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Bulletin (New Series) of the American Mathematical Society
This journal is devoted to articles of the following types:
Research-Expository Surveys
These are, by definition, papers that present a clear and insightful exposition of signif-icant aspects of contemporary mathematical research. Gibbs lectures, Progress in Math-ematics lectures, and retiring presidential addresses will be included in this section.
Research Reports
These are brief, timely reports on important mathematical developments. They arenormally solicited and often written by a disinterested expert.
Book Reviews
Book Reviews are accepted for publication by invitation only. Unsolicited manuscriptswill not be considered.
Submission information. See Information for Authors at the end of this issue.Publisher Item Identifier. The Publisher Item Identifier (PII) appears at the top
of the first page of each article published in this journal. This alphanumeric string ofcharacters uniquely identifies each article and can be used for future cataloging, searching,and electronic retrieval.
Postings to e-MATH. Articles are posted to e-MATH individually soon after proofis returned from authors and before appearing in an issue.
Subscription information. Bulletin (New Series) of the American Mathematical
Society is published quarterly. The Bulletin is also accessible electronically, startingwith the January 1992 issue, from www.ams.org/publications/. For paper delivery, sub-scription prices for Volume 37 (2000) are $300 list, $240 institutional member, $180 in-dividual member. The subscription price for members is included in the annual dues. Alate charge of 10% of the subscription price will be imposed upon orders received fromnonmembers after January 1 of the subscription year. Subscribers outside the UnitedStates and India must pay a postage surcharge of $8.00; subscribers in India must pay apostage surcharge of $13.00. Expedited delivery to destinations in North America is $12;elsewhere $23.
Back number information. For back issues see the AMS Catalog of Publications.Subscriptions and orders should be addressed to the American Mathematical Society,
P.O. Box 845904, Boston, MA 02284-5904. All orders must be accompanied by payment.
Other correspondence should be addressed to P.O. Box 6248, Providence, RI 02940-6248.Copying and reprinting. Material in this journal may be reproduced by any means
for educational and scientific purposes without fee or permission with the exception ofreproduction by services that collect fees for delivery of documents and provided that thecustomary acknowledgment of the source is given. This consent does not extend to otherkinds of copying for general distribution, for advertising or promotional purposes, or forresale. Requests for permission for commercial use of material should be addressed to theAssistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence,RI 02940-6248. Requests can also be made by e-mail to [email protected].
Excluded from these provisions is material in articles for which the author holds copy-right. In such cases, requests for permission to use or reprint should be addressed directlyto the author(s). (Copyright ownership is indicated in the notice in the lower right-handcorner of the first page of each article.)
Bulletin (New Series) of the American Mathematical Society is published quarterly by the Amer-ican Mathematical Society at 201 Charles Street, Providence, RI 02904-2213. Periodicals postage ispaid at Providence, Rhode Island, and additional mailing offices. Postmaster: Send address changes toBulletin, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248.
c© 2000 American Mathematical Society. All rights reserved.
This journal is indexed in Science Citation Indexr, Science Citation
IndexTM–Expanded, ISI Alerting ServicesSM, CompuMath Citation
Indexr, and Current Contentsr/Physical, Chemical & Earth Sciences.Printed in the United States of America.
∞© The paper used in this journal is acid-free and falls withinthe guidelines established to ensure permanence and durability.
10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00
INDEX TO VOLUME 37 (2000)
research-expository surveys
Baouendi, M. S., Ebenfelt, P., and Rothschild, Linda Preiss. Local geometric properties of real
submanifolds in complex space, 309
Birkhoff, G. D. A mathematical critique of some physical theories, 65Brouwer, Dr. L. E. J. Intuitionism and formalism, 55
Di Francesco, P. Folding and coloring problems in mathematics and physics, 251Ebenfelt, P. See Baouendi, M. S.Einstein, Albert. Elementary derivation of the equivalence of mass and energy, 39Fiske, President Thomas S. Mathematical progress in America, 3Fulton, William. Eigenvalues, invariant factors, highest weights, and Schubert calculus, 209Hilbert, David. Mathematical problems, 407McMullen, Curtis T. From dynamics on surfaces to rational points on curves, 119
Morawetz, Cathleen Synge. Variations on conservation laws for the wave equation, 141Morris, W., and Soltan, V. The Erdos-Szekeres problem on points in convex position – a survey,
437Parshall, Karen Hunger. Perspectives on American mathematics, 381Pierpont, Professor James. The history of mathematics in the nineteenth century, 9Poincare, Professor H. The present and the future of mathematical physics, 25Rothschild, Linda Preiss. See Baouendi, M. S.Saari, Donald G. Introductory comments, 1
. Note from the Editor, 379Soltan, V. See Morris, W.Van Vleck, Edward B. Current tendencies of mathematical research, 45
book reviews
Ahlfors, L. V. Complex analysis, reviewed by A. C. Schaeffer, 111Alexandroff, Paul. Einfachste Grundbegriffe der Topologie, with an introduction by David Hilbert,
reviewed by P. A. Smith, 90Alexandroff, Paul, and Hopf, Heinz. Topologie I, reviewed by A. W. Tucker, 93
Andrews, George E., Askey, Richard, and Roy, Ranjan. Special functions, reviewed by Jet Wimp,499
Arnold, V. I., and Khesin, B. A. Topological methods in hydrodynamics, reviewed by SteveShkoller, 175
Askey, Richard. See Andrews, George E.Bade, William G. See Dunford, Nelson
Baouendi, M. Salah, Ebenfelt, Peter, and Rothschild, Linda Preiss. Real submanifolds in complex
space and their mappings, reviewed by Nancy K. Stanton, 511Bartle, Robert G. See Dunford, Nelson
Bass, H.; Buium, A.; and Cassidy, P. J. (editors). Selected works of Ellis Kolchin, reviewed byAndy R. Magid, 337
Birkhoff, G. D. Dynamical systems, reviewed by B. O. Koopman, 88
Bourbaki, N. Elements de mathematique, reviewed by E. Artin, 112Buium, A. See Bass, H.Burnside, W. Theory of groups of a finite order, reviewed by G. A. Miller, 80Caratheodory, C. Funktionentheorie, reviewed by Maurice Heins, 108
Cartan, Elie. La theorie des groupes finis et continus et la geometrie differentielle traitees par
methode du repere mobile, reviewed by Hermann Weyl, 96Cartan, Henri, and Eilenberg, Samuel. Homological algebra, reviewed by S. Mac Lane, 113Cassidy, P. J. See Bass, H.Chevalley, C. Introduction to the theory of algebraic functions of one variable, reviewed by A.
