Report

This has been a basic research project in mathematics, with the goal of combining the fellow’s expertise and the human resources already available at the Discrete Geometry group of Freie Universität Berlin (and other groups both at FU Berlin and TU Berlin), plus the new members funded via the Einstein Fellowship.

The research topics are among the current hot trends in Discrete Geometry, and fall in two research lines: the combinatorics and geometry of lattice polytopes and simplicial
and cell complexes.

Polytopes, the technical word for what we commonly call polyhedra, are among the most ancient objects of study by mathematicians, dating back to mesopotamian and greek scholars. In three dimensions our knowledge of them is quite complete, but in higher dimensions there are many things we would like to understand better. Some of the questions studied in the project were:
• How large can the combinatorial diameter of a polytope be, in terms of its number of facets? The Hirsch conjecture (stating that a number of steps equal to the number of facets allows one to travel from any vertex to any other vertex) was disproved by the Fellow in 2010 but the polynomial version of it is wide open, and would have important consequences for the complexity analysis of linear programming, in particular to the 9th of Steven Smale’s “mathematical problems for the 21st century”.
• Lattice polytopes, that is, polytopes with integer coordinates for their vertices, are important in algebraic geometry (they are the Newton polytopes of multivariate polynomials) and optimization (they are the feasibility regions of integer linear programs). In both contexts different geometric properties are of interest. We have studied questions such as the maximum width they can have in fixed dimension if you forbid them from having interior lattice points, how much do you need to dilate them in order to cover space with them via integer translations, or a lattice analogue of Hilbert’s third problem: what conditions are necessary and sufficient for two lattice polytopes to have the property that one can be cut in a finite number of pieces and the pieces rearranged to form the other one. In the context of lattice polytopes the natural meaning of “rearranged” is that unimodular transformations (affine transformations preserving the lattice) are allowed.
• A simplicial complex can be thought of as a combinatorial abstraction of the (partially ordered) set of faces of a poytope. We have studied extremal questions of them, in particular an Erdös-Ko-Rado property that we conjecture all pure, flag simplicial complexes to have. This would be a vast generalization of a Conjecture of Kalai regarding the associahedron (a triangulated sphere whose facets are in bijection to complete binary trees and other structures, that appears profusely in Coxeter combinatorics). We have also studied separation properties of 3- dimensional spheres, aiming at using tem to get lower bounds on the $f$-vectors of some special classes of triangulated 4-manifolds (stacked, neighborly, uniform, transitive, etc.).

Four international workshops were organized by the Einstein group, with about 350 participants in total. The project has led to three PhD theses: Jorge Olarte (FU-Berlin, 2019), Giulia Codenotti (FU-Berlin, 2020) and Francisco Criado (TU-Berlin, 2020), and more than 40 publications.