Bjorn Engquist. Wilfried Schmid Editors

MATHEMATICS UNLIMITED -200I AND BEYOND

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Bjorn Engquist Wilfried Schmid Editors

MATHEMATICS UNLIMITED-2001 AND BEYOND With 179 Figures Including 95 in Colour 91 Portraits and II Tables

Springer

BJORN ENGQUIST

Center for Parallel Computers Royal Institute of Technology (KTH) Lindstedtsvagen 25 10044 Stockholm, Sweden engquist@nada.kth.se and Department of Mathematics University of California at Los Angeles 3148 Murphy Hall, 7619A MSB Los Angeles, CA 90095-1555, USA engquist@ math.ucla.edu

WILFRIED SCHMID

Department of Mathematics Harvard University Cambridge, MA 02138-2901, USA schmid @math.harvard.edu

A Collector's Edition of this book was produced under the same title as a set in 2 volumes with the ISBN 3-540-67099-8

Library of Congress Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek- CIP-Einheitsaufnahme Mathematics unlimited- 2001 and beyond/Bjiim Engquist; Wilfried Schmid ed.­Berlin; Heidelberg; New York; Barcelona; Hong Kong ; London; Milan; Paris ; Singapore; Tokyo: Springer, 2001 ISBN-13: 978-3-540-66913-5 e-ISBN-13: 978-3-642-56478-9 DOl: 10.1007/978-3-642-56478-9

Mathematics Subject Classification (2000): 00-02, 03-XX, 05-XX, 11-XX, 14-XX, 34-XX, 35-XX, 46-XX, 47-XX, 49-XX, 51-XX, 52-XX, 53-XX, 55-XX, 57-XX, 58-XX, 60-XX, 65-XX, 68-XX, 70-XX, 73-XX, 76-XX, 81-XX, 83-XX, 85-XX, 90-XX, 92-XX, 94-XX

This work is subject to copyright. All rights are reserved, whether the whole or part of the mate­rial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+ Media Business GmbH

©Springer-Verlag Berlin Heidelberg 2001 Softcover reprint of the hardcover 1st edition 2001 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective Jaws and regulations and therefore free for general use.

Production details are found at the end of the book Printed on acid-free paper SPIN 10862218 46/3111- 5 4 3 2 1

Preface

Ein neues Lied, ein besseres Lied, 0 Freunde, will ich Euch dichten! *

A new song, a better song, My friends, I intend to compose you!

What are the important developments in present-day mathematics? Where is mathematics headed? Our anthology attempts to shed light on these questions.

At the dawn of the 20th century, it was possible for one sage individual to survey the whole of mathematics: Hilbert's presentation of twenty-three prob­lems in 1900 not only gave a sense of the direction of mathematics, but also helped it move forward.

The scope of mathematics has expanded enormously over the last hundred years. Scientific and technological advances, in particular the explosive growth of computing power, have created numerous opportunities for mathematics and mathematicians. The core areas of mathematics did not suffer as a result of the proliferating applications; quite to the contrary, "pure mathematics" is thriving, with the invention of powerful theories, the solution of celebrated problems, and the emergence of unforeseen connections between different areas of math­ematics and physics, chemistry, computer science.

Can one eminent mathematician, or a small group of eminent mathemati­cians, afford an overview of the breadth of today's mathematics? We think not, we present a composite of many individual views, out of necessity and con­viction. The contributors are leading experts in their fields, ranging all the way from mathematical logic to applications of mathematics in the film industry.

Our volume is not an encyclopedia. Completeness was not the main goal, nor was it an achievable goal. We do hope to provide the reader with a view of the great variety and the vitality of mathematics as we enter the new millennium.

Los Angeles Cambridge, Massachusetts October 2000

Bjorn Engquist Wilfried Schmid

*Taken from Heinrich Heine (1797-1856): Deutschland. Ein Wintermiirchen. We thank Georgina Paul, Department of German Studies, University of Warwick, Warwick, for the translation.

