Ehrhart polynomials: inequalities and extremal constructions

May 9 to May 13, 2022

at the

American Institute of Mathematics, San Jose, California

organized by

Matthias Beck, Benjamin Braun, Katharina Jochemko, and Fu Liu

Original Announcement

This workshop will be devoted to Ehrhart polynomials and quasi-polynomials. These objects are invariants of lattice and rational polytopes that are the focus of Ehrhart Theory. Since Ehrhart's original work in the late 1960's, Ehrhart theory has developed into a key topic at the intersection of polyhedral geometry, number theory, commutative algebra, algebraic geometry, enumerative combinatorics, and integer programming. The goal of the proposed workshop is to bring together an international and diverse team of experts and young researchers in order to make substantial breakthroughs on existing open problems and to identify new research directions.

The main topics for the workshop are:

Classification problems: What are the possible coefficient vectors of Ehrhart polynomials for lattice polytopes of fixed dimension, especially in dimensions three and four? Is it possible to classify Ehrhart polynomials for sufficiently special families of lattice polytopes, for example, lattice zonotopes?
Inequalities for polynomial coefficients: What properties of a lattice polytope imply that the Ehrhart polynomial has positive coefficients? For example, do generalized permutahedra have this property? What geometric or arithmetic properties of lattice polytopes imply having a unimodal Ehrhart h*-vector? For example, do lattice polytopes with the integer decomposition property behave in this manner?
Extremal constructions: How can we demonstrate the existence of (possibly high-dimensional) lattice polytopes with extreme properties for their Ehrhart polynomials? What are efficient strategies for creating examples/counterexamples? What are computationally feasible choices of classes of lattice polytopes to search for examples/counterexamples?

Material from the workshop

A list of participants.

The workshop schedule.

A list of open problems.

Workshop Videos

Papers arising from the workshop:

Sums of weighted lattice points of polytopes

by Jesús A. De Loera, Laura Escobar, Nathan Kaplan, and Chengyang Wang

Examples and counterexamples in Ehrhart theory

by Luis Ferroni, Akihiro Higashitani

Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem

by Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

Local $h^*$-polynomials for one-row Hermite normal form simplices

by Esme Bajo, Benjamin Braun, Giulia Codenotti, Johannes Hofscheier, Andrés R. Vindas-Meléndez

Minimal free resolutions of numerical semigroup algebras via Apéry specialization

by Benjamin Braun, Tara Gomes, Ezra Miller, Christopher O'Neill, Aleksandra Sobieska

Local $h^*$-polynomials for one-row Hermite normal form simplices

by Esme Bajo, Benjamin Braun, Giulia Codenotti, Johannes Hofscheier, Andrés R. Vindas-Meléndez