Enumerative and Topological Properties of Matroids

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The PI plans to continue his research into theenumerative and topological properties of matroids. Introduced in 1935 by Whitney, matroids are acombinatorial abstraction of one of the fundamentalstructures of mathematics, linear independence. Theenumerative aspect of the work involves combiningtechniques from commutative algebra, topology andcombinatorics in order to study h-vectors of brokencircuit and independence complexes. Thesecombinatorial invariants appear in applications tohyperplane arrangements, linear coding theory, networkreliability and quotients of spheres. Topologicalproperties of matroids will be explored using theinvestigator's recent discovery that matroids can berepresented as arrangements of homotopy spheres. Thispoint of view indicates possible applications tocombinatorial projective planes, oriented matroids andclassical bundle theories.This research is in the general area of combinatorics.One of the central themes in combinatorics is theenumeration of discrete objects with complicatedstructure. For instance, this proposal includesmethods for estimating the reliability of largenetworks of the type that might be found in the designof telephone or other similar communication systems.