Matroid theory began in the 1930's as an abstraction of the ideas of linear independence in linear algebra, algebraic independence in field theory, and cycle-free edge sets in graph theory. There is a two-way interplay between matroid theory and its many fields of application which yields a rich theory with powerful applications. For instance, linear algebra contributes the ideas of flats (generalizing subspaces) and rank (generalizing dimension); graph theory contributes the idea of circuits. Flats and rank then become new tools for exploring graphs; circuits shed light on linear independence.
The 1960's and 70's witnessed an explosive growth in the field, spurred partly by the discovery of connections with optimization: for instance, matroids are the simplicial complexes on which the greedy algorithm produces optimal solutions, and they are the combinatorial structure behind linear programming. More recently, the subject has been shown to have major connections with coding theory, arrangements of hyperplanes, the rigidity of bar and joint frameworks, and many other areas.
Much of my work is on Dowling lattices, which are roughly to groups what projective geometries (essentially, lattices of subspaces of a vector space) are to fields. Some of my other work is on the critical problem, which forms the theoretical framework for numerous important problems, including the fundamental problem of linear coding theory, Tutte's 5-flow conjecture, and Hadwiger's conjecture. My research is expanding into the areas of extremal matroid theory (how large can a matroid be if we exclude certain other matroids from being contained in it?) and representations (which matroids arise from sets of vectors in a vector space?).