WHEN: 22.01.15 at 14:15
WHERE: Seminar Room, Arnimallee 2, FU Berlin

Speaker: Matthias Henze (FU Berlin)

Congruence Arguments in the Geometry of Numbers and a General
Discrete Minkowksi-type Theorem

Abstract: One of the most fruitful results from Minkowski's geometric
viewpoint on number theory is his so called 1st Fundamental Theorem.
It says that the volume of every o-symmetric n-dimensional convex body
whose only interior lattice point is the origin is bounded from above
by the volume of the orthogonal n-cube of edge length two. Minkowski
also obtained a discrete analog by identifying the n-cube as a
maximizer of the number of lattice points in the boundary of such
convex bodies. Whereas the volume inequality has been generalized to
any number of interior lattice points already by van der Corput in the
1930s, a corresponding result for the discrete case remained to be
proven. Using congruence arguments for lattice points and an
inequality in additive combinatorics, we determine a best possible
relation of this kind. In the talk, we will moreover highlight the
usefulness of considering congruences on lattice points in the
geometry of numbers.
This is joint work with Bernardo González Merino.