# Graduiertenkolleg: Methods for Discrete Structures

Deutsche Forschungsgemeinschaft
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# Monday Lecture and Colloquium

Monday, January 25, 2010

Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136
room MA 041
10623 Berlin

Lecture - 14:15

### Lattice Packing and Covering

Abstract:
The lecture deals with lattice packing and covering of Euclidean balls and smooth convex bodies in Euclidean d-space. After a short survey of classical results, the following topics will be investigated in more detail:
Uniqueness of densest lattice packings Characterizations of lattice packings with refined extremum properties of the density Characterizations of lattice coverings of balls with refined extremum properties of the density Kissing numbers

The characterizations are in the sense of Voronoi's classical criterion: a lattice packing of balls has maximum density if and only if it is eutactic and perfect. The results for balls can be expressed also in terms of positive definite quadratic forms (homogeneous and inhomogeneous problem)

Colloquium - 16:00

### Expected Frobenius Numbers

Abstract:
Given a primitive positive integer vector $a\in\mathbb{Z}^n_{>0}$, the largest integer that cannot be represented as a non-negative integer combination of the coefficients of $a$ is called the Frobenius number of $a$. In a series of papers V. Arnold initiated the research to study the average size of Frobenius numbers, and in a recent paper, Bourgain&Sinai showed that the probability of a ''large'' Frobenius number is ''comparable small''. Based on an approach using methods from Geometry of Numbers we can strengthen this result in such a way that we can estimate the average size of Frobenius numbers. Together with a discrete reverse arithmetic-geometric-mean inequality, this allows us to show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound.

The material presented here is part of joint works with Iskander Aliev and Iskander Aliev & Aicke Hinrichs.