Summer Graduate School

The Ehrhart polynomial of a lattice polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic polynomials?, we give a brief survey of attempts during the last half century to classify Ehrhart polynomials. It turns out that this classification problem is related to that of a whole family of polynomials in combinatorics.

We will present some new results for Ehrhart polynomials of zonotopes, i.e., projections of (higher dimensional) cubes. This includes a combinatorial description in terms of refined descent statistics of permutations and a formula in matroidal terms which complements a well-known zonotopal identity of Stanley (1991). Finally, we give a complete description of the convex hull of the Ehrhart coefficients of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

New results in this talk comes from joint work with Katharina Jochemko (KTH) and Emily McCullough (University of San Francisco).

Generalized permutahedra are a beautiful family of polytopes with a rich combinatorial structure. We explore the Hopf algebraic structure of this family. We then use this structure to unify old results, prove new results, and answer open questions about many objects of interest, such as graphs, matroids, posets, trees, set partitions, building sets, hypergraphs, and simplicial complexes. In particular, we shed new light on the basic problem of inverting a power series.

The talk will be based on joint work with Marcelo Aguiar, and will assume no previous knowledge of Hopf algebras or generalized permutahedra.