Enumerative and Topological Properties of Matroids

Cornell University, Ithaca NY

Investigators

Abstract

The PI plans to continue his research into the enumerative and topological properties of matroids. Introduced in 1935 by Whitney, matroids are a combinatorial abstraction of one of the fundamental structures of mathematics, linear independence. The enumerative aspect of the work involves combining techniques from commutative algebra, topology and combinatorics in order to study h-vectors of broken circuit and independence complexes. These combinatorial invariants appear in applications to hyperplane arrangements, linear coding theory, network reliability and quotients of spheres. Topological properties of matroids will be explored using the investigator's recent discovery that matroids can be represented as arrangements of homotopy spheres. This point of view indicates possible applications to combinatorial projective planes, oriented matroids and classical bundle theories. This research is in the general area of combinatorics. One of the central themes in combinatorics is the enumeration of discrete objects with complicated structure. For instance, this proposal includes methods for estimating the reliability of large networks of the type that might be found in the design of telephone or other similar communication systems.