The lattice packing problem in dimension 9 by Voronoi’s algorithm

In 1908 Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates the finitely many perfect lattices up to similarity in a fixed dimension. However, due to the high complexity of the algorithm this enumeration had, until now, only been completed up to dimension 8. In this talk we will present our work on a full enumeration of all 2,237,251,040 perfect lattices in dimension 9 via Voronoi's algorithm. As a corollary, this shows that the laminated lattice gives the densest lattice packing in dimension 9. Furthermore, as a byproduct of the computation, we classify the set of possible kissing numbers in dimension 9. We will discuss Voronoi's algorithm and the many algorithmic, implementation, and parallelization efforts that were required for this computation to succeed. This is joint work with Mathieu Dutour Sikirić.