Spectral bounds in extremal discrete geometry

点击翻译按钮获取中文摘要

Extremal / optimal structures in discrete geometry are fundamental to many areas in mathematics, physics, (quantum) information theory, and materials science. Famous examples are densest packings of spheres, densest packings of tetrahedra, and the chromatic number of the Euclidean plane. The problem of optimal packings of tetrahedra goes back to the ancient Greeks, the sphere packing problem was first mentioned by Kepler, and determining the chromatic number of the plane is known as the Hadwiger-Nelson problem. Despite the long history of these problems, there are only few mathematical tools available to tackle them.Studying extremal structures in discrete geometry, we are facing twobasic tasks:Constructions: How to construct structures which are conjecturally optimal?Obstructions: How to prove that a given structure is indeed optimal?For the constructions researchers in mathematics and engineering found many heuristics which often work well in practice. The main objective of this proposal is the development and the validation of (computational) tools for the obstructions. For the special case ofsphere packings the PI developed a blend of tools coming from infinite-dimensional semidefinite optimization and harmonic analysis,together with computational techniques coming from real algebraic geometry and polynomial optimization. The results obtained arefrequently the best-known, for example for the sphere packing problem, the kissing number problem or the measurable chromatic number of Euclidean space.The aim of this proposal is to go beyond sphere packings to packings of more complex geometric shapes (like tetrahedra) and to go beyond measurable colorings of Euclidean spaces to measurable colorings of more complex geometries (like Riemannian symmetric spaces).To achieve this one has to improve current computational techniques. In particular, the goal is to 1. extend the current methods so that they can deal with more complex geometries,2. build on stronger, computationally more expensive, tools from combinatorial optimization.This will allow to apply mathematical optimization to a much wider range of challenging optimization problems in discrete geometry.