Weil, 103Chevalley, Claude. The theory of Lie groups, I, reviewed by P. A. Smith, 105Chriss, Neil, and Ginzburg, Victor. Representation theory and complex geometry, reviewed by
Ivan Mirkovic, 343
Courant, R., and Hilbert, D. Methoden der mathematischen Physik, Vol. 2, reviewed by HermannWeyl, 97
INDEX TO VOLUME 37 (2000)
Cox, David A., and Katz, Sheldon. Mirror symmetry and algebraic geometry, reviewed by V.Batyrev, 473
Coxeter, H. S. M. Regular polytopes, reviewed by C. B. Allendoerfer, 107Curtis, Charles W. Pioneers of representation theory, reviewed by J. E. Humphreys, 359Dieudonne, J. See Grothendieck, A.
Donkin, Stephen. The q-Schur algebra, reviewed by Brian Parshall, 467van den Dries, Lou. Tame topology and o-minimal structures, reviewed by David Marker, 351Dunford, Nelson and Schwartz, Jacob T., with the assistance of William G. Bade and Robert G.
Bartle. Linear operators. Part II. Spectral theory, reviewed by Gian-Carlo Rota, 118Ebenfelt, Peter. See Baouendi, M. SalahEdgar, G. Integral, probability, and fractal measures, reviewed by Lars Olsen, 481Eilenberg, S., and Steenrod, N. Foundations of algebraic topology, reviewed by E. H. Spanier,
114Eilenberg, Samuel. See Cartan, HenriEtingof, Pavel I; Frenkel, Igor B.; and Kirillov, Alexander, A., Jr. Lectures on representation
theory and Knizhnik-Zamolodchikov equations, reviewed by Vyjayanthi Chari, 161Evans, L. C. Partial differential equations, reviewed by Jeffrey Rauch, 363Feller, William. An introduction to probability theory and its applications, Vol. I, reviewed by
J. Wolfowitz, 109
. Festschrift zur Feier des 100 Geburtstages Eduard Kummers mit Briefen an seine Mutter und
an Leopold Kronecker, reviewed by L. E. Dickson, 82Frenkel, Igor B. See Etingof, Pavel I.Gelfand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V. Discriminants, resultants and multidi-
mensional determinants, reviewed by Fabrizio Catanese, 183Ginzburg, Victor. See Chriss, Neil
Godel, Kurt. Consistency of the axiom of choice and of the generalized continuum-hypothesis
with the axioms of set theory, reviewed by C. C. Torrance, 102
Goursat, Edouard. Cours d’analyse mathematique, reviewed by W. F. Osgood, 81
Grothendieck, A. Elements de geometrie algebrique, rediges avec la collaboration de J. Dieu-
donne, reviewed by S. Lang, 115Hardy, G. H., and Wright, E. M. An introduction to the theory of numbers, reviewed by E. T.
Bell, 99Hilbert, D. See Courant, R.Hilbert, David. Poincare’s review of Hilbert’s Foundations of geometry, reviewed by H. Poincare,
77Hodge, W. V. D. The theory and applications of harmonic integrals, reviewed by D. J. Struik,
102Hopf, Heinz. See Alexandroff, Paul
Jost, J. Riemannian geometry and geometric analysis, reviewed by Werner Ballmann, 459Kapranov, M. M. See Gelfand, I. M.
Katz, Sheldon. See Cox, David A.
Khesin, B. A. See Arnold, V. I.Kirillov, Alexander A., Jr. See Etingof, Pavel I.
