6th polymake conference and developer meeting at TU Berlin (St. Nicholas special)

David Bremner: Building Algorithmic Polytopes – Slides

In this talk I will discuss ongoing work to develop a compiler from a simple ALGOL-like pseudocode to polynomial sized linear programs. These LPs can compute the output for any input (of a given size) to the corresponding algorithm by a trivial encoding of the input into the objective function. The talk will cover the structure of the inequalities needed to simulate a simple bit-oriented register machine supporting arithmetic and arrays, and a limited kind of integrality guarantee needed to solve these systems as linear, rather than integer linear programs. I'll also give an overview of the current compiler implementation, and time permitting a demo.
This talk shares some background and motivation with my recent MDS seminar on “Succinct linear programs for easy problems”, but it should be accessible to people that missed that, and mostly new to people that saw the previous talk.

Matthias Beck: Very ample and Koszul segmental fibrations – Slides

A lattice polytope P ⊂ Rⁿ is the convex hull of finitely many points in Zⁿ. There is a natural hierarchy of structural sophistication for lattice polytopes, with various concepts motivated from toric geometry and commutative algebra. We will discuss three such concepts in this hierarchy, occupying a point of origin (normality), the bottom (very ampleness), and the top spot (Koszul property). More specifically, we explore a simple construction for lattice polytopes with a twofold aim. On the one hand, we derive an explicit series of very ample 3-dimensional polytopes with arbitrarily large deviation from the normality property, measured via the highest discrepancy degree between the corresponding Hilbert functions and Hilbert polynomials. On the other hand, we describe a large class of Koszul polytopes of arbitrary dimensions, containing many smooth polytopes and extending the previously known class of Nakajima polytopes.
This is joint work with Jessica Delgado, Joseph Gubeladze, and Mateusz Michalek.