at the

American Institute of Mathematics, San Jose, California

organized by

Joseph Gubeladze, Christian Haase, and Diane Maclagan

The first of these is the question of which lattice polytopes *P* have the
property that every lattice point in the dilation 2*P* is a sum of two
lattice points in *P*. With 2 replaced by *k* for all *k>2*, this asks whether
the corresponding polarized toric variety is projectively normal. An
important open case is whether this property holds whenever the toric
variety is smooth.

A closely related question on a smooth projectively normal toric variety is whether it is necessarily defined by quadrics. A stronger property would be the existence of quadratic regular unimodular triangulation of the underlying polytope, in which case the homogenous coordinate ring of the variety is even Koszul. We will also search for broader combinatorial interpretations of this property.

If the polytope *P* is sufficiently dilated such a (quadratic/regular)
unimodular triangulation is known to exist, but the proof is not
effective. We will seek a concrete bound on the constant *c* for which the
polytope *cP* has these properties. It is unclear whether there is a
uniform bound depending only on the dimension.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Few smooth d-polytopes with N lattice points

by Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Francisco Santos, and Hal Schenck

Convex normality of rational polytopes with long edges

by Joseph Gubeladze

A simple combinatorial criterion for projective toric manifolds with dual defect

by Alicia Dickenstein and Benjamin Nill