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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 significant 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 are normally solicited and often written by a disinterested expert. Book Reviews Book Reviews are accepted for publication by invitation only. Unsolicited manuscripts will 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 of characters 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 proof is 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, starting with the January 1992 issue, from www.ams.org/publications/. For paper delivery, subscription 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. A late charge of 10% of the subscription price will be imposed upon orders received from nonmembers after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of $8.00; subscribers in India must pay a postage 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 of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant 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 copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner 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 is paid at Providence, Rhode Island, and additional mailing oﬃces. Postmaster: Send address changes to Bulletin, 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 Index r, Science Citation TM SM Index –Expanded, ISI Alerting Services , CompuMath Citation Index r, 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 within the 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, 65 Brouwer, Dr. L. E. J. Intuitionism and formalism, 55 Di Francesco, P. Folding and coloring problems in mathematics and physics, 251 Ebenfelt, P. See Baouendi, M. S. Einstein, Albert. Elementary derivation of the equivalence of mass and energy, 39 Fiske, President Thomas S. Mathematical progress in America, 3 Fulton, William. Eigenvalues, invariant factors, highest weights, and Schubert calculus, 209 Hilbert, David. Mathematical problems, 407 McMullen, Curtis T. From dynamics on surfaces to rational points on curves, 119 Morawetz, Cathleen Synge. Variations on conservation laws for the wave equation, 141 Morris, W., and Soltan, V. The Erd˝os-Szekeres problem on points in convex position – a survey, 437 Parshall, Karen Hunger. Perspectives on American mathematics, 381 Pierpont, Professor James. The history of mathematics in the nineteenth century, 9 Poincar´e,Professor H. The present and the future of mathematical physics, 25 Rothschild, Linda Preiss. See Baouendi, M. S. Saari, Donald G. Introductory comments, 1 . Note from the Editor, 379 Soltan, 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, 111 Alexandroff, Paul. Einfachste Grundbegriffe der Topologie, with an introduction by David Hilbert, reviewed by P. A. Smith, 90 Alexandroff, 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 Steve Shkoller, 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, 511 Bartle, Robert G. See Dunford, Nelson Bass, H.; Buium, A.; and Cassidy, P. J. (editors). Selected works of Ellis Kolchin, reviewed by Andy R. Magid, 337 Birkhoff, G. D. Dynamical systems, reviewed by B. O. Koopman, 88 Bourbaki, N. Elémentsde´ mathématique, reviewed by E. Artin, 112 Buium, A. See Bass, H. Burnside, W. Theory of groups of a finite order, reviewed by G. A. Miller, 80 Carathéodory, C. Funktionentheorie, reviewed by Maurice Heins, 108 Cartan, Elie.´ La théoriedes groupes finis et continus et la géométriedifférentielle traitéespar méthode du repère mobile, reviewed by Hermann Weyl, 96 Cartan, Henri, and Eilenberg, Samuel. Homological algebra, reviewed by S. Mac Lane, 113 Cassidy, P. J. See Bass, H. Chevalley, C. Introduction to the theory of algebraic functions of one variable, reviewed by A. Weil, 103 Chevalley, Claude. The theory of Lie groups, I, reviewed by P. A. Smith, 105 Chriss, Neil, and Ginzburg, Victor. Representation theory and complex geometry, reviewed by Ivan Mirković,343 Courant, R., and Hilbert, D. Methoden der mathematischen Physik, Vol. 2, reviewed by Hermann Weyl, 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, 107 Curtis, Charles W. Pioneers of representation theory, reviewed by J. E. Humphreys, 359 Dieudonné,J. See Grothendieck, A. Donkin, Stephen. The q-Schur algebra, reviewed by Brian Parshall, 467 van den Dries, Lou. Tame topology and o-minimal structures, reviewed by David Marker, 351 Dunford, 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, 118 Ebenfelt, Peter. See Baouendi, M. Salah Edgar, G. Integral, probability, and fractal measures, reviewed by Lars Olsen, 481 Eilenberg, S., and