This project centers on two combinatorial models for manifolds and vector bundles: Gelfand and MacPherson's theory of combinatorial differential manifolds and matroid bundles and Mnev and Mani's theory of locally polytopal manifolds. The primary focus is on the bundle theory and classifying spaces within each model. In both of these models, a bundle-theoretic perspective leads to deep and unexpected new relations between topology and combinatorics: the primary objective of this research is to expand on these relations. The project's goals include both combinatorial applications in topology (for instance, combinatorial formulas for topological invariants) and topological results in combinatorics (particularly on the topology of posets of oriented matroids and polytopes). The classifying spaces for Gelfand and MacPherson's theory derive from certain partially ordered sets of oriented matroids, and have close ties to the real Grassmannian. Among the objectives of this research are a closer knowledge of the topology of the combinatorial Grassmannian and applications of this knowledge both to real vector bundles and to various combinatorial problems. Mnev and Mani's theory also appears to give rise to a classifying space of interest in both topology and combinatorics, relating PL manifolds and convex polytopes. The author will also study a new type of ``non-Euclidean'' pathology in oriented matroids and its implications for combinatorial differential manifolds. Much of the power of these combinatorial models for topology derives from the elegance with which topological concepts translate into combinatorial analogs. One can, for instance, derive combinatorial formulas for topological invariants by simply translating each element of a topological formula into its combinatorial equivalent. The translation from combinatorics to topology has proven equally useful: these models have provided a new framework in which to view various combinatorial topics. Both models have revealed a vector-bundle perspective on various combinatorial problems that originally appeared to have no connection to real vector bundles, leading to unexpected topological proofs. This project will build on recent results in this area whose depth lies in uniting the combinatorial side of mathematics, rooted in counting things, to the topological side, rooted in plasticity and spatial continuity. ***