Klain, Daniel A., and Rota, Gian-Carlo. Introduction to geometric probability, reviewed byGunter M. Ziegler, 205
Kolmogoroff, A. Grundbegriffe der Wahrscheinlichkeitsrechnung, reviewed by H. L. Rietz, 91Krasil’shchik, I. S., and Vinogradov, A. M. Symmetries and conservation laws for differential
equations of mathematical physics, reviewed by Peter J. Olver, 369Landau, Edmund. Vorlesungen uber Zahlentheorie, reviewed by G. D. Birkhoff, 86Laugwitz, Detlef. Bernhard Riemann, 1826–1866: Turning points in the conception of mathe-
matics, reviewed by Roger Cooke, 477Lebesgue, H. Lecons sur l’integration et la recherche des fonctions primitives, reviewed by
T. H. Hildebrandt, 87Lie, Sophus. Vorlesungen uber continuirliche Gruppen mit geometrischen und anderen Anwen-
dungen, reviewed by J. M. Brooks, 78. Gesammelte Abhandlungen, reviewed by R. D. Carmichael, 79
Mac Lane, Saunders. Homology, reviewed by David A. Buchsbaum, 117
INDEX TO VOLUME 37 (2000)
Mathas, Andrew. Iwahori-Hecke algebras and Schur algebras of the symmetric group, reviewedby Brian Parshall, 467
Mordell, L. J. Three lectures on Fermat’s last theorem, reviewed by G. E. Wahlin, 84Morgenstern, Oskar. See von Neumann, JohnMurnaghan, Francis D. The theory of group representations, reviewed by R. Brauer, 100Murty, M. Ram, and Murty, V. Kumar. Non–vanishing of L–functions and applications, reviewed
by Dorian Goldfeld, 155Murty, V. Kumar. See Murty, M. Ram
Navarro, Gabriel. Characters and blocks of finite groups, reviewed by David Gluck, 169von Neumann, John, and Morgenstern, Oskar. Theory of games and economic behavior, reviewed
by A. H. Copeland, 103Osgood, Dr. W. F. Lehrbuch der Funktionentheorie, reviewed by E. B. Van Vleck, 83Petersen, P. Riemannian geometry, reviewed by Werner Ballmann, 459Ramakrishnan, D., and Valenza, R. J. Fourier analysis on number fields, reviewed by Jeremy
Teitelbaum, 373Reidemeister, K. Knotentheorie, reviewed by P. A. Smith, 90Reidemeister, Kurt. Einfuhrung in die kombinatorische Topologie, reviewed by P. A. Smith, 90Riesz, F., and Sz.-Nagy, B. Lecons d’analyse functionelle, reviewed by E. R. Lorch, 111Rota, Gian-Carlo. See Klain, Daniel A.
Rothschild, Linda Preiss. See Baouendi, M. SalahRoy, Ranjan. See Andrews, George E.Sakai, T. Riemannian geometry, reviewed by Werner Ballmann, 459Schwartz, Jacob T. See Dunford, NelsonSeifert, H., and Threlfall, W. Lehrbuch der Topologie, reviewed by A. W. Tucker, 92Smart, Nigel P. The algorithmic resolution of Diophantine equations, reviewed by Maurice
Mignotte, 207Srinivasan, Bhama. Editor’s introduction, 75
Steenrod, N. See Eilenberg, S.Steinitz, Ernst. Vorlesungen uber die Theorie der Polyeder unter Einschluss der Elemente der
Topologie, edited and completed by Hans Rademacher, reviewed by A. W. Tucker, 92Sz.-Nagy, B. See Riesz, F.Threlfall, W. See Seifert, H.Titchmarsh, E. C. The theory of the Riemann zeta-function, reviewed by N. Levinson, 110Traub, J. F., and Werschulz, A. G. Complexity and information, reviewed by Mark A. Kon, 199Valenza, R. J. See Ramakrishnan, D.Veblen, Oswald. The Cambridge Colloquium, 1916, Part II. Analysis Situs, reviewed by S. Lef-
schetz, 85Vinogradov, A. M. See Krasil’shchik, I. S.
van der Waerden, B. L. Moderne algebra, reviewed by O. Ore, 89Weil, Andre. Foundations of algebraic geometry, reviewed by Oscar Zariski, 106Werschulz, A. G. See Traub, J. F.
Weyl, Hermann. The classical groups, reviewed by N. Jacobson, 100Wright, E. M. See Hardy, G. H.
Zariski, Oscar. Algebraic surfaces, reviewed by Solomon Lefschetz, 94Zelevinsky, A. V. See Gelfand, I. M.Zygmund, A. Trigonometric series, 2d. ed., reviewed by R. Salem, 96Zygmund, Antoni. Trigonometric series, reviewed by J. D. Tamarkin, 95
TITLE PAGE
VOLUME 37
PROVIDENCE, RHODE ISLAND USA
ISSN 0273-0979
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YA M E R I C A N M A T H E M A T I C A L S O C I E T Y
ULLETINB OF THE
( N E W S E R I E S )
2000
EDITORS
Donald G. Saari, Chief EditorBhama Srinivasan, Book Reviews
ASSOCIATE EDITORS
John C. Baez Philip ProtterL. Craig Evans Hugo RossiJohn M. Franks Daniel RubermanEric Friedlander Audrey A. TerrasCraig L. Huneke David VoganDouglas A. Lind Alex J. WilkieBarry Mazur Wolfgang ZillerHaynes R. Miller
Bulletin (New Series) of the American Mathematical Society
This journal is devoted to articles of the following types:
Research-Expository Surveys
These are, by definition, papers that present a clear and insightful exposition of signif-icant aspects of contemporary mathematical research. Gibbs lectures, Progress in Math-ematics lectures, and retiring presidential addresses will be included in this section.
Research Reports
These are brief, timely reports on important mathematical developments. They arenormally solicited and often written by a disinterested expert.
