Geometry for the unrepresented (matroids)

Matroids are basic structures in combinatorics: they are set systems that encode an "independence" property of a finite set. A key example -- but not the only example! -- is linear independence on a finite set of vectors. When a matroid arises this way, we call it (linearly) representable.

A set of vectors is geometric data, and so given a representable matroid we can build several useful geometric spaces and use geometry to constrain enumerative properties. What about matroids that are not representable? Developments of the last decade have provided surrogates for the geometric spaces, such as tropical objects, from which we can recover similar constraints. I'll talk about some of this.