Contents

STUART S. ANTMAN

Ivo BABUSKA · J. TINSLEY 0DEN

JOHN c. BAEZ. JAMES DOLAN

DAVID H. BAILEY JONATHAN M. BORWEIN

BADI H. BALTAGI

OLE E. BARNDORFF-NIELSEN RICHARD D. GILL· PETER E. JuPP

MASSIMO BERTOLINI HENRI DARMON

JEAN-PIERRE BouRGUIGNON

ROGER BROCKETT

HERNAN CENDRA JERROLD E. MARSDEN TuDORS. RATIU

ACHIM BACHEM interviewed

Nonlinear Continuum Physics I

Computational Mechanics: Where Is It Going? 23

From Finite Sets to Feynman Diagrams 29

Experimental Mathematics: Recent Developments and Future Outlook 5 I

The Mathematical Aspects of Econometrics 67

Quantum Information 83

The p-Adic £-Functions of Modular Elliptic Curves 109

A Basis for a New Relationship Between Mathematics and Society I7I

New Issues in the Mathematics of Control I89

Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems 22I

by VAsco ALEXANDER SCHMIDT Mathematics: From the Outside Looking in 275

ARJEH M. CoHEN Communicating Mathematics Across the Web 283

HENRI CoHEN Computational Aspects of Number Theory 30I

J. BRIAN CONREY

PETER CONSTANTIN

MARK DAVIS

ERIK DE SCHUTTER

RICK DURRETT

WEINANE

BJORN ENGQUIST . GENE GOLUB

GERD FALTINGS

L-Functions and Random Matrices 33 I

Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics 353

Mathematics of Financial Markets 36 I

Computational Neuroscience: More Math Is Needed to Understand the Human Brain 38 I

Probability Theory - An Introduction to Its Applications 393

Selected Problems in Materials Science 407

From Numerical Analysis to Computational Science 433

Diophantine Equations 449

VIII Contents

LENNART CARLESON interviewed by BJORN ENGQUIST After the 'Golden Age': What Next? 455

GERALD FARIN

JORG FRAUENDIENER ROGER PENROSE

ODED GOLDREICH

MIKHAEL GROMOV

JOZEF GRUSKA

PETER HALL

NIGEL HITCHIN

GERHARD HUISKEN

CHRISTOPHER R. JOHNSON

Twistors and General Relativity 4 79

Computational Complexity 507

Possible Trends in Mathematics in the Coming Decades 525

Quantum Computing Challenges 529

Statistical Science- Evolution, Motivation and Direction 565

Global Differential Geometry 577

Evolution Equations in Geometry 593

YARDEN LIVNAT ·LEONID ZHUKOV Computational Field Visualization 605 DAVID HART· GORDON KINDLMANN

CHRISTOPHER K. R.T. JONES

DAVID EISENBUD interviewed by SARA ROBINSON

JAY JORGENSON. SERGE LANG

FRANK KELLY

CLAUDIA KLi'rPPELBERG

TosHIYUKI KoBAYASHI

NEAL KOBLITZ

Whither Applied Nonlinear Dynamics? 63I

Mathematics Comes from Many Sources... 647

The Ubiquitous Heat Kernel 655

Mathematical Modelling of the Internet 685

Developments in Insurance Mathematics 703

Discontinuous Groups for Non-Riemannian Homogeneous Spaces 723

Cryptography 749

MAXIM KoNTSEVICH ·DoN ZAGIER Periods 77I

HANS PETTER LANGTANGEN ASLAK TVEITO

MALCOLM S. LONGAIR

WOLFGANG LUCK

KISHORE B. MARA THE

MATTHIAS KRECK interviewed

How Should We Prepare the Students of Science and Technology for a Life in the Computer Age? 809

Astrophysics and Cosmology in the 21st Century 827

L 2-Invariants and Their Applications to Geometry, Group Theory and Spectral Theory 859

A Chapter in Physical Mathematics: Theory of Knots in the Sciences 873

by VAsco ALEXANDER ScHMIDT The Unity of Mathematics 889

DAVID R. MORRISON

YIANNIS N. MOSCHOVAKIS

RISTO M. NIEMINEN

ALFIO QUARTERONI

Geometric Aspects of Mirror Symmetry 899

What Is an Algorithm? 919

From Number Crunching to Virtual Reality: Mathematics, Physics and Computation 937

Modeling the Cardiovascular System: A Mathematical Challenge 961

DouG RoBLE · ToNY CHAN

MARIE-FRANc;:orsE RoY

KYOJI SAITO

PETER SCHUSTER

DIRK SELLE . WOLF SPINDLER

BERNHARD PREIM

HEINZ-OTTO PEITGEN

DENIS SERRE

MASATAKE MORI

and KYOJI SAITO interviewed by TOSHIKAZU SUNADA

JOEL SPENCER

DANIEL W. STROOCK

GERARD VANDER GEER

JAN VAN LEEUWEN

Julf WIEDERMANN

ANATOLY M. VERSHIK

MARCELO VIANA

HANS VON STORCH

JIN-SONG VON STORCH

PETER MULLER

GUNTER M. ZIEGLER

Contents

Math in the Entertainment Industry 97 I

Three Problems in Real Algebraic Geometry and Their Descendants 99 I

Primitive Automorphic Forms I 003

Mathematical Challenges from Molecular Evolution I o I 9

Mathematical Methods in Medical Imaging: Analysis of Vascular Structures for Liver Surgery Planning I 039

