Random Lattices with Symmetries

What is the densest lattice sphere packing in the d-dimensional Euclidean space? In this talk we will investigate this question as dimension d goes to infinity and we will focus on the lower bounds for the best packing density, or in other words on the existence results. I will give a historical overview of the existence of dense lattice packings by H. Minkowski, E. Hlawka, C. L. Siegel, C. A. Rogers,  and more recently by S. Vance and A. Venkatesh. For a long time the best asymptotic  lower bounds on packing density were known for lattice packings. Recently, a new asymptotic lower bound for sphere packings in high dimensions was proven by M.Campos, M. Jensen, M. Michelin, and J. Sahasrabudhe. This new bound is proven by a combinatorial method and a dense packing guaranteed by their theorem is not necessarily a lattice packing. In the final part of the lecture I will present a recent work done in collaboration with V. Serban and N. Gargava, and Ilaria Viglino on the moments of the number of lattice points in a bounded set for random lattices constructed from a number field.