Steenrod, N. Foundations of algebraic topology, reviewed by E. H. Spanier, 114 Eilenberg, Samuel. See Cartan, Henri Etingof, Pavel I; Frenkel, Igor B.; and Kirillov, Alexander, A., Jr. Lectures on representation theory and Knizhnik-Zamolodchikov equations, reviewed by Vyjayanthi Chari, 161 Evans, L. C. Partial differential equations, reviewed by Jeffrey Rauch, 363 Feller, 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, 82 Frenkel, 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, 183 Ginzburg, Victor. See Chriss, Neil Gödel,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 mathématique, reviewed by W. F. Osgood, 81 Grothendieck, A. Elémentsde´ géométriealgébrique,rédigésavec la collaboration de J. Dieu- donné, reviewed by S. Lang, 115 Hardy, G. H., and Wright, E. M. An introduction to the theory of numbers, reviewed by E. T. Bell, 99 Hilbert, D. See Courant, R. Hilbert, David. Poincaré’sreview of Hilbert’s Foundations of geometry, reviewed by H. Poincaré, 77 Hodge, W. V. D. The theory and applications of harmonic integrals, reviewed by D. J. Struik, 102 Hopf, Heinz. See Alexandroff, Paul Jost, J. Riemannian geometry and geometric analysis, reviewed by Werner Ballmann, 459 Kapranov, 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 by Günter M. Ziegler, 205 Kolmogoroff, A. Grundbegriffe der Wahrscheinlichkeitsrechnung, reviewed by H. L. Rietz, 91 Krasil’shchik, I. S., and Vinogradov, A. M. Symmetries and conservation laws for differential equations of mathematical physics, reviewed by Peter J. Olver, 369 Landau, Edmund. Vorlesungen über Zahlentheorie, reviewed by G. D. Birkhoff, 86 Laugwitz, Detlef. Bernhard Riemann, 1826–1866: Turning points in the conception of mathematics, reviewed by Roger Cooke, 477 Lebesgue, H. Le¸cons sur l’intégration et la recherche des fonctions primitives, reviewed by T. H. Hildebrandt, 87 Lie, Sophus. Vorlesungen über 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, reviewed by Brian Parshall, 467 Mordell, L. J. Three lectures on Fermat’s last theorem, reviewed by G. E. Wahlin, 84 Morgenstern, Oskar. See von Neumann, John Murnaghan, Francis D. The theory of group representations, reviewed by R. Brauer, 100 Murty, M. Ram, and Murty, V. Kumar. Non–vanishing of L–functions and applications, reviewed by Dorian Goldfeld, 155 Murty, V. Kumar. See Murty, M. Ram Navarro, Gabriel. Characters and blocks of finite groups, reviewed by David Gluck, 169 von Neumann, John, and Morgenstern, Oskar. Theory of games and economic behavior, reviewed by A. H. Copeland, 103 Osgood, Dr. W. F. Lehrbuch der Funktionentheorie, reviewed by E. B. Van Vleck, 83 Petersen, P. Riemannian geometry, reviewed by Werner Ballmann, 459 Ramakrishnan, D., and Valenza, R. J. Fourier analysis on number fields, reviewed by Jeremy Teitelbaum, 373 Reidemeister, K. Knotentheorie, reviewed by P. A. Smith, 90 Reidemeister, Kurt. Einführungin die kombinatorische Topologie, reviewed by P. A. Smith, 90 Riesz, F., and Sz.-Nagy, B. Le¸consd’analyse functionelle, reviewed by E. R. Lorch, 111 Rota, Gian-Carlo. See Klain, Daniel A. Rothschild, Linda Preiss. See Baouendi, M. Salah Roy, Ranjan. See Andrews, George E. Sakai, T. Riemannian geometry, reviewed by Werner Ballmann, 459 Schwartz, Jacob T. See Dunford, Nelson Seifert, H., and Threlfall, W. Lehrbuch der Topologie, reviewed by A. W. Tucker, 92 Smart, Nigel P. The algorithmic resolution of Diophantine equations, reviewed by Maurice Mignotte, 207 Srinivasan, Bhama. Editor’s introduction, 75 Steenrod, N. See Eilenberg, S. Steinitz, Ernst. Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie, edited and completed by Hans Rademacher, reviewed by A. W. Tucker, 92 Sz.-Nagy, B. See Riesz, F. Threlfall, W. See Seifert, H. Titchmarsh, E. C. The theory of the Riemann zeta-function, reviewed by N. Levinson, 110 Traub, J. F., and Werschulz, A. G. Complexity and information, reviewed by Mark A. Kon, 199 Valenza, R. J. See Ramakrishnan, D. Veblen, Oswald. The Cambridge Colloquium, 1916, Part II. Analysis Situs, reviewed by S. Lef- schetz, 85 Vinogradov, A. M. See Krasil’shchik, I. S. van der Waerden, B. L. Moderne algebra, reviewed by O. Ore, 89 Weil, André. Foundations of algebraic geometry, reviewed by Oscar Zariski, 106 Werschulz, A. G. See Traub, J. F. Weyl, Hermann. The classical groups, reviewed by N. Jacobson, 100 Wright, E. M. See Hardy, G. H. Zariski, Oscar. Algebraic surfaces, reviewed by Solomon Lefschetz, 94 Zelevinsky, A. V. See Gelfand, I. M. Zygmund, A. Trigonometric series, 2d. ed., reviewed by R. Salem, 96 Zygmund, Antoni. Trigonometric series, reviewed by J. D. Tamarkin, 95 TITLE PAGE

