Algebraic combinatorics and its application to algebraic geometry and low dimensional topology

自20世纪80年代S以来，在与数学物理的深刻联系的推动下，现代几何学经历了一段相当长的显著而激烈的活动时期。出于同样的原因，现代纽结理论也有了进步，部分原因是在数学物理和纽结理论中都出现了一个特殊的基本方程--杨-巴克斯特方程。随着这些领域研究的进展，越来越明显的是，在关于单个节点不变量的问题的深处，存在着关于Vassiliev不变量的新的极其复杂的问题，以及受复杂关系约束的新对象或图的集合。这些新问题本质上是组合性的，可以抽象地表达，而不涉及它们的原始上下文。这种图的一个例子是来自量子场论的费曼图。 本研究方案的目的是为解决以这种方式产生的组合问题而取得进展。为此，我建议研究四个极其复杂的问题，这些问题本身就是值得注意的，因为它们背后隐藏着丰富的结构，一旦理解，将导致强大的数学方法论，具有广泛的适用性。我建议使用组合结构来转换新的对象或图，然后使用精心构建的代数的性质，以及进一步的数学转换，来获得关于原始几何和拓扑问题本身的有形和具体的信息。这个问题的核心是理解如何用相对较新的代数(如平面代数、带状Hopf代数和色代数)来表示对分解和凝聚编码复杂结构的组合图来说是自然的运算。然后，目的是使用这种代数中的基本定理来帮助从图示系统中提取重要的新结构信息。

Since about the 1980's there has been a long period of remarkable and intense activity in modern geometry, spurred on by the deep connexions with mathematical physics. For the same reason, there were advances in Modern Knot Theory, prompted partly by the occurrence of a particular fundamental equation, the Yang-Baxter Equation, in both mathematical physics and knot theory. As research progressed in these areas, it became increasingly apparent that deeply within questions about individual knot invariants lay new questions of great complexity about Vassiliev invariants, and aggregates of new objects, or diagrams, constrained by complex relations. These new questions are essentially combinatorial in nature, and may be expressed abstractly, without reference to their original context. An instance of such a diagram is the Feynman diagram from quantum field theory. The purpose of this Research Proposal is to make advances into the solution of combinatorial questions that arise in this way. To do so, I propose to examine four questions of great complexity, that are notable in their own right, with the knowledge that hidden within them is rich structure that, once understood, will lead to powerful mathematical methodology of wide applicability. I propose to use combinatorial constructions to transform the new objects, or diagrams, and then to use properties of carefully constructed algebras, and further mathematical transformations, to elicit tangible and concrete information about the original geometric and topological questions themselves. At the heart of the matter is the understanding of how operations that are natural to dissecting and agglomerating combinatorial diagrams encoding complex structure may be represented by algebraic operations governed by relatively new algebras, such as planar algebras, ribbon Hopf algebras and chromatic algebras. The intent then is to use fundamental theorems in such algebras to assist in the extraction of significant new structural information from the diagrammatic systems.