Systems of Conservation Laws: A Challenge for the XXIst Century I o6 I

RIMS for the 21st Century I o8 I

Discrete Probability I 095

Probability Theory- A Somewhat Narrow Perspective I 105

Error-Correcting Codes and Curves over Finite Fields I I I5

The Turing Machine Paradigm in Contemporary Computing I I39

Randomization of Algebra and Algebraization of Probability -An Attempt at Prediction I I 57

Dynamical Systems: Moving into the Next Century I 167

Noise in the Climate System- Ubiquitous, Constitutive and Concealing I 179

Questions About Polytopes I I95

Biographies I 2 I 3

IX

Contributors

STUART S. ANTMAN Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015, USA ssa@ math. umd.edu

Ivo BABUSKA The Texas Institute for Computational and Applied Mathematics The University of Texas Austin, TX 78712, USA

ACHIM BACHEM Deutsches Zentrum fur Luft­und Raumfahrt e. V. (DLR) Vorstandsbereich Raurnfahrt Konigswintererstrasse 522-524 53227 Bonn, Germany achim. bachem@ dlr.de

JOHNC.BAEZ Department of Mathematics University of California Riverside, CA 92521, USA baez@ math. ucr.edu

DAVID H. BAILEY Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA dhbailey@lbl.gov

BAD! H. BALTAGI Texas A & M University Department of Economics College Station, TX 77843-4228, USA badi @econ. tamu.edu

OLE E. BARNDORFF-NIELSEN MaPhySto Center for Mathematical Physics and Stochastics

University of Aarhus Ny Munkegade 8000 Aarhus C, Denmark oebn @imf.au.dk

MASSIMO BERTOLINI Dipartimento di Matematica Universita di Pavia Strada Ferrata l 27100 Pavia, Italy massimo@ dimat. unipv.it

JONATHAN BORWEIN Centre for Experimental and Constructive Mathematics (CECM) Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6, Canada jborwein @cecm.sfu.ca

JEAN-PiERRE BouRGUIGNON Institut des Hautes Etudes Scientifiques Le Bois-Marie 35, route des Chartres 91440 Bures-sur-Yvette, France jpb@ihes.fr

ROGER BROCKETT Electrical Engineering and Computer Science Room 345 Maxwell Dworkin Laboratory Harvard University 33 Oxford Street Cambridge, MA 02138, USA brockett@ hrl.harvard.edu

LENNART CARLESON Department of Mathematics Royal Institute of Technology (KTH) Lindstedtsvagen 25 10044 Stockholm, Sweden carleson@math.kth.se

HERNAN CENDRA Universidad Nacional del Sur Av. Alem 1254 8000 Bahia Blanca, Argentina uscendra@ criba.edu.ar

TONY CHAN Department of Mathematics University of California at Los Angeles 3148 Murphy Hall, 7619A MSB Los Angeles, CA 90095-1555 USA chan@ ipam. ucla.edu

ARJEH M. COHEN Department of Mathematics and Computing Science Eindhoven University of Technology P. 0. Box 513 5600 MB Eindhoven The Netherlands amc@win.tue.nl

HENRI COHEN Laboratoire A2X U.M.R. 5465 du C.N.R.S. Universite Bordeaux I 351 Cours de la Liberation 33405 Talence Cedex, France henri.cohen @math.u-bordeaux.fr

J. BRIAN CONREY American Institute of Mathematics 360 Portage Avenue Palo Alto, CA 94306, USA conrey@aimath.org

PETER CONSTANTIN Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637, USA const@ math. uchicago.edu

XII

HENRIDARMON

McGill University Mathematics Department Burnside Hall 805 Sherbrooke Street West Montreal, QC H3A-2K6, Canada darmon @math.mcgill.ca

MARK H. A. DAVIS

Department of Mathematics Imperial College London SW7 2BZ, Great Britain mark.davis@ ic.ac. uk

ERIK DE SCHUTTER

Theoretical Neurobiology Born-Bunge Foundation University of Antwerp Universiteitsplein 1 2610 Antwerp, Belgium erik@bbf.uia.ac.be

