance classes as a metric space, under several natural metrics. We will also establish new structure in the knot concordance group, a sort of Wang sequence, arising from the failure of several conjectures of Kauffman.In its broadest terms this project is about the mathematical structure of the "shape" of 3- and 4-dimensional objects. Understanding complex shapes has many applications. The geometric structure of proteins and other complex molecules is crucial to drug development. The geometric configuration of cellular DNA is important in determining the precise mechanisms of cellular processes. The geometric shape of organs (especially the brain) is vital to the field of medical imaging. The prediction of shape from incomplete data (satellite imaging, sensor networks) has important military and commercial applications. Additionally, this project will increase the participation of U.S. citizens, permanent residents and women in mathematical research and education. This project focusses on knot theory, which is known to parametrize all 3-dimensional manifolds.

本项目将研究结和链接的一致性及其对3-和4-流形的影响。我们将利用规范理论、Heegaard Floer同调和von Neumann签名的技术，通过最近定义的双极过滤来研究拓扑切片结的光滑协调类。我们将尝试使用非交换定位技术来展示结调和群的初级分解。我们还将研究在若干自然度量下作为度量空间的一致性类集。我们还将在结谐和群中建立新的结构，这是一种王序列，源于考夫曼的几个猜想的失败。从广义上讲，这个项目是关于三维和四维物体的“形状”的数学结构。理解复杂的形状有很多应用。蛋白质和其他复杂分子的几何结构对药物开发至关重要。细胞DNA的几何结构在确定细胞过程的精确机制方面是重要的。器官（尤其是大脑）的几何形状对医学成像领域至关重要。从不完全数据（卫星成像、传感器网络）中预测形状具有重要的军事和商业应用。此外，该项目将增加美国公民、永久居民和妇女在数学研究和教育中的参与。这个项目的重点是结理论，众所周知，它可以参数化所有的三维流形。