Leonardo Saud Maia Leite: An invitation to Ehrhart theory

OBS: Note the different location!

Ehrhart theory, developed by the mathematician Eugène Ehrhart in the 1960s, explores the combinatorial properties of lattice polytopes. The central idea of the theory is to study the function \(L_P(n)\) which counts the number of lattice points inside a scaled version of a polytope \(P\). Specifically, Ehrhart’s theorem states that for a lattice polytope \(P\), the function \(L_P(n)\) is a polynomial in \(n\), whose degree is equal to the dimension of the polytope. Ehrhart polynomials are naturally connected to other areas of Combinatorics, such as order polynomials of posets, chromatic polynomials of graphs and Diophantine equations.

In this expository talk, we introduce Ehrhart theory, exploring its main ideas such as the Ehrhart polynomial and the \(h^*\)-vector, and we provide some important examples. In particular, we study order polytopes and present some of our contributions to the area. This is a joint work with Teemu Lundström.