made accessible to the growing community of computer enthusiastic mathemati-
cians. Underlying many of the presented computational results are in particular
two such programs: A program for rigorous determinant maximization (including
semidefinite programming) allowing exact certified error bounds, and secondly, a
program for polyhedral representation conversion under symmetries.
Computer assisted mathematical explorations and proofs are of increasing im-
portance in many areas of modern mathematics. Even close to the topics of this
book there have been amazing developments recently. An example is the proof by
Hales [129], [130] of the famous Kepler conjecture. Several exciting results have
been obtained in the context of linear and semidefinite programming bounds for
spherical codes and point sets in Euclidean spaces. There is the new sphere pack-
ing bound by Cohn and Elkies [60], and based on it, the proof by Cohn and Kumar
[63] (see also [61]) that the Leech lattice gives the best lattice sphere packing in
24 dimensions. There is the proof of Musin [185] showing that the kissing number
in four dimensions is 24 (see [197] for an excellent survey). Shortly after, Bachoc
and Vallentin [6], gave more general, new bounds on the size of spherical codes.
Their works are followed by similar approaches for other problems, using semidef-
inite programming. As in some parts of this book, these works involve numerical
computations which are then turned into mathematical rigorous proofs. Often nu-
merical quests and subsequent mathematical analysis lead to new mathematical
insights. A fascinating example is the study of universally optimal point configura-
tions, recently invoked by Cohn and Kumar [62] (see also [9] and [264]). Although
all of this is happening literally next door to the topics of this book, I decided to
keep it focused as it is. Adequate treatments will hopefully fill other books in the
near future. For now I encourage the reader to study the great original works.
Acknowledgments. This book grew out of lectures held at an Oberwolfach
Seminar on Sphere Packings and at the University of Magdeburg, together with
parts of research articles which were previously published, in a similar or partially
different form (see [45], [225], [226], [228], [229]). I thank my coauthors
David Bremner, Francisco Santos Leal and in particular Mathieu Dutour Sikiri´c
and Frank Vallentin for their many contributions and their shared enthusiasm for
the subject. I thank Frank, Mathieu, Henry Cohn, Slava Grishukhin, Jeff Lagarias
and Jacques Martinet for their very helpful feedback on prior versions.
I am grateful to my teachers Ulrich Betke and org M. Wills and thank many
other colleagues for fruitful communications on topics related to this book; among
them David Avis, Christine Bachoc, Eiichi Bannai, Yves Benoist, Anne-Marie
Berg´ e, Andras and Karoly Bezdek, Karoly or¨ ozky Jr. and Sr., Jin-Yi Cai, Bob
Connelly, Renaud Coulangeon, Jesus DeLoera, Antoine and Michel Deza, Nikolai
P. Dolbilin, Noam Elkies, Bob Erdahl, Komei Fukuda, Lenny Fukshansky, Rajinder
J. Hans-Gill, Robert L. Griess Jr., Peter Gritzmann, Peter M. Gruber, Thomas C.
Hales, Jonathan Hanke, Martin Henk, Jen¨ o Horvath, Michael Joswig, Abhinav Ku-
mar, Wlodzimierz Kuperberg, Peter McMullen, Oleg Musin, Gabriele Nebe, Cor-
dian Riener, Konstantin Rybnikov, Rudolf Scharlau, Claus-Peter Schnorr, Fran¸cois
Sigrist, Warren D. Smith, Sal Torquato, Stephanie Vance, Boris Venkov, unter M.
Ziegler, Chuanming Zong and Stefan van Zwam. I thank the Deutsche Forschungs-
gemeinschaft (DFG) and the AMS editors for their support.
September 2008 Achill Sch¨urmann
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