Lattice Packing of Spheres in High Dimensions Using a Stochastically Evolving Ellipsoid-Boaz Klartag @videosfromIAS

Uploaded March 2026 | Updated September 2026, 3 weeks ago

Special Member's Seminar
2:00pm|Simonyi 101
Topic: Lattice Packing of Spheres in High Dimensions Using a Stochastically Evolving Ellipsoid
Speaker: Boaz Klartag
Affiliation: Institute for Advanced Study
Date: March 06, 2026

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 greater than 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 n / 2^n, up to logarithmic factors. 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.

Lattice Packing of Spheres in High Dimensions Using a Stochastically Evolving Ellipsoid-Boaz Klartag

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