M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ

N ΕΙΣΙΤΩ S A O

C C I

R E

E T

VOLUME 37 ΑΓΕΩΜΕ

Y 2000

M A

F O 8 U 88 NDED 1 ULLETIN ( N E W S E R I E S ) B OF THE A 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

EDITORS Donald G. Saari, Chief Editor Bhama Srinivasan, Book Reviews

ASSOCIATE EDITORS John C. Baez Philip Protter L. Craig Evans Hugo Rossi John M. Franks Daniel Ruberman Eric Friedlander Audrey A. Terras Craig L. Huneke David Vogan Douglas A. Lind Alex J. Wilkie Barry Mazur Wolfgang Ziller Haynes R. Miller

PROVIDENCE, RHODE ISLAND USA ISSN 0273-0979 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 significant 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 are normally solicited and often written by a disinterested expert. Book Reviews Book Reviews are accepted for publication by invitation only. Unsolicited manuscripts will not be considered.

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 nineteenth century ...... 9 Professor H. Poincar´e, The present and the future of mathematical physics 25 Albert Einstein, Elementary derivation of the equivalence of mass and energy ...... 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. Poincaré (Reviewer), Poincaré’s review of Hilbert’s Foundations of geometry, by David Hilbert ...... 77 J. M. Brooks (Reviewer), Vorlesungen über continuirliche Gruppen mit geometrischen 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 mathématique, by Edouard´ Goursat ...... 81 L. E. Dickson (Reviewer), Festschrift zur Feier des 100 Geburtstages Eduard Kummers 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. Analysis Situs, by Oswald Veblen ...... 85 G. D. Birkhoff (Reviewer), Vorlesungen über Zahlentheorie, by Edmund Landau ...... 86 T. H. Hildebrandt (Reviewer), Le¸conssur l’intégrationet la recherche des fonctions 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 Paul Alexandroff, with an introduction by David Hilbert; Einführungin die kombinatorische Topologie, by Kurt Reidemeister; Knotentheorie, by K. Reidemeister ...... 90 H. L. Rietz (Reviewer), Grundbegriffe der Wahrscheinlichkeitsrechnung, by A. Kolmogoroff ...... 91 A. W. Tucker (Reviewer), Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie, by Ernst Steinitz; edited and completed by Hans Rademacher; Lehrbuch der Topologie, by H. Seifert and 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 théoriedes groupes finis et continus et la géométriedifférentielle traitéespar méthode du repèremobile, by Elie´ Cartan ...... 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 the generalized continuum-hypothesis with the axioms of set theory, by Kurt Gödel...... 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, by John von Neumann and Oskar Morgenstern ...... 103 A. Weil (Reviewer), Introduction to the theory of algebraic functions of one variable, 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 André Weil ...... 106 C. B. Allendoerfer (Reviewer), Regular polytopes, by H. S. M. Coxeter . 107 Maurice Heins (Reviewer), Funktionentheorie, by C. Carathéodory ...... 108 J. Wolfowitz (Reviewer), An introduction to probability theory and its applications, 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), Le¸consd’analyse functionelle, by F. Riesz and B. Sz.-Nagy ...... 111 E. Artin (Reviewer), Eléments´ de mathématique, by N. Bourbaki ...... 112 S. Mac Lane (Reviewer), Homological algebra, by Henri Cartan and Samuel Eilenberg ...... 113 E. H. Spanier (Reviewer), Foundations of algebraic topology, by S. Eilenberg and N. Steenrod ...... 114 S. Lang (Reviewer), Eléments´ de géométriealgébrique, by A. Grothendieck, rédigésavec la collaboration de J. Dieudonné ...... 115 David A. Buchsbaum (Reviewer), Homology, by Saunders Mac Lane . . . 117 Gian-Carlo Rota (Reviewer), Linear operators. Part II. Spectral theory, by Nelson Dunford and Jacob T. Schwartz, with the assistance of William G. 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 on curves ...... 119 Cathleen Synge Morawetz, Variations on conservation laws for the wave equation ...... 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 and Knizhnik-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 Gabriel Navarro ...... 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. Traub and A. G. Werschulz ...... 199 G¨unter M. Ziegler (Reviewer), Introduction to geometric probability, by Daniel A. Klain and Gian-Carlo Rota ...... 205 Maurice Mignotte (Reviewer), The algorithmic resolution of Diophantine equations, by Nigel P. Smart ...... 207

