x PREFACE

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 J¨ 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 B¨ 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, G¨ 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

[95],