Research Statement Algebraic Matroids: Structure and Applications

Matroids were introduced in the early 20th century as a way of uniting disparate notions of “independence” from across mathematics. Among these notions were linear independence of vectors and graphic independence – defined by acyclicity on the subgraph corresponding to a set of edges. Algebraic independence over a field k, defined by the non-existence of polynomial relations with coefficients in k among elements of a set, is another such notion. Van der Waerden first defined algebraic matroids [vdW91], and they were studied by MacLane in an early matroid paper [ML38]. They were largely neglected after then until the 1970’s. Ingleton and Main showed the existence of non-algebraic matroids in [IM75], and Bernt Lindström treated algebraic representability in several papers (e.g. [Lin83, Lin86, Lin88]), as did some others ([DL87, Gor88]).