Mathematical Sciences: Polynomial Invariants in the Theory of Knots

这个项目研究了由状态和模型、量子群和拓扑量子场论产生的结、3-流形和高维流形的不变量。这包括琼斯多项式的结构，瓦西里耶夫不变量，量子场论和分子生物学的联系，以及在拓扑学中使用功能集成的技术和猜想。在过去的十年中，通过从数学物理和分子生物学、化学和理论物理的一系列相互关系中注入的大量技术，结理论经历了一次觉醒。通过引入结点和链路的多项式不变量的状态和模型，首席研究员是这种关系与物理学（统计力学）开始的关键人物之一。在状态总和模型中，将结视为一个微型物理系统，并将结的拓扑性质视为物理相互作用的平均值。这些模型已经发展到包含量子场论的技术，并且在拓扑学及其在生物学和化学分子相互作用研究中的应用方面非常有用。在拓扑学中，这些技术适用于三维和四维空间的结构以及结。这意味着这些技术对于三维结构的构建至关重要——从分子到通信网络，从原子到星系。所有科学领域都以空间结构为基础。这个项目的拓扑研究的特别之处在于，它阐明了大型网络中的拓扑组成部分，其中结是主要的例子。这种网络是RNA和DNA等重要结构的基础，拓扑结构是相互作用的关键。事实上，从这位首席研究员的观点来看，拓扑学在这些学科中的引入只是一个更大的相互作用的开始，在这个相互作用中，拓扑学被视为研究复杂系统递归和循环架构的基础。一个星系、一个社会、一个有机体和一个物理系统都有一个共同的稳定性，这种稳定性是由循环相互作用的封闭系统产生的。传统上，这种相互作用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，提供了新的问题和许多新的问题。所有这些问题都在由首席研究员编辑和贡献的新书“结和应用”中进行了处理。＊＊＊

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. ***