Algebraic combinatorics and its applications to algebraic geometry

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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. As research progressed in these areas, it became increasingly apparent that deeply within certain questions lay new questions of great complexity about aggregates of new objects constrained by complex relations. These new questions are essentially combinatorial in the sense that they can be expressed in purely combinatorial terms, without reference to the original setting in geometry and mathematical physics. 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 three questions of great  complexity in algebraic geometry, that are notable in their own right, in the conviction that hidden within them is rich structure that, once understood, will lead to powerful mathematical methodology of  wider applicability.  I propose to use combinatorial constructions to transform the new interrelated objects, and then to use properties of carefully constructed algebras and further mathematical transformations to elicit tangible and concrete information about the original geometric questions themselves. This will require the development of the algebraic and analytic methodology.  Because of the level of generality that these abstractions permit, the methodology developed here will be applicable to other questions in this area.  At the foundation of this approach is the research and experience that I have already accumulated in algebraic combinatorics. (For mathematicians: The questions I shall examine are: i) the structure of the double Hurwitz numbers for ramified covers of the sphere, ii) Faber's intersection number conjecture for the moduli space of smooth curves, and iii) certain invariants of 3-manifolds.)