Preface

The Geometry of Positive Definite Quadratic Forms is a rich and old subject

which arose in the arithmetic studies of quadratic forms. Through the seminal

works of Minkowski and Voronoi a century ago, the geometric viewpoint became

predominant. The study of arithmetical and inhomogeneous minima of positive

definite quadratic forms turned into a study of lattice sphere packings and cover-

ings. Lattices and, more generally, periodic (point) sets are by now widespread in

mathematics and its applications. The important monograph “Sphere packings,

Lattices and Groups” [69] by Conway and Sloane, with its over 100 pages of ref-

erences, shows exemplarily the influence on other mathematical disciplines. This

becomes particularly apparent for the 24-dimensional Leech lattice and its connec-

tions to number theory, group theory, coding theory and mathematical physics.

Since the complexity of problems grows with the dimension, it is no surprise that

over the past decades more and more computer support was used to study higher

dimensional lattices and more general structures. Still, the Geometry of Positive

Definite Quadratic Forms is an essential tool, not only in the study of lattice sphere

packings and coverings.

One aim of this book is to give a nearly self-contained introduction to this

beautiful subject. We present the known material with new proofs, which then

admit natural generalizations. These extentions of the known theory were mainly

targeted to support the study of extreme periodic sets. However, it turned out

that the resulting new theory has other applications as well, as for example, the

classification of totally real thin number fields. On the way, always an eye is kept on

computability; algorithms are developed that allow computer assisted treatments.

Using tools from combinatorial, from linear and from convex optimization, many

diﬃcult problems become accessible now. This is, for example, demonstrated in the

search for new currently best-known lattice sphere coverings and in the classification

of 8-dimensional perfect lattices, which previously was thought to be impossible

with the known methods.

Although this book deals with classical topics which have been worked on ex-

tensively by numerous authors, it shows exemplarily how computers may help to

gain new insights. On the one hand it is shown how computer assisted (sometimes

heuristic) exploration helps to discover new exceptional structures. In many cases

these would probably not have been found without a computer. On the other hand

several computer assisted proofs are given, which deal with extraordinarily large

data or involve large enumerations. It is shown how proofs can be obtained from

numerical results, by postprocessing of roundoff solutions. All of these aspects of

computer mathematics are nowadays supported by a growing functionality of com-

puter algebra systems and by an increasing number of reliable small programs for

specific purposes. In some cases one has to combine, to supplement and to improve

on existing software tools. If solutions for basic tasks are obtained they should be

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