Speaker
Description
This talk is based on joint work with Anders Södergren, motivated by the question of whether the densest sphere packings in high dimensions are ordered or disordered. A strong form of translational order is the structure of a lattice (i.e. Bravais lattice), where the packing is just a single sphere per fundamental cell repeated periodically. In high dimensions, there are many possibilities for the shape of the cell, leading to a rich space of lattices. A natural way to explore this space is to choose a lattice at random according to the invariant measure with respect to the special linear group.
We show that, with high probability as the dimension of space grows, there is enough room to insert additional spheres into the packing given by a random lattice. In this sense, most lattices fail quite badly to give dense packings.