Collaborative Research: Algebraic Structures and Cohomology Theories Associated to Knottings

新的状态和不变量的纽结在3维空间和纽结表面在4维空间被定义，在一个状态和的形式，由主要研究者和合作者，使用自分配操作称为quandles和他们的着色结和表面图。状态和的权重是由quandle上同调理论推导出来的。已经发现了一些应用程序的各种性质的节点和表面。该项目研究quandles，李代数，余代数，交叉模及其上同调理论之间的关系，以开发流形不变量等应用。它还建议使用几何和图解的方法来分析特定的量子化、量子群和上同调理论。四维空间中的曲面也可以打结。纽结理论研究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。在一张纸上画出的纽结图，以及从图中很容易计算出的数值量，在纽结理论中得到了广泛的应用。主要研究者和他们的合作者已经从结图中开发出了代数系统，这些结图给出了结的视觉表示的密切反映。这些图表和相关版本的代数简明地编码了节点和物理系统之间的深层联系。当前的项目开发了图的代数系统和其他与物理学中的标准模型密切相关的已建立的代数系统（李代数和交叉模块）之间的新联系。这些技术也将应用于分类的背景下-一个身份被如何识别的指令所取代的过程。

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.