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Discrete Math

A Reverse Minkowski Theorem 

Noah Stephens-Davidowitz, Princeton/IAS 

Location:  HILL 705
Date & time: Monday, 09 October 2017 at 2:00PM - 3:00PM

Abstract:  A classical problem in the geometry of numbers asks us to estimate how many lattice points lie in some ball around the origin. Minkowski’s celebrated theorem gives us a tight lower bound for this number that depends only on the determinant of the lattice. (One can think of the determinant as the limit of the density of lattice points inside large balls--i.e., the "global density" of the lattice. So, Minkowski’s theorem gives a lower bound on a lattice’s “local density” based on its “global density.”) We resolve a conjecture due to Dadush that gives a nearly tight converse to Minkowski’s theorem—an upper bound on the number of lattice points in a ball that depends only on the determinants of sublattices. This theorem has numerous applications, from complexity theory, to the geometry of numbers, to the behavior of Brownian motion on flat tori.

Based on joint work with Oded Regev.

 

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