Workshop E2: Experimentation with, construction of, and enumeration of optimal geometric structures

Date: March 25 - 28, 2008
Venue: HIM lecture hall, Poppelsdorfer Allee 45
Organizers: Henry Cohn, Mathieu Dutour Sikiric, Achill Schürmann, Frank Vallentin

In this workshop we explored methods for constructing conjectural optimal point configurations for different extremality problems. In some cases, computer assisted enumeration and classification of "locally extreme" structures enables to give computational proofs of difficult theorems.

Topics and Goal:
Remarkable structures, such as universally optimal spherical codes, can be found by computer searches based on simulating energy minimization. Currently, this approach is limited to "small" examples. So it is desirable to improve on current computational techniques to find new structures, e.g. new best-known kissing configurations. Moreover, it is desirable to develop similar tools for Kelvin's problem and optimal spherical coverings. We want to develop new computational tools for spherical t-designs. For example, currently there is no known systematic way to prove that a spherical t-design is locally unique. Spherical and Euclidean t-designs can be used for example for cubature formulas. Based on a novel algorithm due to Dutour and Rybnikov (2007), one can classify and search for extreme Delaunay polytopes. Is it possible to solve the lattice covering problem in dimension 6, based on this? We want to study the relation of extreme Delaunay polytopes and local lattice covering maxima. Using this one can hope for a computer assisted proof of an open number theoretical conjecture of Minkowski. The classification of perfect forms allows in principle a solution of the lattice sphere packing problem in a given dimension. Is it possible to extend the successful classification from 8 to 9 dimensions? Going to higher dimensions, one can try to obtain classification results for more restrictive notions of perfectness.