Geometric and topological combinatorics

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The primary research aim of this project is to study the interplay between topology, geometry and combinatorics. Three areas will receive special emphasis: triangulations of compact manifolds (with and without boundary) and closely related spaces, finite linear quotients of spheres, and matroids. Compact manifolds are one of the cornerstones of modern mathematics and physics. In the last century a tremendous amount of research has been directed toward unlocking their topology and geometry. However, questions of a combinatorial character have remained largely unanswered. What is the minimum number of facets required to triangulate a given manifold? How do various topological invariants determine how complicated it is to represent the manifold on a computer? What methods are there to construct manifolds with particular combinatorial properties? Conversely, 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. 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? Matroids are a combinatorial abstraction of linear independence with a variety of applications including hyperplane arrangements, linear optimization, reliability, order restricted statistical inference, and the aforementioned linear quotients of spheres. The project will concentrate on enumerative properties of matroids. The questions above are representative of the research that will be conducted by the PI during the project. How does one study complicated spaces and structures? One approach is to approximate them with simpler objects. For instance, an n-dimensional object 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.