Ehrhart polynomials: inequalities and extremal constructions
May 9 to May 13, 2022
American Institute of Mathematics,
San Jose, California
and Fu Liu
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
- 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.