Collaborative Research: Algebraic Structures and Cohomology Theories Associated to Knottings

合作研究：与结相关的代数结构和上同调理论

• 批准号: 0603876
• 负责人: Masahiko Saito
• 金额: $ 9.69万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603876&HistoricalAwards=false
• 关键词: Collaborative; Research; Algebraic; Structures; Cohomology

New state-sum invariants for knots in 3-dimensional space and knotted surfaces in 4-dimensional space were defined, in a state-sum form, by the principal investigator and collaborators, using self-distributive operations called quandles and their colorings of knot and surface diagrams. The weights of the state-sum are derived from quandle cohomology theories. A number of applications to various properties of knots and surfaces have been discovered. The project investigates relationships among quandles, Lie algebras, coalgebras, crossed modules and their cohomology theories in order to develop applications such as manifold invariants. It also proposes to use geometric and diagrammatic methods to analyse specific categorifications, quantum groups, and cohomology theories.A knot is a circle situated in space. Surfaces in four-dimensional space can also be knotted. Knot theory studies such knotted circles and surfaces, and has provided models and applications to DNA theory, molecular configurations, and physics. Knot diagrams drawn on a piece of paper, and numerical quantities that are easily computable from diagrams, have been extensively used in knot theory. The principal investigators and their collaborators have developed algebraic systems from the knot diagrams that give a close reflection of the visual representations of knots. The algebra of these diagrams and related versions concisely encode deep connections among knots and physical systems. The current project develops new connections between the algebraic system of diagrams and other established algebraic systems (Lie algebras and crossed modules) that are closely associated with the standard model in physics. The techniques will also be applied in the context of categorification --- a process by which identity is replaced by an instruction of how to identify.

首席调查员和合作者使用称为Quandles及其颜色及其的色彩及其颜色，以一种状态和合作者的形式定义了三维空间中的结节和打结表面的新的状态和状态不变。表面图。状态-AM的权重来自QUANDLE的共同体理论。已经发现了许多针对结和表面各种特性的应用。该项目调查了难题，谎言代数，山地，越过模块及其共同体学理论之间的关系，以开发诸如流形不变的应用。它还建议使用几何学和示意图分析特定的分类，量子组和协同理论。A结是位于空间中的一个圆。在四维空间中的表面也可以打结。结理论研究了这种打结的圆圈和表面，并为DNA理论，分子构型和物理学提供了模型和应用。在一张纸上绘制的结图，并且很容易从图表中计算的数量数量已在结理论中广泛使用。主要研究人员及其合作者从结图开发了代数系统，这些系统对结的视觉表示有了密切的反映。这些图表和相关版本的代数简洁地编码结和物理系统之间的深度连接。当前的项目在图表的代数系统与其他已建立的代数系统（LIE代数和交叉模块）之间建立了新的连接，这些连接与物理学的标准模型密切相关。这些技术也将在分类的背景下应用 - 一个过程被身份替换为如何识别的指令。