Book Reviews
Book Reviews are accepted for publication by invitation only. Unsolicited manuscriptswill not be considered.
Submission information. See Information for Authors at the end of this issue.Publisher Item Identifier. The Publisher Item Identifier (PII) appears at the top
of the first page of each article published in this journal. This alphanumeric string ofcharacters uniquely identifies each article and can be used for future cataloging, searching,and electronic retrieval.
Postings to e-MATH. Articles are posted to e-MATH individually soon after proofis returned from authors and before appearing in an issue.
Subscription information. Bulletin (New Series) of the American Mathematical
Society is published quarterly. The Bulletin is also accessible electronically, startingwith the January 1992 issue, from www.ams.org/publications/. For paper delivery, sub-scription prices for Volume 37 (2000) are $300 list, $240 institutional member, $180 in-dividual member. The subscription price for members is included in the annual dues. Alate charge of 10% of the subscription price will be imposed upon orders received fromnonmembers after January 1 of the subscription year. Subscribers outside the UnitedStates and India must pay a postage surcharge of $8.00; subscribers in India must pay apostage surcharge of $13.00. Expedited delivery to destinations in North America is $12;elsewhere $23.
Back number information. For back issues see the AMS Catalog of Publications.Subscriptions and orders should be addressed to the American Mathematical Society,
P.O. Box 845904, Boston, MA 02284-5904. All orders must be accompanied by payment.
Other correspondence should be addressed to P.O. Box 6248, Providence, RI 02940-6248.Copying and reprinting. Material in this journal may be reproduced by any means
for educational and scientific purposes without fee or permission with the exception ofreproduction by services that collect fees for delivery of documents and provided that thecustomary acknowledgment of the source is given. This consent does not extend to otherkinds of copying for general distribution, for advertising or promotional purposes, or forresale. Requests for permission for commercial use of material should be addressed to theAssistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence,RI 02940-6248. Requests can also be made by e-mail to [email protected].
Excluded from these provisions is material in articles for which the author holds copy-right. In such cases, requests for permission to use or reprint should be addressed directlyto the author(s). (Copyright ownership is indicated in the notice in the lower right-handcorner of the first page of each article.)
Bulletin (New Series) of the American Mathematical Society is published quarterly by the Amer-ican Mathematical Society at 201 Charles Street, Providence, RI 02904-2213. Periodicals postage ispaid at Providence, Rhode Island, and additional mailing offices. Postmaster: Send address changes toBulletin, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248.
c© 2000 American Mathematical Society. All rights reserved.
This journal is indexed in Science Citation Indexr, Science Citation
IndexTM–Expanded, ISI Alerting ServicesSM, CompuMath Citation
Indexr, and Current Contentsr/Physical, Chemical & Earth Sciences.Printed in the United States of America.
∞© The paper used in this journal is acid-free and falls withinthe guidelines established to ensure permanence and durability.
10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
CONTENTS
Vol. 37, No. 1 January 2000
Research-Expository Surveys
Donald G. Saari, Introductory comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
President Thomas S. Fiske, Mathematical progress in America . . . . . . . . . 3
Professor James Pierpont, The history of mathematics in the nineteenthcentury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Professor H. Poincare, The present and the future of mathematical physics 25
Albert Einstein, Elementary derivation of the equivalence of mass andenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Edward B. Van Vleck, Current tendencies of mathematical research . . . . 45
Dr. L. E. J. Brouwer, Intuitionism and formalism . . . . . . . . . . . . . . . . . . . . . . . 55
G. D. Birkhoff, A mathematical critique of some physical theories . . . . . . . 65
Book Reviews
Bhama Srinivasan, Editor’s introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
H. Poincare (Reviewer), Poincare’s review of Hilbert’s Foundations ofgeometry, by David Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
J. M. Brooks (Reviewer), Vorlesungen uber continuirliche Gruppen mitgeometrischen und anderen Anwendungen, by Sophus Lie . . . . . . . . . . . . 78
R. D. Carmichael (Reviewer), Gesammelte Abhandlungen, by Sophus Lie 79
G. A. Miller (Reviewer), Theory of groups of a finite order, by W. Burnside 80
W. F. Osgood (Reviewer), Cours d’analyse mathematique, by EdouardGoursat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
L. E. Dickson (Reviewer), Festschrift zur Feier des 100 Geburtstages EduardKummers mit Briefen an seine Mutter und an Leopold Kronecker . . . . 82
E. B. Van Vleck (Reviewer), Lehrbuch der Funktionentheorie, by Dr. W.F. Osgood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
G. E. Wahlin (Reviewer), Three lectures on Fermat’s last theorem, by L. J.Mordell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
S. Lefschetz (Reviewer), The Cambridge Colloquium, 1916, Part II. AnalysisSitus, by Oswald Veblen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
G. D. Birkhoff (Reviewer), Vorlesungen uber Zahlentheorie, by EdmundLandau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
T. H. Hildebrandt (Reviewer), Lecons sur l’integration et la recherche desfonctions primitives, by H. Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B. O. Koopman (Reviewer), Dynamical systems, by G. D. Birkhoff . . . . . . 88
O. Ore (Reviewer), Moderne algebra, by B. L. van der Waerden . . . . . . . . . . 89
P. A. Smith (Reviewer), Einfachste Grundbegriffe der Topologie, by PaulAlexandroff, with an introduction by David Hilbert; Einfuhrung in diekombinatorische Topologie, by Kurt Reidemeister; Knotentheorie, by K.Reidemeister . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
