Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Lara Pudwell**, Rutgers University, lpudwell {at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Minkowski's 1905 Theorem That Good Lattice Sphere Packings Exist

Speaker: **Neil Sloane**, AT&T Labs

Date: Thursday, November 13, 2008 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

In 1905 Minkowski showed that there are lattice packings of equal spheres that are reasonably dense. This is one of the classical results in the "Geometry of Numbers". I will present one of the standard proofs of this, or rather of Hlawka's generalization. The argument is quite delicate, and predates the probabilistic method in combinatorics and the random coding argument in information theory. It gives some insight into how to choose good random lattices. The talk will contain nothing new, although we are hoping to apply the methods to a problem in data compression.