Vol. 37, No. 3 July 2000

Research-Expository Surveys William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus ...... 209 P. Di Francesco, Folding and coloring problems in mathematics and physics 251 M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Local geometric 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 Mirković (Reviewer), Representation theory and complex geometry, by Neil Chriss and Victor Ginzburg ...... 343 David Marker (Reviewer), Tame topology and o-minimal structures, by Lou van den Dries ...... 351 J. E. Humphreys (Reviewer), Pioneers of representation theory, by Charles W. Curtis ...... 359 Jeffrey Rauch (Reviewer), Partial differential equations, by L. C. Evans . . . 363 Peter J. Olver (Reviewer), Symmetries and conservation laws for differential equations 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 Erd˝os-Szekeres problem on points in convex position – a survey ...... 437

Book Reviews Werner Ballmann (Reviewer), Riemannian geometry and geometric analysis, by J. Jost; Riemannian geometry, by P. Petersen; Riemannian geometry, by T. Sakai ...... 459 Brian Parshall (Reviewer), The q-Schur algebra, by Stephen Donkin; Iwahori-Hecke algebras and Schur algebras of the symmetric group, by Andrew Mathas ...... 467 V. Batyrev (Reviewer), Mirror symmetry and algebraic geometry, by David A. Cox and Sheldon Katz ...... 473 Roger Cooke (Reviewer), Bernhard Riemann, 1826–1866: Turning points in the 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, Richard Askey, and Ranjan Roy ...... 499 Nancy K. Stanton (Reviewer), Real submanifolds in complex space and their mappings, by M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild ...... 511 Editorial Board for Research-Expository Surveys John C. Baez Douglas A. Lind Craig L. Huneke Donald G. Saari, Chair

Editorial Board for Research Reports John M. Franks Hugo Rossi Eric Friedlander Daniel Ruberman Barry Mazur Donald G. Saari, Chair Haynes R. Miller David Vogan