H. L. Rietz (Reviewer), Grundbegriffe der Wahrscheinlichkeitsrechnung, byA. Kolmogoroff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A. W. Tucker (Reviewer), Vorlesungen uber die Theorie der Polyeder unterEinschluss der Elemente der Topologie, by Ernst Steinitz; edited andcompleted by Hans Rademacher; Lehrbuch der Topologie, by H. Seifertand W. Threlfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A. W. Tucker (Reviewer), Topologie I, by Paul Alexandroff and Heinz Hopf 93
Solomon Lefschetz (Reviewer), Algebraic surfaces, by Oscar Zariski . . . . . 94
J. D. Tamarkin (Reviewer), Trigonometric series, by Antoni Zygmund . . . 95
R. Salem (Reviewer), Trigonometric series, 2d. ed.,, by A. Zygmund . . . . . . 96
Hermann Weyl (Reviewer), La theorie des groupes finis et continus et la
geometrie differentielle traitees par methode du repere mobile, by ElieCartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Hermann Weyl (Reviewer), Methoden der mathematischen Physik, Vol. 2,by R. Courant and D. Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
E. T. Bell (Reviewer), An introduction to the theory of numbers, by G. H.Hardy and E. M. Wright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
N. Jacobson (Reviewer), The classical groups, by Hermann Weyl . . . . . . . . 100
R. Brauer (Reviewer), The theory of group representations, by Francis D.Murnaghan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C. C. Torrance (Reviewer), Consistency of the axiom of choice and of thegeneralized continuum-hypothesis with the axioms of set theory, by KurtGodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
D. J. Struik (Reviewer), The theory and applications of harmonic integrals,by W. V. D. Hodge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A. H. Copeland (Reviewer), Theory of games and economic behavior, byJohn von Neumann and Oskar Morgenstern . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A. Weil (Reviewer), Introduction to the theory of algebraic functions of onevariable, by C. Chevalley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
P. A. Smith (Reviewer), The theory of Lie groups, I, by Claude Chevalley 105
Oscar Zariski (Reviewer), Foundations of algebraic geometry, by AndreWeil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C. B. Allendoerfer (Reviewer), Regular polytopes, by H. S. M. Coxeter . 107
Maurice Heins (Reviewer), Funktionentheorie, by C. Caratheodory . . . . . . 108
J. Wolfowitz (Reviewer), An introduction to probability theory and itsapplications, Vol. I, by William Feller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
N. Levinson (Reviewer), The theory of the Riemann zeta-function, by E.C. Titchmarsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A. C. Schaeffer (Reviewer), Complex analysis, by L. V. Ahlfors . . . . . . . . . 111
E. R. Lorch (Reviewer), Lecons d’analyse functionelle, by F. Riesz and B.Sz.-Nagy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
E. Artin (Reviewer), Elements de mathematique, by N. Bourbaki . . . . . . . . 112
S. Mac Lane (Reviewer), Homological algebra, by Henri Cartan and SamuelEilenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
E. H. Spanier (Reviewer), Foundations of algebraic topology, by S. Eilenbergand N. Steenrod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
S. Lang (Reviewer), Elements de geometrie algebrique, by A. Grothendieck,rediges avec la collaboration de J. Dieudonne . . . . . . . . . . . . . . . . . . . . . . . . 115
David A. Buchsbaum (Reviewer), Homology, by Saunders Mac Lane . . . 117
Gian-Carlo Rota (Reviewer), Linear operators.Part II. Spectral theory, byNelson Dunford and Jacob T. Schwartz, with the assistance of WilliamG. Bade and Robert G. Bartle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Vol. 37, No. 2 April 2000
Research-Expository Surveys
Curtis T. McMullen, From dynamics on surfaces to rational points oncurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Cathleen Synge Morawetz, Variations on conservation laws for the waveequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Book Reviews
Dorian Goldfeld (Reviewer), Non–vanishing of L–functions and applications, by M. Ram Murty and V. Kumar Murty . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Vyjayanthi Chari (Reviewer), Lectures on representation theory andKnizhnik-Zamolodchikov equations, by Pavel I. Etingof, Igor B. Frenkel,and Alexander A. Kirillov, Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
David Gluck (Reviewer), Characters and blocks of finite groups, by GabrielNavarro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Steve Shkoller (Reviewer), Topological methods in hydrodynamics, by V.I. Arnold and B. A. Khesin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Fabrizio Catanese (Reviewer), Discriminants, resultants and multidimen-sional determinants, by I. M. Gelfand, M. M. Kapranov, and A. V.Zelevinsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Mark A. Kon (Reviewer), Complexity and information, by J. F. Trauband A. G. Werschulz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Gunter M. Ziegler (Reviewer), Introduction to geometric probability, byDaniel A. Klain and Gian-Carlo Rota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Maurice Mignotte (Reviewer), The algorithmic resolution of Diophantineequations, by Nigel P. Smart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Vol. 37, No. 3 July 2000
Research-Expository Surveys
William Fulton, Eigenvalues, invariant factors, highest weights, andSchubert calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
P. Di Francesco, Folding and coloring problems in mathematics and physics 251
M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Localgeometric properties of real submanifolds in complex space . . . . . . . . . . . 309
Book Reviews
Andy R. Magid (Reviewer), Selected works of Ellis Kolchin, by H. Bass,A. Buium, and P.J. Cassidy (editors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Ivan Mirkovic (Reviewer), Representation theory and complex geometry,by Neil Chriss and Victor Ginzburg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
David Marker (Reviewer), Tame topology and o-minimal structures, by Louvan den Dries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