JAMES DOLAN

Department of Mathematics University of California Riverside, CA 92521, USA jdolan@ math. ucr.edu

RICK DURRETT

Department of Mathematics 523 Malott Hall Cornell University Ithaca, NY 14853, USA rtdl @comell.edu

WEINAN E

Department of Mathematics and Program in Applied and Computational Mathematics Princeton University 218 Fine Hall Princeton, NJ 08544-1000, USA weinan@ math.princeton.edu and Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012, USA

DAVID EISENBUD

Mathematical Sciences Research Institute (MSRI) 1000 Centennial Drive Berkeley, CA 94720, USA de@msri.org

Contributors

BJORN ENGQUIST

Center for Parallel Computers Royal Institute of Technology (KTH) Lindstedtsvagen 25 10044 Stockholm, Sweden engquist@nada.kth.se and Department of Mathematics University of California at Los Angeles 3148 Murphy Hall, 7619A MSB Los Angeles, CA 90095-1555 USA engquist@ math. ucla.edu

GERD FALTINGS

Max-Planck-Institut fiir Mathematik Vivatsgasse 7 53111 Bonn, Germany gerd@mpim-bonn.mpg.de

GERALD FARIN

Computer Science and Engineering Arizona State University Tempe, AZ 85287-5406, USA farin@asu.edu

JORG FRAUENDIENER

lnstitut fiir Theoretische Astrophysik Universitat Tiibingen Auf der Morgenstelle 10 72076 Tiibingen, Germany joergf@tat.physik.uni-tuebingen.de

RICHARD D. GILL

Mathematical Institute University of Utrecht Box 80010 3058 TA Utrecht, The Netherlands gill @math.ruu.nl

ODED GOLDREICH

Department of Computer Science Weizmann Institute of Science Rehovot, Israel oded@ wisdom. weizmann.ac.il

GENE GOLUB

Stanford University Computer Science Department Gates 2B MC 9025 Stanford, CA 94305-9025, USA golub@ sccm.stanford.edu

MIKHAEL GROMOV

Institut des Hautes Etudes Scientifiques Le Bois-Marie 35, route des Chartres 91440 Bures-sur-Yvette, France gromov@ihes.fr

JOZEF GRUSKA

Faculty of Informatics Masaryk University Botanicka 68a 60200 Bmo, Czech Republic gruska@ informatics.muni.cz

PETER HALL

Centre for Mathematics and its Applications The Australian National University Canberra, A.C.T. 0200 Australia peter. hall@ anu.edu.au

DAVID HART

Scientific Computing and Imaging Institute School of Computing University of Utah Salt Lake City, UT 84112, USA dahart@cs.utah.edu

NIGEL HITCHIN

Mathematical Institute University of Oxford 24-29 St Giles' Oxford OXl 3LB, Great Britain hitchin @maths.ox.ac.uk

GERHARD HUISKEN

Mathematisches Institut Universitiit Tiibingen Auf der Morgenstelle 10 72076 Tiibingen, Germany gerhard.huisken@ uni-tuebingen.de

CHRISTOPHER R. JOHNSON

Scientific Computing and Imaging Institute School of Computing Merrill Engineering Building 50 South Campus Central Dr. Room 3490 University of Utah Salt Lake City, UT 84112-9205, USA crj @cs.utah.edu

CHRISTOPHER JONES

Division of Applied Mathematics Lefschetz Center for Dynamical Systems Brown University 182 George Street Providence, RI 02912, USA ckrtj @cfm.brown.edu

JAY JORGENSON

Department of Mathematics City College of New York I CUNY Convent Avenue at !38th New York, NY 10031, USA jjorgenson @mindspring.com

PETER E. JUPP

School of Mathematics and Statistics University of St Andrews North Haugh St. Andrews, Fife KY16 9SS Great Britain pej @st-andrews.ac.uk

FRANK KELLY

Center for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 OWE, Great Britain F.P.Kell y@ statslab.cam.ac. uk

GORDON KINDLMANN

Scientific Computing and Imaging Institute School of Computing University of Utah Salt Lake City, UT 84112, USA gk@cs.utah.edu

CLAUDIA KLi'rPPELBERG

Center of Mathematical Sciences Munich University of Technology 80290 Munich, Germany cklu@ma.tum.de

TOSHIYUKI KOBAYASHI

Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Kumaba, Muguro Tokyo 153-8914, Japan toshi @ms.u-tokyo.ac.jp

