Random Lattices as Sphere Packings

Explore the fascinating world of random lattices and sphere packings in this 33-minute lecture by Nihar Gargava from EPFL at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the history of lattice-sum functions, starting with Siegel's 1945 discovery of the expected value over unit covolume lattices. Learn about Venkatesh's 2012 extension to lattices with cyclic symmetry groups and its impact on optimal sphere packing density bounds. Examine new analogues of Siegel's mean value theorem and their applications in improving lattice packing bounds across multiple dimensions. Discover how these results can be made effective and investigate various extensions of the theory. Gain insights into the joint work of Nihar Gargava and Vlad Serban, pushing the boundaries of our understanding of random lattices and their role in sphere packing problems.