Event time:
Thursday, January 29, 2026 - 4:00pm
Speaker:
Boaz Klartag
Speaker affiliation:
Weizmann Institute
Event description:
We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose density is at least c n^2 / 2^n. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of at most n log n / 2^n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.