J. E. Humphreys (Reviewer), Pioneers of representation theory, byCharles W. Curtis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Jeffrey Rauch (Reviewer), Partial differential equations, by L. C. Evans . . . 363
Peter J. Olver (Reviewer), Symmetries and conservation laws for differentialequations of mathematical physics, by I. S. Krasil’shchik and A. M.Vinogradov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Jeremy Teitelbaum (Reviewer), Fourier analysis on number fields, by D.Ramakrishnan and R. J. Valenza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Vol. 37, No. 4 October 2000
Research-Expository Surveys
Donald G. Saari, Note from the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Karen Hunger Parshall, Perspectives on American mathematics . . . . . . . . 381
David Hilbert, Mathematical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
W. Morris and V. Soltan, The Erdos-Szekeres problem on points in convexposition – a survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Book Reviews
Werner Ballmann (Reviewer), Riemannian geometry and geometricanalysis, by J. Jost; Riemannian geometry, by P. Petersen; Riemanniangeometry, by T. Sakai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Brian Parshall (Reviewer), The q-Schur algebra, by Stephen Donkin;Iwahori-Hecke algebras and Schur algebras of the symmetric group, byAndrew Mathas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
V. Batyrev (Reviewer), Mirror symmetry and algebraic geometry, by DavidA. Cox and Sheldon Katz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Roger Cooke (Reviewer), Bernhard Riemann, 1826–1866: Turning points inthe conception of mathematics, by Detlef Laugwitz . . . . . . . . . . . . . . . . . . 477
Lars Olsen (Reviewer), Integral, probability, and fractal measures, by G.Edgar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Jet Wimp (Reviewer), Special functions, by George E. Andrews, RichardAskey, and Ranjan Roy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Nancy K. Stanton (Reviewer), Real submanifolds in complex space andtheir mappings, by M. Salah Baouendi, Peter Ebenfelt, and Linda PreissRothschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Editorial Board for Research-Expository Surveys
John C. BaezCraig L. Huneke
Douglas A. LindDonald G. Saari, Chair
Editorial Board for Research Reports
John M. FranksEric FriedlanderBarry MazurHaynes R. Miller
Hugo RossiDaniel RubermanDonald G. Saari, ChairDavid Vogan
Editorial Board for Book Reviews
L. Craig EvansPhilip ProtterBhama Srinivasan, Chair
Audrey A. TerrasAlex J. WilkieWolfgang Ziller
Chief Editor: Donald G. Saari
Editorial Information
As of June 30, 2000, the backlog for this journal was approximately 0 issues.This estimate is the result of reducing the number of manuscripts for this journalin the Providence office that have not yet gone to the printer on the above date bythe number of articles assigned to issues to be published in four months (the timenecessary for editing and composing a typical issue). In an effort to make articlesavailable as quickly as possible, articles are posted to e-MATH individually soonafter proof is returned from authors and before appearing in an issue.
A Consent to Publish and Copyright Agreement is required before a paper will bepublished in this journal. After a paper is accepted for publication, the Providenceoffice will send a Consent to Publish and Copyright Agreement to all authors of thepaper. By submitting a paper to this journal, authors certify that the results havenot been submitted to nor are they under consideration for publication by anotherjournal, conference proceedings, or similar publication.
Information for Authors
Initial submission. For Research-Expository Surveys and Research Reports,copies of the manuscript—three copies for Surveys and four copies for Reports—may be submitted to Professor Donald G. Saari, Department of Mathematics. Mul-tipurpose Science and Technology Building, University of California, Irvine, Irvine,CA 92697-3875. Manuscripts may also be submitted to other members of the re-spective Editorial Boards, with a copy of the cover letter and of the manuscript alsosent to Professor Donald G. Saari. The author should keep one copy. Research-Expository Surveys and Research Reports can also be submitted electronically inAMS-LATEX or AMS-TEX to [email protected]. IF an editor is agreeable, anelectronic manuscript prepared in TEX or LATEX may be submitted by pointing toan appropriate URL on a preprint or e-print server.
For Research-Expository Surveys and Research Reports, the first page mustconsist of a descriptive title, followed by an abstract that summarizes the articlein language suitable for workers in the general field (algebra, analysis, etc.). Thedescriptive title should be short but informative; useless or vague phrases such as“some remarks about” or “concerning” should be avoided. The abstract shouldbe a brief technical description of the new material. Then, for Research Reports
there should be an introductory section addressed to nonexperts describing themotivation, background, and significance of the results announced. Following thestatement of results, there should be a sketch of proofs that may be addressed toexperts. Included with the footnotes in each paper should be the 2000 Mathe-
matics Subject Classification representing the primary and secondary subjects ofthe article. The classifications are accessible from www.ams.org/msc/. The listof classifications is also available in print starting with the 1999 annual index ofMathematical Reviews. The Mathematics Subject Classification footnote may befollowed by a list of key words and phrases describing the subject matter of thearticle and taken from it. Journal abbreviations used in bibliographies are listedin the latest Mathematical Reviews annual index. The series abbreviations arealso accessible from www.ams.org/publications/. To help in preparing and ver-ifying references, the AMS offers MR Lookup, a Reference Tool for Linking, atwww.ams.org/mrlookup/. When the manuscript is submitted, authors should sup-ply the editor with electronic addresses if available. These will be printed after thepostal address at the end of each article.
