Jörg Wills -Universität Siegen

Lattice-Polytopes - old and new

Abstract: Estimates for the number of lattice points in convex bodies have a long tradition and trace back to Gauss and Minkowski. Moreover there are various applications to integer optimization.

Erhart's polynomial (1962) counts the number of lattice points in lattice polytopes and their integer multiples, and bridges convex and discrete geometry. Since its discovery there were many investigations to understand this polynomial. In recent years one has focussed on the geometric interpretation of the complex zeros of the Erhart polynomial. We present new results and open problems.

Colloquium - 16:00

Florian Zickfeld -Technische Universität Berlin

Representations of Orthogonal Surfaces

Abstract: We investigate different representations of orthogonal surfaces.

Orthogonal surfaces are closely related with Schnyder Woods on 3-connected planar graphs. Given a Schnyder Wood, we get an associated orthogonal surface by counting faces.

An orthogonal surface is coplanar if all the generating minima lie on a plane of the form x+y+z=c. We show that given a "normalized" coplanar surface, and a Schnyder Wood, we can find weights for the bounded faces of the Schnyder Wood such that weighted face counting yields this surface.

The other setting we consider is the following: for every minimum and maximum m on the orthogonal surface, we are given the height h(m)=m_x+m_y+m_z. We conjecture that this information together with the combinatorial information about the surface uniquely determines the surface. We give a proof for stacked triangulations and present some approaches for the general problem.