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

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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.