Matroids

Matroids are a broad class of objects that underlie everything from concrete problems such as job assignments to finite affine geometry such as the Fano plane. One place matroids appear is in algebraic geometry.

Intersection theory is a branch of algebraic geometry which studies how curves (and more) intersect inside a space. One classical result from this field is Bezout's theorem. Intersection theory is one of the most studied branches of algebraic geometry today with much still unknown about the general case. Tools and examples are needed to test conjectures and matroids give rise to a rich class of examples with concrete descriptions. 

Associated with each matroid is an algebraic variety such that the combinatorics of the matroid perfectly encode the intersection theory on that variety. By understanding the chains of flats of a matroid, we can prove statements about the intersection theory of the corresponding variety. Current work by the Matlab group at the University of Kentucky has produced a decomposition theorem for expressing arbitrary matroidal subvarieties associated with modular cuts in terms of principal ones.