For Book Reviews the first page must include the title of the book being reviewed;the name(s) of the author(s); publisher; city of publication; year of publication;number of pages, including front matter; price if known; and ISBN. There shouldalso be a footnote with the 2000 Mathematics Subject Classification representingthe primary and secondary subjects of the book under review. The classificationsare accessible from www.ams.org/msc/ and are also available in print starting withthe 1999 annual index of Mathematical Reviews. To help in preparing and ver-ifying references, the AMS offers MR Lookup, a Reference Tool for Linking, atwww.ams.org/mrlookup/.
Electronically prepared manuscripts. The AMS encourages electronicallyprepared manuscripts, with a strong preference for AMS-LATEX. To this end, theSociety has prepared AMS-LATEX author packages for each AMS publication. Au-thor packages include instructions for preparing electronic manuscripts, the AMS
Author Handbook, samples, and a style file that generates the particular designspecifications of that publication series. Articles properly prepared using the AMS-LATEX style file and the \label and \ref commands automatically enable extensiveintra-document linking to the bibliography and other elements of the article forsearching electronically on the Web. Because linking must often be added manu-ally to electronically prepared manuscripts in other forms of TEX, using AMS-LATEXalso reduces the amount of technical intervention once the files are received by theAMS. This results in fewer errors in processing and saves the author proofreadingtime. AMS-LATEX papers also move more efficiently through the production stream,helping to minimize publishing costs.
AMS-LATEX is the highly preferred format of TEX, but author packages are alsoavailable in AMS-TEX. Those authors who make use of these style files from thebeginning of the writing process will further reduce their own efforts. Manuscriptsprepared electronically in LATEX or plain TEX are normally not acceptable due tothe high amount of technical time required to insure that the file will run properlythrough the AMS in-house production system. LATEX users will find that AMS-LATEX is the same as LATEX with additional commands to simplify the typesettingof mathematics, and users of plain TEX should have the foundation for learningAMS-LATEX.
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At the time of submission, authors should indicate if the paper has been preparedusing AMS-LATEX or AMS-TEX and provide the Editor with a paper manuscriptthat matches the electronic manuscript. The final version of the electronic man-uscript should be sent to the Providence office immediately after the paper hasbeen accepted for publication. The author should also send the final version of thepaper manuscript to the Editor, who will forward a copy to the Providence office.Editors will require authors to send their electronically prepared manuscripts to theProvidence office in a timely fashion. Electronically prepared manuscripts can besent via email to [email protected] (Internet) or on diskette to the ElectronicPrepress Department, American Mathematical Society, P.O. Box 6248, Providence,RI 02940-6248. When sending a manuscript electronically, please be sure to includea message indicating in which publication the paper has been accepted. No cor-rections will be accepted electronically. Authors must mark their changes on theirproof copies and return them to the Providence office. Complete instructions onhow to send files are included in the author package.
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Research-Expository Surveys
This section is intended for reasonably broad expository surveys of a currentlyactive area of mathematical research. To stress this fact, the name of the sectionhas been changed from “Research-Expository Papers” to “Research-Expository Sur-veys”. The level of exposition is expected to be such that a graduate student—or aresearch mathematician from a remote area—will be able to read an article in orderto get an overview of a subject. A well-written article will include motivating prob-lems and examples, some indication of the historical development of the subject,and of course the results and open problems that make it an interesting and excit-ing area of mathematics. In most cases proofs should be at most briefly sketched,and there should be a good bibliography whose main aim should be helping thosewishing to pursue the subject further. Usually Research-Expository Surveys willbe at most 40 pages long.
Certain of the Society’s Lecture Series, including the Gibbs Lectures, Collo-quium Lectures, and Progress in Mathematics Lectures, traditionally appear in theResearch-Expository Surveys, and those invited to give these lectures are informedof this tradition and of the above guidelines. Research-Expository Surveys are alsofrequently solicited by the Editorial Board, but unsolicited manuscripts are alsoencouraged; in particular, those invited to give invited hour addresses at meetingsof the Society should consider writing up their lecture in a manner that meetsthe requirements of the preceding paragraph and submitting their manuscripts forconsideration by the Editorial Board for the Research-Expository Surveys.
Research Reports
A Research Report is a timely, short (4–8 pages) report on a recent accomplish-ment in mathematical research. It should be written so as to be understandableby mathematicians who are not experts in the subject matter of the Report. Thecontext giving significance to the research being reported on should be made clear,and the consequences or potential consequences should be described. Usually onlythe barest hint of the methods of proof should be given, although elements of theproof which are novel should be included. In any case, references should be given sothat an interested reader can find the details. It is expected that Research Reportswill frequently be written by specialists other than the individuals immediatelyinvolved in the research.
Research Reports are normally solicited by the “Editorial Board for ResearchReports”. They cordially invite suggestions for topics and reporters. Unsolicitedmanuscripts, submitted to a member of the Editorial Board for Research Reports,will also be considered.
