From Topology to Combinatorics and Back

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ABSTRACTPrincipal Investigator: Swartz, Edward B. Proposal Number: DMS - 0900912 Institution: Cornell UniversityTitle: From Topology to Combinatorics and BackThe primary research aim of this project is to study the interplay between topology, geometry and combinatorics. Four areas will receive special emphasis: triangulations of compact manifolds and closely related spaces, Cohen-Macaulay complexes, finite linear quotients of spheres, and matroids. In the last century a tremendous amount of research has been directed toward unlocking the topology and geometry of manifolds. However, questions of a combinatorial character have remained largely unanswered. What is the minimum number of facets required to triangulate a given manifold? What methods are there to construct manifolds with particular combinatorial properties? Given limits on the combinatorics of a triangulation, what does that imply about the possible geometries on the manifold? Similar questions, where even less is known, apply to spaces with singularities which include algebraic varieties, quotients of group actions, or limits of Riemannian manifolds. Cohen-Macaulay complexes and posets are one of the fundamental structures of algebraic combinatorics with applications ranging from partitions to network reliability. Their enumerative properties were worked out in the 1970's by the ground breaking work of Hochster, Reisner and Stanley. What happens if further constraints are put on these spaces? As previously shown by the PI, complexes with more structure, such as finite buildings, geometric lattices and matroid complexes, have strong restrictions on their f-vectors. How common are these additional structures and constraints? Finite linear quotients of spheres lie at the intersection of representation theory, topology, geometry and combinatorics. How are the topology and geometry of the quotient space related to the combinatorics and algebra of the representation theoretic data? How does one study complicated spaces and structures? One approach is to approximate them with simpler objects. For instance, an n-dimensional space might be represented as a collection of n-simplices - the higher dimensional analogues of triangles and tetrahedrons. How complicated are these representations? How many of the smaller pieces are needed? How is this reflected in the shape and geometry of the original object? Are there computationally practical ways of producing these models? These are types of questions addressed in this research.