NEAL KOBLITZ

Department of Mathematics Box 354350

Contributors

University of Washington Seattle, WA 98195, USA koblitz@math.washington.edu

MAXIM KONTSEVICH

Institut des Hautes Etudes Scientifiques Le Bois-Marie 35, route de Chartes 91440 Bures-sur-Yvette, France maxim@ihes.fr

MATTHIAS KRECK

Mathematisches Forschungsinstitut Oberwolfach Lorenzenhof 77709 Oberwolfach-Walke, Germany kreck@mfo.de and Mathematisches Institut Universitiit Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg, Germany kreck@mathi.uni-heidelberg.de

SERGE LANG

Yale University Department of Mathematics 10, Hillhouse Avenue P. 0. Box 208283 New Haven, CT 06520-8283, USA

HANS PETTER LANGTANGEN

Department of Informatics University of Oslo P. 0. Box 1080 Blindem 0316 Oslo, Norway hpl @ifi. uio.no

YARDEN LIVNAT

Scientific Computing and Imaging Institute School of Computing University of Utah Salt Lake City, UT 84112, USA livnat@cs.utah.edu

MALCOLM S. LONGAIR

Cavendish Laboratory Madingley Road Cambridge CB3 OHE, Great Britain msl @mrao.cam.ac.uk

WOLFGANG LucK

Fachbereich Mathematik Universitiit Miinster

Einsteinstrasse 62 48149 Miinster, Germany Iueck@ math. uni-muenster. de

KISHORE B. MARA THE

Department of Mathematics · City University of New York Brooklyn, NY 11210-2889 USA kbm@ sci. brooklyn.cuny.edu and Max-Planck-lnstitut fiir Mathematik in den Wissenschaften Inse1strasse 22-26 04103 Leipzig, Germany kishore.marathe@mis.mpg.de

JERROLD E. MARSDEN

Control and Dynamical Systems 107-81 California Institute of Technology Pasadena, CA 91125-8100, USA marsden @cds.caltech.edu

MASATAKE MORI

RIMS Research Institute for Mathematical Sciences Kyoto University 606-8502 Kyoto, Japan mmori @kurims.kyoto-u.ac.jp

DAVID R. MORRISON

Center for Geometry and Theoretical Physics Duke University Durham, NC 27708, USA drm@math.duke.edu

YIANNIS N. MoscHOVAKIS

Department of Mathematics University of California at Los Angeles Los Angeles, CA 90095-1555 USA ynm @math.ucla.edu and Department of Mathematics University of Athens Athens, Greece

PETER MuLLER

Department of Oceanography University of Hawaii I 000 Pope Road Honolulu, HI 96822, USA pmuller@iniki.soest.hawaii.edu

XIII

XIV

RISTO M. NIEMINEN

Laboratory of Physics Helsinki University of Technology 02150 Espoo, Finland mi @fyslab.hut.fi

J. TINSLEY 0DEN

The Texas Institute for Computational and Applied Mathematics The University of Texas Austin, TX 78712, USA

HEINZ-OTTO PEITGEN

Me Vis - Center for Medical Diagnostic Systems and Visualization at the University of Bremen Universitatsallee 29 28359 Bremen, Germany peitgen@mevis.uni-bremen.de

RoGER PENROSE

Mathematical Institute University of Oxford 24-29 St Giles' Oxford OX1 3LB, Great Britain rouse@maths.ox.ac.uk and Gresham College Barnard's Inn Hall, Holbom London EC1N 2HH, Great Britain and Center for Gravitational Physics and Geometry Pennsylvania State University University Park, PA 16802, USA

BERNHARD PREIM

Me Vis - Center for Medical Diagnostic Systems and Visualization at the University of Bremen Universitatsallee 29 28359 Bremen, Germany preim@mevis.uni-bremen.de

ALFIO QUARTERONI

Departement de Mathematiques Ecole Polytechnique Fecterale de Lausanne 1015 Lausanne, Switzerland alfio.quarteroni @epfl.ch

TuDORS. RATIU

Department de Mathematiques Ecole Polytechnique Fecterale de Lausanne 1015 Lausanne, Switzerland tudor.ratiu @epfl.ch

Contributors

SARA ROBINSON

50 Beale St., 14th floor San Francisco, CA 94105, USA aras @cs.berkeley.edu

DouG RoBLE

Digital Domain 300 Rose Avenue Venice, CA 90291, USA doug.roble@d2.com

MARIE-FRANyOISE RoY

IRMAR, CNRS UMR 6625 Universite de Rennes I Campus de Beaulieu 35042 Rennes Cedex, France mfroy@maths.univ-rennes1.fr