The American Mathematical Society’s electronic-only journal,Electronic Research Announcements of the AMS (ERA-AMS), is available on the World Wide Web at www.ams.org/era.
ERA-AMS publishes high-quality research announcements ofsignificant advances in all branches of mathematics. Authorsmay submit manuscripts to any editor. All papers are reviewed,and the entire Editorial Board must approve the acceptance ofany paper. Papers are posted as soon as they are accepted andprocessed by the AMS.
AMERICAN MATHEMATICAL SOCIETY
AMERICAN MATHEMATICAL SOCIETY
OF THE
Electronic
ResearchAnnouncements
Most recent articlesBrian Smith and Gilbert Weinstein, On the connectedness of thespace of initial data for the Einstein equations
A. V. Grishin, On non-Spechtianness of the variety of associativerings that satisfy the identity x32
= 0
Michael Hutchings, Frank Morgan, Manuel Ritoré, andAntonio Ros, Proof of the double bubble conjecture
A. Giambruno and M. Zaicev, Minimal varieties of algebras ofexponential growth
Danny Calegari, Geometry and topology of R -covered foliations
S. R. Bullett and W. J. Harvey, Mating quadratic maps withKleinian groups via quasiconformal surgery
A. A. Kirillov, Family algebras
Alejandro Adem and Jeff H. Smith, On spaces with periodiccohomology
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, LaurelLangford, and Masahico Saito, State-sum invariants of knottedcurves and surfaces from quandle cohomology
Z. Reichstein and B. Youssin, Parusinski's “Key Lemma” viaalgebraic geometry
MANAGING EDITOR
Svetlana Katok
EDITORIAL BOARD
Stuart Antman
Douglas Arnold
David Benson
Dmitri Burago
Mark Freidlin
James Glimm
Ronald Graham
Yitzhak Katznelson
David Kazhdan
Alexander Kechris
Alexandre Kirillov
Frances Kirwan
Krystyna Kuperberg
Robert Lazarsfeld
Gregory Margulis
Hugh Montgomery
Walter Neumann
Klaus Schmidt
Richard Schoen
Masamichi Takesaki
Michael Taylor
Thomas Wolff
Zhihong (Jeff) Xia
Don Zagier
Efim Zelmanov
www.ams.org/era
AMERICAN MATHEMATICAL SOCIETY
MANAGING EDITOR
Linda Keen
City University of New York
EDITORIAL BOARD MEMBERS
K.Astala
University of Jyväskylä
F.W. Gehring
University of Michigan
Tan Lei
University of Warwick
M. Lyubich
SUNY at Stony Brook
Most recent articlesA combination theorem for covering correspondences and
an application to mating polynomial maps with Kleinian
groups
Shaun Bullett
A census of rational maps
Eva Brezin, Rosemary Byrne, Joshua Levy,Kevin Pilgrim and Kelly Plummer
The role of the Ahlfors five islands theorem in complex
dynamics
Walter Bergweiler
Unbounded components in parameter space of rational
maps
Peter M. Makienko
Examples of uniformly quasiregular mappings
Kirsi Peltonen
Nonlinear automorphisms of plane domains
Timo Erkama
Ford and Dirichlet domains for cyclic subgroups of PSL2(C)
acting on H3
R and ∂H3
R
Todd A. Drumm and Jonathan A. Poritz
This journal provides a forum for mathematical work inrelated fields broadly described as conformal geometryand dynamics.This includes complex dynamics (and realdynamics using complex techniques), Kleinian groups,hyperbolic geometry,Teichmüller theory, andquasiconformal mappings.
www.ams.org/ecgd
CONFORMALGEOMETRY DYNAMICS
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A N E L E C T R O N I C J O U R N A L O F T H E A M S
www.ams.org/ert
AMERICAN MATHEMATICAL SOCIETY
MANAGING
EDITOR
David VoganMassachusettsInstitute of Technology
EDITORIAL
BOARD
MEMBERS
A. KnappSUNY at Stony Brook
J. LepowskyRutgers University
G. LusztigMassachusettsInstitute of Technology
D. MilicicUniversity of Utah
B. SpehCornell University
Most recent articlesOn Laguerre polynomials, Bessel functions, Hankel transform
and a series in the unitary dual of the simply-connectedcovering group of Sl(2,R)Bertram Kostant
On square-integrable representations of classical p-adicgroups IIChris Jantzen
Large Schubert varietiesMichel Brion and Patrick Polo
Harish-Chandra modules for quantum symmetric pairsGail Letzter
Symmetric polynomials and Uq(sl2)Naihuan Jing
Verifying Kottwitz' conjecture by computerBill Casselman
Stable nilpotent orbital integrals on real reductive Lie algebrasRobert E. Kottwitz
Involutions in Weyl groupsRobert E. Kottwitz
Differential operators on some nilpotent orbitsT. Levasseur and J.T. Stafford
The adjoint representation of a reductive group and
hyperplane arrangements
J. Matthew Douglass
This journal is devoted to research in representationtheory. It seeks to maintain a high standard for expositionas well as for mathematical content. Papers are posted inseveral different electronic formats on e-MATH via theWorld Wide Web.
REPRESENTATION
THEORYA N E L E C T R O N I C J O U R N A L O F T H E A M S
