Mathematics Unlimited - 2001 and Beyond PDF
Preview Mathematics Unlimited - 2001 and Beyond
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 WILFRIED SCHMID Center for Parallel Computers Department of Mathematics Royal Institute of Technology (KTH) Harvard University Lindstedtsvagen 25 Cambridge, MA 02138-2901, USA 10044 Stockholm, Sweden schmid @math.harvard.edu [email protected] 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 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 mate rial 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 prob lems 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 math ematics and physics, chemistry, computer science. Can one eminent mathematician, or a small group of eminent mathemati cians, 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 con viction. 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 Bjorn Engquist Cambridge, Massachusetts Wilfried Schmid October 2000 *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 Nonlinear Continuum Physics I Ivo BABUSKA · J. TINSLEY 0DEN Computational Mechanics: Where Is It Going? 23 JOHN c. BAEZ. JAMES DOLAN From Finite Sets to Feynman Diagrams 29 DAVID H. BAILEY Experimental Mathematics: JONATHAN M. BORWEIN Recent Developments and Future Outlook 5 I BADI H. BALTAGI The Mathematical Aspects of Econometrics 67 OLE E. BARNDORFF-NIELSEN RICHARD D. GILL· PETER E. JuPP Quantum Information 83 MASSIMO BERTOLINI The p-Adic £-Functions of Modular HENRI DARMON Elliptic Curves 109 JEAN-PIERRE BouRGUIGNON A Basis for a New Relationship Between Mathematics and Society I7I ROGER BROCKETT New Issues in the Mathematics of Control I89 HERNAN CENDRA JERROLD E. MARSDEN Geometric Mechanics, Lagrangian Reduction, TuDORS. RATIU and Nonholonomic Systems 22I ACHIM BACHEM interviewed 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 L-Functions and Random Matrices 33 I PETER CONSTANTIN Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics 353 MARK DAVIS Mathematics of Financial Markets 36 I ERIK DE SCHUTTER Computational Neuroscience: More Math Is Needed to Understand the Human Brain 38 I RICK DURRETT Probability Theory - An Introduction to Its Applications 393 WEINANE Selected Problems in Materials Science 407 BJORN ENGQUIST . GENE GOLUB From Numerical Analysis to Computational Science 433 GERD FALTINGS Diophantine Equations 449 VIII Contents LENNART CARLESON interviewed by BJORN ENGQUIST After the 'Golden Age': What Next? 455 GERALD FARIN JORG FRAUENDIENER ROGER PENROSE Twistors and General Relativity 4 79 ODED GOLDREICH Computational Complexity 507 MIKHAEL GROMOV Possible Trends in Mathematics in the Coming Decades 525 JOZEF GRUSKA Quantum Computing Challenges 529 PETER HALL Statistical Science-Evolution, Motivation and Direction 565 NIGEL HITCHIN Global Differential Geometry 577 GERHARD HUISKEN Evolution Equations in Geometry 593 CHRISTOPHER R. JOHNSON YARDEN LIVNAT ·LEONID ZHUKOV Computational Field Visualization 605 DAVID HART· GORDON KINDLMANN CHRISTOPHER K. R.T. JONES Whither Applied Nonlinear Dynamics? 63I DAVID EISENBUD interviewed by SARA ROBINSON Mathematics Comes from Many Sources... 647 JAY JORGENSON. SERGE LANG The Ubiquitous Heat Kernel 655 FRANK KELLY Mathematical Modelling of the Internet 685 CLAUDIA KLi'rPPELBERG Developments in Insurance Mathematics 703 TosHIYUKI KoBAYASHI Discontinuous Groups for Non-Riemannian Homogeneous Spaces 723 NEAL KOBLITZ Cryptography 749 MAXIM KoNTSEVICH ·DoN ZAGIER Periods 77I HANS PETTER LANGTANGEN How Should We Prepare the Students of Science and Technology ASLAK TVEITO for a Life in the Computer Age? 809 MALCOLM S. LONGAIR Astrophysics and Cosmology in the 21st Century 827 WOLFGANG LUCK L 2-Invariants and Their Applications to Geometry, Group Theory and Spectral Theory 859 KISHORE B. MARA THE A Chapter in Physical Mathematics: Theory of Knots in the Sciences 873 MATTHIAS KRECK interviewed by VAsco ALEXANDER ScHMIDT The Unity of Mathematics 889 DAVID R. MORRISON Geometric Aspects of Mirror Symmetry 899 YIANNIS N. MOSCHOVAKIS What Is an Algorithm? 