Studying subschemes of affine/projective space via matroids

Given a homogeneous ideal I in a polynomial ring, one may apply the following combinatorial operation: for each degree d, make a list of all subsets S of the set of degree-d monomials such that S is the set of nonzero coefficients of an element of I. For each d, this set of subsets is a combinatorial object called a matroid. As d varies, the resulting sequence of matroids is called the tropicalization of I.

I will discuss some of the many questions one can ask about tropicalizations of ideals, and how they are related to some classical questions in combinatorial algebraic geometry, such as the classification of torus orbits on Hilbert schemes of points in C^2. Some unexpected combinatorial objects appear: e.g. when studying tropicalizations of subschemes of P^1, one is led to Schur polynomials and binary necklaces.