From Topology to Combinatorics and Back

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The primary theme of this project is to study the interplay between topology, geometry and combinatorics. Four areas will receive special emphasis: triangulations of compact manifolds, Cohen-Macaulay complexes, finite linear quotients of spheres, and matroids. It has been over 100 years since the introduction of the Euler-Poincare formula, yet the combinatorial properties of triangulations of compact manifolds remain largely unknown. Indeed, a complete characterization of all possible f-vectors (which record the number of faces in each dimension) is not known for any manifold of dimension five or more. The possible f-vectors of Cohen-Macaulay simplicial complexes have been known since the foundational work of Hochster, Reisner and Stanley in the 1970's. However, the classification of face counts for important subclasses, such as spheres and doubly Cohen-Macaulay complexes, remains a fundamental open problem in algebraic and topological combinatorics. Even though the representation theory of finite groups is an extremely well developed subject, tools for computing many topological invariants, such as Betti numbers, for the corresponding spherical quotients are practically nonexistent. Matroids are a combinatorial abstraction of linear independence which can be modeled through a variety of topological spaces. Their enumerative properties have many applications including coloring and flows on graphs, network reliability, and the topology of hyperplane arrangementsTwo recurrent and complementary ideas in mathematics are the approximation of continuous phenomena using discrete data, and modeling the latter with smooth objects, such as spheres or polyhedra. The guiding principle of this project is that the combination of these two techniques is a powerful method of approaching both situations.For instance, compact manifolds, which are frequently presented as solutions to nonsingular polynomial equations in several variables, can also be examined by using simplicial complexes which are easier to encode into a computer. In turn, this allows a deeper study of the shape and geometry of the original space.