What is a matroid?

A matroid is what remains when you remember only the linear dependence or independence among a set of vectors in a vector space, or you remember only the incidences among a collection of points, lines, planes, etc. and forget their angles and distances. The properties satisfied by incidence or linear (in)dependence give rise to the axioms of a matroid. In fact, there are multiple systems of axioms, each based on a different structure from linear algebra like linear independence, bases, rank, spanning sets, etc., but they all magically end up defining the same objects. And the objects are more general than what you started with: there are matroids which cannot be realized by a set of vectors over any field, and in a precise sense almost every matroid is not realizable. I'll give an introduction to these objects and a few of the things you can do with them; this can be viewed as a pre-talk for my talk May 8 in the Discrete Math seminar.