919 RISTO M. NIEMINEN From Number Crunching to Virtual Reality: Mathematics, Physics and Computation 937 ALFIO QUARTERONI Modeling the Cardiovascular System: A Mathematical Challenge 961 Contents IX DouG RoBLE · ToNY CHAN Math in the Entertainment Industry 97 I MARIE-FRANc;:orsE RoY Three Problems in Real Algebraic Geometry and Their Descendants 99 I KYOJI SAITO Primitive Automorphic Forms I 003 PETER SCHUSTER Mathematical Challenges from Molecular Evolution I o I 9 DIRK SELLE . WOLF SPINDLER BERNHARD PREIM Mathematical Methods in Medical Imaging: HEINZ-OTTO PEITGEN Analysis of Vascular Structures for Liver Surgery Planning I 039 DENIS SERRE Systems of Conservation Laws: A Challenge for the XXIst Century I o6 I MASATAKE MORI and KYOJI SAITO interviewed RIMS for the 21st Century I o8 I by TOSHIKAZU SUNADA JOEL SPENCER Discrete Probability I 095 DANIEL W. STROOCK Probability Theory-A Somewhat Narrow Perspective I 105 GERARD VANDER GEER Error-Correcting Codes and Curves over Finite Fields I I I5 JAN VAN LEEUWEN Julf WIEDERMANN The Turing Machine Paradigm in Contemporary Computing I I39 ANATOLY M. VERSHIK Randomization of Algebra and Algebraization of Probability - An Attempt at Prediction I I 57 MARCELO VIANA Dynamical Systems: Moving into the Next Century I 167 HANS VON STORCH JIN-SONG VON STORCH Noise in the Climate System-Ubiquitous, Constitutive PETER MULLER and Concealing I 179 GUNTER M. ZIEGLER Questions About Polytopes I I95 Biographies I 2 I 3 Contributors STUART S. ANTMAN University of Aarhus HERNAN CENDRA Department of Mathematics Ny Munkegade Universidad Nacional del Sur and Institute for Physical Science 8000 Aarhus C, Denmark Av. Alem 1254 and Technology oebn @imf.au.dk 8000 Bahia Blanca, Argentina University of Maryland uscendra@ criba.edu.ar College Park, MD 20742-4015, USA MASSIMO BERTOLINI ssa@ math. umd.edu Dipartimento di Matematica TONY CHAN Universita di Pavia Department of Mathematics Ivo BABUSKA Strada Ferrata l University of California at Los Angeles The Texas Institute for Computational 27100 Pavia, Italy 3148 Murphy Hall, 7619A MSB and Applied Mathematics massimo@ dimat. unipv.it Los Angeles, CA 90095-1555 The University of Texas USA Austin, TX 78712, USA JONATHAN BORWEIN chan@ ipam. ucla.edu Centre for Experimental and ACHIM BACHEM Constructive Mathematics (CECM) ARJEH M. COHEN Deutsches Zentrum fur Luft Department of Mathematics Department of Mathematics und Raumfahrt e. V. (DLR) and Statistics and Computing Science Vorstandsbereich Raurnfahrt Simon Fraser University Eindhoven University of Technology Konigswintererstrasse 522-524 Burnaby, BC V5A 1S6, Canada P. 0. Box 513 53227 Bonn, Germany jborwein @cecm.sfu.ca 5600 MB Eindhoven achim. bachem@ dlr.de The Netherlands JEAN-PiERRE BouRGUIGNON [email protected] JOHNC.BAEZ Institut des Hautes Etudes Scientifiques Department of Mathematics Le Bois-Marie HENRI COHEN University of California 35, route des Chartres Riverside, CA 92521, USA 91440 Bures-sur-Yvette, France Laboratoire A2X baez@ math. ucr.edu [email protected] U.M.R. 5465 du C.N.R.S. Universite Bordeaux I DAVID H. BAILEY ROGER BROCKETT 351 Cours de la Liberation 33405 Talence Cedex, France Lawrence Berkeley Electrical Engineering henri.cohen @math.u-bordeaux.fr National Laboratory and Computer Science Berkeley, CA 94720, USA Room 345 [email protected] Maxwell Dworkin Laboratory J. BRIAN CONREY Harvard University American Institute of Mathematics BAD! H. BALTAGI 33 Oxford Street 360 Portage Avenue Texas A & M University Cambridge, MA 02138, USA Palo Alto, CA 94306, USA Department of Economics brockett@ hrl.harvard.edu [email protected] College Station, TX 77843-4228, USA badi @econ. tamu.edu LENNART CARLESON PETER CONSTANTIN Department of Mathematics Department of Mathematics OLE E. BARNDORFF-NIELSEN Royal Institute of Technology (KTH) University of Chicago MaPhySto Lindstedtsvagen 25 5734 University Avenue Center for Mathematical Physics 10044 Stockholm, Sweden Chicago, IL 60637, USA and Stochastics [email protected] const@ math. uchicago.edu
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