Editorial Board for Book Reviews L. Craig Evans Audrey A. Terras Philip Protter Alex J. Wilkie Bhama Srinivasan, Chair Wolfgang 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 journal in the Providence office that have not yet gone to the printer on the above date by the number of articles assigned to issues to be published in four months (the time necessary for editing and composing a typical issue). In an effort to make articles available as quickly as possible, articles are posted to e-MATH individually soon after proof is returned from authors and before appearing in an issue. A Consent to Publish and Copyright Agreement is required before a paper will be published in this journal. After a paper is accepted for publication, the Providence office will send a Consent to Publish and Copyright Agreement to all authors of the paper. By submitting a paper to this journal, authors certify that the results have not been submitted to nor are they under consideration for publication by another journal, 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 also sent to Professor Donald G. Saari. The author should keep one copy. Research- Expository Surveys and Research Reports can also be submitted electronically in AMS-LATEX or AMS-TEX to [email protected]. IF an editor is agreeable, an electronic manuscript prepared in TEX or LATEX may be submitted by pointing to an appropriate URL on a preprint or e-print server. For Research-Expository Surveys and Research Reports, the first page must consist of a descriptive title, followed by an abstract that summarizes the article in language suitable for workers in the general field (algebra, analysis, etc.). The descriptive title should be short but informative; useless or vague phrases such as “some remarks about” or “concerning” should be avoided. The abstract should be a brief technical description of the new material. Then, for Research Reports there should be an introductory section addressed to nonexperts describing the motivation, background, and significance of the results announced. Following the statement of results, there should be a sketch of proofs that may be addressed to experts. Included with the footnotes in each paper should be the 2000 Mathe- matics Subject Classification representing the primary and secondary subjects of the article. The classifications are accessible from www.ams.org/msc/. The list of classifications is also available in print starting with the 1999 annual index of Mathematical Reviews. The Mathematics Subject Classification footnote may be followed by a list of key words and phrases describing the subject matter of the article and taken from it. Journal abbreviations used in bibliographies are listed in the latest Mathematical Reviews annual index. The series abbreviations are also accessible from www.ams.org/publications/. To help in preparing and verifying references, the AMS offers MR Lookup, a Reference Tool for Linking, at www.ams.org/mrlookup/. When the manuscript is submitted, authors should sup- ply the editor with electronic addresses if available. These will be printed after the postal 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 should also be a footnote with the 2000 Mathematics Subject Classification representing the primary and secondary subjects of the book under review. The classifications are accessible from www.ams.org/msc/ and are also available in print starting with the 1999 annual index of Mathematical Reviews. To help in preparing and verifying references, the AMS offers MR Lookup, a Reference Tool for Linking, at www.ams.org/mrlookup/.

