A General Theory of Codes|
Abstract: Oriented matroids can be viewed as combinatorial models for real hyperplane arrangements. In various applications, one wishes to consider the combinatorial analog not only to a single hyperplane arrangement, but to a continuous family of such arrangements. There is a nice combinatorial analog to such families, in which the analog to continuous change is described by weak maps.
Oriented matroids have a beautiful topological aspect, given by the Topological Representation Theorem. This theorem allows one to view oriented matroids as simply particularly nice cell decompositions of spheres. This deep connection between topology and combinatorics has led to remarkable applications of oriented matroids in various aspects of topology. One would hope that a "continuous family of oriented matroids" would have an associated continous family of topological spheres -- indeed, this hope is central to various programs in combinatorial topology. As it turns out, this hope can be fulfilled, but in a more sophisticated way than one might have expected. In this talk we will discuss the topological interpretation of weak maps and applications of this to topology.