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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