  This article is a survey of matroid theory aimed at algebraic geometers.
Matroids are combinatorial abstractions of linear subspaces and hyperplane
arrangements. Not all matroids come from linear subspaces; those that do are
said to be representable. Still, one may apply linear algebraic constructions
to non-representable matroids. There are a number of different definitions of
matroids, a phenomenon known as cryptomorphism. In this survey, we begin by
reviewing the classical definitions of matroids, develop operations in matroid
theory, summarize some results in representability, and construct polynomial
invariants of matroids. Afterwards, we focus on matroid polytopes, introduced
by Gelfand-Goresky-MacPherson-Serganova, which give a cryptomorphic definition
of matroids. We explain certain locally closed subsets of the Grassmannian,
thin Schubert cells, which are labeled by matroids, and which have applications
to representability, moduli problems, and invariants of matroids following
Fink-Speyer. We explain how matroids can be thought of as cohomology classes in
a particular toric variety, the permutohedral variety, by means of Bergman
fans, and apply this description to give an exposition of the proof of
log-concavity of the characteristic polynomial of representable matroids due to
the author with Huh.

Algebraic Geometry;Combinatorics