Enumerative Problems Concerning Oriented Matroids

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An oriented matroid is a combinatorial abstraction which can be used to encapsulate features of a nite point set in Euclidean space which are of interest in combinatorial geometry. It is planned to study certain enumeration problems involving oriented matroids by making use of the theory of valuations on distributive lattices. Indeed some progress along these lines has already been made. It is hoped that information on the range of certain important functions for example, the f-vectors of convex polytopes, or the number of "k-sets" of a finite point set will be generated. Additionally it is hoped that this research will enable the extension to oriented matroids in general of substantial results already obtained for uniform oriented matroids.Combinatorial geometry is a subject with a venerable history which in recent years has become an even more important arena for research, due to its applications in the area of algorithms for geometrical structures. Research in combinatorial geometry often leads to better algorithms in computational geometry, which in turn leads, for example, to improved design of computer chips, better algorithms for image analysis and processing, and so forth.