This award supports the continuing research of Jason Rush at the University of Washington. In prior work, Rush introduced a family of metrics, the use of which has led to improvement of the Minkowski-Hlawka lower bounds on lattice packing density for various families of bodies in Euclidean n-space. In this project, he will continue the study of properties of these metrics and their application to problems of constructing dense lattice packings for additional types of geometric bodies, to multiple packing and covering, and to the construction of error-correcting codes. This research is in the field of the geometry of numbers. The geometry of numbers is a blending of number theory - the study of properties of the whole numbers - and the geometry of variously shaped bodies in Euclidean n-space. An example of a fundamental problem in this field is the determination of what configuration of spheres in Euclidean n-space will contain the maximum possible volume. Such sphere packings are closely related to efficient error-correcting codes, such as those used to store data on compact discs, and also have applications to problems in information transmission which occur for example in the construction of modems.