Electronically prepared manuscripts. The AMS encourages electronically prepared manuscripts, with a strong preference for AMS-LATEX. To this end, the Society 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 design specifications of that publication series. Articles properly prepared using the AMS- LATEX style file and the \label and \ref commands automatically enable extensive intra-document linking to the bibliography and other elements of the article for searching electronically on the Web. Because linking must often be added manu- ally to electronically prepared manuscripts in other forms of TEX, using AMS-LATEX also reduces the amount of technical intervention once the files are received by the AMS. This results in fewer errors in processing and saves the author proofreading time. 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 also available in AMS-TEX. Those authors who make use of these style files from the beginning of the writing process will further reduce their own efforts. Manuscripts prepared electronically in LATEX or plain TEX are normally not acceptable due to the high amount of technical time required to insure that the file will run properly through the AMS in-house production system. LATEX users will find that AMS- LATEX is the same as LATEX with additional commands to simplify the typesetting of mathematics, and users of plain TEX should have the foundation for learning AMS-LATEX. Authors may retrieve an author package from e-MATH starting from www.ams. org/tex/ or via FTP to ftp.ams.org (login as anonymous, enter username as password, and type cd pub/author-info). The AMS Author Handbook and the Instruction Manual are available in PDF format following the author packages link from www.ams.org/tex/. The author package can also be obtained free of charge by sending email to [email protected] (Internet) or from the Publication Divi- sion, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248. When requesting an author package, please specify AMS-LATEX or AMS-TEX, Mac- intosh or IBM (3.5) format, and the publication in which your paper will appear. Please be sure to include your complete mailing address. At the time of submission, authors should indicate if the paper has been prepared using AMS-LATEX or AMS-TEX and provide the Editor with a paper manuscript that matches the electronic manuscript. The final version of the electronic manuscript should be sent to the Providence office immediately after the paper has been accepted for publication. The author should also send the final version of the paper manuscript to the Editor, who will forward a copy to the Providence office. Editors will require authors to send their electronically prepared manuscripts to the Providence office in a timely fashion. Electronically prepared manuscripts can be sent via email to [email protected] (Internet) or on diskette to the Electronic Prepress Department, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248. When sending a manuscript electronically, please be sure to include a message indicating in which publication the paper has been accepted. No cor- rections will be accepted electronically. Authors must mark their changes on their proof copies and return them to the Providence office. Complete instructions on how to send files are included in the author package. Electronic graphics. Comprehensive instructions on preparing graphics are available at www.ams.org/jourhtml/graphics.html. A few of the major requirements are given here. Submit files for graphics as EPS (Encapsulated PostScript) files. This includes graphics originated via a graphics application as well as scanned photographs or other computer-generated images. If this is not possible, TIFF files are acceptable as long as they can be opened in Adobe Photoshop or Illustrator. No matter what method was used to produce the graphic, it is necessary to provide a paper copy to the AMS. Authors using graphics packages for the creation of electronic art should also avoid the use of any lines thinner than 0.5 points in width. Many graphics packages allow the user to specify a “hairline” for a very thin line. Hairlines often look acceptable when proofed on a typical laser printer. However, when produced on a high-resolution laser imagesetter, hairlines become nearly invisible and will be lost entirely in the final printing process. Screens should be set to values between 15% and 85%. Screens which fall outside of this range are too light or too dark to print correctly. Variations of screens within a graphic should be no less than 10%. Secure manuscript tracking on the Web and via email. Authors can track their manuscripts through the AMS journal production process using the personal AMS ID and Article ID printed in the upper right-hand corner of the Consent to Publish form sent to each author who publishes in AMS journals. Access to the tracking system is available from www.ams.org/mstrack/ or via email sent to [email protected]. To access by email, on the subject line of the message simply enter the AMS ID and Article ID. To track more than one manuscript by email, choose one of the Article IDs and enter the AMS ID and the Article ID followed by the word all on the subject line. An explanation of each production step is provided on the Web through links from the manuscript tracking screen. Questions can be sent to [email protected]. TEX files available. Beginning with the January 1992 issue of the Bulletin and the January 1996 issues of Transactions, Proceedings, Mathematics of Computation, and the Journal of the AMS, TEX files can be downloaded from e-MATH, starting from www.ams.org/journals/. Authors without Web access may request their files at the address given below after the article has been published. For Bulletin papers published in 1987 through 1991 and for Transactions, Proceedings, Mathematics of Computation, and the Journal of the AMS papers published in 1987 through 1995, TEX files are available upon request for authors without Web access by sending email to [email protected] or by contacting the Electronic Prepress Department, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248. The request should include the title of the paper, the name(s) of the author(s), the name of the publication in which the paper has or will appear, and the volume and issue numbers if known. The TEX file will be sent to the author making the request after the article goes to the printer. If the requestor can receive Internet email, please include the email address to which the file should be sent. Otherwise please indicate a diskette format and postal address to which a disk should be mailed. Note: Because TEX production at the AMS sometimes requires extra fonts and macros that are not yet publicly available, TEX files cannot be guaranteed to run through the author’s version of TEX without errors. The AMS regrets that it cannot provide support to eliminate such errors in the author’s TEX environment. Inquiries. Any inquiries concerning a paper that has been accepted for publication that cannot be answered via the manuscript tracking system mentioned above should be sent to [email protected] or directly to the Electronic Pre- press Department, American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248.

