Connections between Euler Characteristic Invariants, H-Vector Problems, and Algebro-Geometric Properties of (Poly)matroids

We explore combinatorial questions using tools from algebraic geometry/topology (or the converse). The first direction we start with involves combinatorial constructions approximating and characterizing properties of varieties. More specifically, we started out with (approximate) relations in the Grothendieck ring of varieties. This involved an arithmetic statistics-type result showing that the Fano variety of k-planes contained in a given variety are determined mostly by symmetric products of points in the initial variety. This involves using a motivic limit/approximate relation of Galkin--Shinder in the Grothendieck ring of varieties. Moving to *exact* relations, we showed that the original relation of Galkin--Shinder can be used to characterize cubic hypersurfaces using a projective geometry construction (and intersections of two quartic or quadric hypersurfaces after weakening assumptions). Exact relations in this ring also gave a transition to combinatorial invariants. Expressions in the Grothendieck ring of varieties for configuration spaces of points led to our transition to combinatorial problems. More specifically, certain generalizations of chromatic polynomials are uniquely defined (up to normalization) by Cooper-de Silva-Sazdanovic. We showed that these can be expressed using h-vectors of certain simplicial complexes (under certain conditions). Note that *any* h-vector appears in such a construction. In addition, we show that there are no nontrivial bounds on ranks of proper flats that cover the underlying set of a matroid satisfying the matroidal analogue of the Cayley-Bacharach property. This gives a negative answer to a question in recent work of Levinson and Ullery. Finally, we also explore connections to combinatorial invariance and Chow rings of matroids.