Mathematical Sciences: Polynomial Invariants in the Theory of Knots

数学科学：结理论中的多项式不变量

• 批准号: 9504471
• 负责人: Louis Kauffman
• 金额: $ 7.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504471&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Polynomial; Invariants; Theory

9504471 Kauffman This project investigates invariants of knots, 3-manifolds and higher dimensional manifolds that arise from state summation models, quantum groups and topological quantum field theory. This includes work on the structure of the Jones polynomial, on Vassiliev invariants, and on connections with quantum field theory and molecular biology, as well as techniques and conjectures about the use of functional integration in topology. In the last decade, knot theory has encountered an awakening through the infusion of a remarkable collection of techniques from mathematical physics and a series of interrelationships with molecular biology, chemistry, and theoretical physics. The principal investigator is one of the key figures in the inception of this relationship with physics (statistical mechanics) via his introduction of the use of state summation models for polynomial invariants of knots and links. In a state summation model the knot is regarded as a miniature physical system, and topological properties of the knot are seen as averages of physical interactions. These models have grown to encompass techniques from quantum field theory and have been useful in topology and its applications to the study of molecular interactions in biology and chemistry. In topology these techniques apply to the structure of three- and four-dimensional spaces as well as to knots. This means that these techniques are essential to the architecture of three- dimensional structures - from molecules to networks of communication, from atoms to galaxies. All fields of science are based on the structure of space. What is particular to this project's topological study is that it articulates the topological components in large networks of which knots are the principal example. Such networks are the basis of significant structures such as RNA and DNA, where topology is the key to interaction. In fact, the introduction of topology in these subjects is, from the viewpoint of this principal investigator, just the beginning of a larger interplay, where topology is seen as the basis for studying the recursive and circular architectures of complex systems. A galaxy, a society, an organism and a physical system all have in common the stability that arises through closed systems of circular interactions. Such interactions have traditionally been treated via cybernetics and computer modeling. These topological investigations add a new dimension to cybernetics and complex systems, providing new questions and a host of new problems. All these issues are treated in the new book "Knots and Applications," edited and contributed to by the principal investigator. ***

9504471考夫曼这个项目研究纽结、三维流形和高维流形的不变量，这些不变量来自态求和模型、量子群和拓扑量子场论。这包括琼斯多项式的结构，瓦西里耶夫不变量，与量子场论和分子生物学的联系，以及关于在拓扑学中使用函数积分的技术和猜想。在过去的十年中，纽结理论通过注入来自数学物理的一系列非凡的技术集合以及与分子生物学、化学和理论物理的一系列相互关系而经历了一次觉醒。这位主要的研究人员是这种与物理学(统计力学)关系建立的关键人物之一，他介绍了节点和链节的多项式不变量的状态求和模型的使用。在状态求和模型中，节点被视为一个微小的物理系统，节点的拓扑性质被视为物理相互作用的平均值。这些模型已经发展到包含来自量子场论的技术，并在拓扑学及其在生物和化学中研究分子相互作用的应用中发挥了作用。在拓扑学中，这些技术适用于三维和四维空间的结构以及节点。这意味着这些技术对三维结构的架构至关重要--从分子到通讯网络，从原子到星系。科学的所有领域都是以空间结构为基础的。这个项目的拓扑研究的特殊之处在于，它阐明了大型网络中的拓扑组件，其中节点是主要的例子。这样的网络是RNA和DNA等重要结构的基础，其中拓扑是相互作用的关键。事实上，从这位首席研究员的角度来看，在这些学科中引入拓扑学只是一个更大的相互作用的开始，拓扑学被视为研究复杂系统的递归和循环结构的基础。一个星系、一个社会、一个有机体和一个物理系统都具有通过封闭的循环相互作用系统产生的稳定性的共同之处。这种相互作用传统上是通过控制论和计算机建模来处理的。这些拓扑研究为控制论和复杂系统增加了一个新的维度，提供了新的问题和一系列新的问题。所有这些问题都在新书《结和应用》中讨论，该书由首席研究员编辑和贡献。***