#
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**

###
Peter Gruber - Wien

### 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**

###
Martin Henk - Magdeburg

###
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.