KYOJI SAITO

RIMS Research Institute of Mathematical Sciences Kyoto University 606-8502 Kyoto, Japan saito@kurims.kyoto-u.ac.jp

WILFRIED SCHMID

Department of Mathematics Harvard University Cambridge, MA 02138-2901, USA schmid @math.harvard.edu

VASCO ALEXANDER SCHMIDT

Stubenrauchstrasse 40 12161 Berlin, Germany vasco@ zedat.fu-berlin.de

PETER SCHUSTER

Institut fiir Theoretische Chemie und Molekulare Strukturbiologie Universitat Wien Wahringerstrasse 17 1 090 Wien, Austria pks @tbi.univie.ac.at and Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM 87501, USA pks@ santafe.edu

DIRK SELLE

Me Vis -Center for Medical Diagnostic Systems and Visualization at the University of Bremen Universitatsallee 29 28359 Bremen, Germany selle@mevis.uni-bremen.de

DENIS SERRE

UMPA, CNRS UMR # 5669 Ecole Normale Superieure de Lyon 69364 Lyon Cedex 07, France serre@umpa.ens-lyon.fr

JOEL SPENCER

Department of Computer Science Courant Institute of Mathematical Sciences New York University Room 829, 251 Mercer Street New York, NY 10012, USA spencer@cs.nyu.edu

WOLF SPINDLER

Me Vis - Center for Medical Diagnostic Systems and Visualization at the University of Bremen Universitatsallee 29 28359 Bremen, Germany spindler@mevis.uni-bremen.de

DANIEL W. STROOCK

Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA dws@math.rnit.edu

ToSHIKAZU SuNADA

Mathematical Institute Tohoku University Sendai, Miyagi, Japan, 980-8578 sunada@math.tohoku.ac.jp

ASLAK TVEITO

Department of Informatics University of Oslo P. 0. Box 1080 Blindem 0316 Oslo, Norway as1ak@ifi.uio.no

GERARD VAN DER GEER

Korteweg-de Vries Instituut Universiteit van Amsterdam P1antage Muidergracht 24 1018 TV Amsterdam, The Netherlands geer@ science.uva.nl

JAN VAN LEEUWEN

Department of Computer Science Utrecht University Padualaan 14 3584 CH Utrecht, The Netherlands jan@cs.uu.nl

ANATOLY VERSHIK

St. Petersburg Mathematical Institute of Russian Academy of Sciences Fontanka 27 St. Petersburg 191011, Russia vershik@pdmi.ras.ru

MARCELO VIANA

lnstituto de Matematica Pura e Aplicade - IMPA Estrada Dona Castorina, 110 Jardim Botiinico Rio de Janeiro- RJ Brasil vi ana@ impa.br

HANS VON STORCH

Institute of Hydrophysics GKSS Research Center Max-Planck-Strasse 1 21502 Geesthacht, Germany hans.von.storch@gkss.de

JIN-SONG VON STORCH

Institute of Meteorology University of Hamburg Bundesstrasse 55 20146 Hamburg, Germany jinsong. von.storch@ gkss.de

Contributors

JnH WIEDERMANN

Institute of Computer Science Academy of Science of the Czech Republic Pod Vodarenskou vezf 218207 Prague 8, Czech Republic wieder@uivt.cas.cz

DoN ZAGIER

Max-Planck-Institut fiir Mathematik Vivatsgasse 7 53111 Bonn, Germany zagier@mpim-bonn.mpg.de

LEONID ZHUKOV

Scientific Computing and Imaging Institute School of Computing University of Utah Salt Lake City, UT 84112, USA zhukov@cs.utah.edu

GUNTER M. ZIEGLER

Department of Mathematics MA6-2 Technical University of Berlin 10623 Berlin, Germany ziegler@math.tu-berlin.de

Translators and Copyeditors:

AMINIA M. BRUEGGEMANN

263 Morris Avenue Providence, RI 02906, USA

SHULAMIT B. SPAIN-GAYER

42 Falkland Street Glasgow Gl2 9QY Great Britain sbsg@cily.demon.co.uk

CHRIS WEEKS

Academic Translation Services Downeycroft, Virginstow Beaworthy Devon EX21 5EA Great Britain cjw6@tutor.open.ac.uk

STEPHEN S. WILSON

Scientific Translator Technical Translation Services 31 Harp Hill Cheltenham GL52 6PY Great Britain ssw@stephenswilson.co.uk www.techtrans.cwc.net

XV