HIM Junior Trimester Program - Computational Mathematics
Exterme Geometric Structures
H. Cohn, M. Dutour Sikirić, A. Schürmann, F. Vallentin

Workshop
Experimentation with, construction of, and enumeration of optimal geometric structures
March 25-28, 2008

Topics and Goals

In this workshop we want to explore 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.

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.

Participants

Frank Bowert, David Bremner, Henry Cohn, Renaud Coulangeon, Michel Deza, Mathieu Dutour Sikirić, Viatcheslav Grishukhin, Jonathan Hanke, Abhinav Kumar, Jacques Martinet, Gabriele Nebe, Cordian Riener, Konstantin Rybnikov, Rudolf Scharlau, Achill Schürmann, Frank Vallentin, Stephanie Vance, Boris Venkov

Scientific program

Tuesday, March 25
10h30-11h30Abhinav Kumar (MIT)
Optimal structures and energy minimization
14h00-15h00Jonathan Hanke (Duke University)Extremal problems for modular forms, quadratic forms, and lattices
17h00-18h00Achill Schürmann (University of Magdeburg)Computational geometry of positive definite quadratic forms

Wednesday, March 26
9h30-10h30Mathieu Dutour Sikiriç (Institute Ruđer Bošković, Zagreb)Practical polyhedral computations under symmetry
11h00-11h40David Bremner (University of New Brunswick)Pivoting under symmetry and symmetric triangulations
13h30-14h15Henry Cohn (Microsoft Research)Unexpected constructions of extremal structures

Thursday, March 27
9h30-10h30Gabriele Nebe (RWH Aachen), Boris Venkov (Steklov Institute St. Petersburg)
Strongly perfect lattices
11h00-11h45Michel Deza (Ecole Normale Supérieure)Some quasi-metrics and orientations of hypercubes
13h30-14h10Bertrand Meyer (University of Bordeaux)Generalised Hermite constants and Voronoi theory
14h20-15h00Stephanie Vance (University of Washington)Mordell's inequality and Hurwitz lattices
16h00-16h40Kostantin Rybnikov (University of Massachusetts Lowell)Geometry in the large (Geometrie im Großen) approach to the packing problem

Friday, March 28
9h30-10h30Renaud Coulangeon (University of Bordeaux)
Spherical designs and zeta functions of lattices
11h00-11h40Viatcheslav Grishukhin (CEMI Russian Academy of Sciences, Moscow)Extremal properties of the root lattice E6 and its dual
13h30-14h10Rudolf Scharlau (University of Dortmund)Remarks on packings by translates of the Barnes-Wall lattice
14h20-15h20Jacques Martinet (University of Bordeaux)The index of a well-rounded sublattice

Problems

Viatcheslav Grishukhin (CEMI Russian Academy of Sciences, Moscow)A special case of Voronoi's conjecture
Achill Schürmann (University of Magdeburg)Lattice vs. non-lattice sphere packings and coverings

Photos