for this workshop

## Ehrhart polynomials: inequalities and extremal constructions

at the

American Institute of Mathematics, San Jose, California

organized by

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

This workshop, sponsored by AIM and the NSF, 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?

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*