Triangulations of oriented matroids
Abstract: An oriented matroid is a combinatorial abstraction of a finite set of points in Euclidean space. Loosely speaking, one derives an oriented matroid from a set of points by considering the convexity relationships between the points. Not every oriented matroid arises in this way, however. Two challenges of the subject are to find oriented matroid analogs to geometric concepts and to find aspects of discrete geometry that cannot be generalized to oriented matroids
One such concept is triangulations of point configurations. A triangulation of a set A of points in Rn is a triangulation of the convex hull of A, linear on each simplex and with 0-simplices a subset of A. So is there a good notion of triangulations of oriented matroids, and if so, is every oriented matroid triangulation a topological ball? The question has arisen in several contexts, including linear programming, convex polytopes, and combinatorial differential manifolds, and has had less-than-satisfactory partial answers for several decades. In the last five years several more workable notions of triangulations of oriented matroid have developed, allowing strong tameness results on topology of triangulations. We will discuss these results and their applications to combinatorial differential manifolds and subdivisions of polytopes.