Research-Expository Surveys This section is intended for reasonably broad expository surveys of a currently active area of mathematical research. To stress this fact, the name of the section has 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 a research mathematician from a remote area—will be able to read an article in order to get an overview of a subject. A well-written article will include motivating problems 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 those wishing to pursue the subject further. Usually Research-Expository Surveys will be 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 the Research-Expository Surveys, and those invited to give these lectures are informed of this tradition and of the above guidelines. Research-Expository Surveys are also frequently solicited by the Editorial Board, but unsolicited manuscripts are also encouraged; in particular, those invited to give invited hour addresses at meetings of the Society should consider writing up their lecture in a manner that meets the requirements of the preceding paragraph and submitting their manuscripts for consideration 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 understandable by mathematicians who are not experts in the subject matter of the Report. The context giving significance to the research being reported on should be made clear, and the consequences or potential consequences should be described. Usually only the barest hint of the methods of proof should be given, although elements of the proof which are novel should be included. In any case, references should be given so that an interested reader can find the details. It is expected that Research Reports will frequently be written by specialists other than the individuals immediately involved in the research. Research Reports are normally solicited by the “Editorial Board for Research Reports”. They cordially invite suggestions for topics and reporters. Unsolicited manuscripts, submitted to a member of the Editorial Board for Research Reports, will also be considered. AMERICAN MATHEMATICAL SOCIETY Electronic esearch nnouncements R OF THE A AMERICAN MATHEMATICAL SOCIETY

MANAGING EDITOR Most recent articles Svetlana Katok Brian Smith and Gilbert Weinstein, On the connectedness of the space of initial data for the Einstein equations EDITORIAL BOARD Stuart Antman A. V. Grishin, On non-Spechtianness of the variety of associative x32 = 0 Douglas Arnold rings that satisfy the identity David Benson Michael Hutchings, Frank Morgan, Manuel Ritoré, and Dmitri Burago Antonio Ros, Proof of the double bubble conjecture Mark Freidlin A. Giambruno and M. Zaicev, Minimal varieties of algebras of James Glimm exponential growth Ronald Graham R Yitzhak Katznelson Danny Calegari, Geometry and topology of -covered foliations David Kazhdan S. R. Bullett and W. J. Harvey, Mating quadratic maps with Alexander Kechris Kleinian groups via quasiconformal surgery Alexandre Kirillov A. A. Kirillov, Family algebras Frances Kirwan Krystyna Kuperberg Alejandro Adem and Jeff H. Smith, On spaces with periodic Robert Lazarsfeld cohomology Gregory Margulis J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Hugh Montgomery Langford, and Masahico Saito, State-sum invariants of knotted Walter Neumann curves and surfaces from quandle cohomology Klaus Schmidt Z. Reichstein and B. Youssin, Parusinski's “Key Lemma” via Richard Schoen algebraic geometry Masamichi Takesaki Michael Taylor Thomas Wolff The American Mathematical Society’s electronic-only journal, Zhihong (Jeff) Xia Electronic Research Announcements of the AMS (ERA-AMS), Don Zagier is available on the World Wide Web at www.ams.org/era. Efim Zelmanov ERA-AMS publishes high-quality research announcements of significant advances in all branches of mathematics. Authors may submit manuscripts to any editor. All papers are reviewed, and the entire Editorial Board must approve the acceptance of any paper. Papers are posted as soon as they are accepted and processed by the AMS.

www.ams.org/era AMERICAN MATHEMATICAL SOCIETY CONFORMAL EOMETRY YNAMICS G AND D A N E LECTRONIC J OURNAL OF THE AMS

Most recent articles MANAGING EDITOR A combination theorem for covering correspondences and Linda Keen an application to mating polynomial maps with Kleinian City University of New York groups Shaun Bullett

EDITORIAL BOARD MEMBERS A census of rational maps Eva Brezin, Rosemary Byrne, Joshua Levy, K.Astala Kevin Pilgrim and Kelly Plummer University of Jyväskylä The role of the Ahlfors ﬁve islands theorem in complex dynamics F.W. Gehring Walter Bergweiler University of Michigan Unbounded components in parameter space of rational maps Tan Lei Peter M. Makienko University of Warwick Examples of uniformly quasiregular mappings Kirsi Peltonen M. Lyubich Nonlinear automorphisms of plane domains SUNY at Stony Brook Timo Erkama

Ford and Dirichlet domains for cyclic subgroups of PSL2(C) 3 3 acting on HR and ∂HR Todd A. Drumm and Jonathan A. Poritz

This journal provides a forum for mathematical work in related ﬁelds broadly described as conformal geometry and dynamics.This includes complex dynamics (and real dynamics using complex techniques), Kleinian groups, hyperbolic geometry,Teichmüller theory, and quasiconformal mappings. www.ams.org/ecgd AMERICAN MATHEMATICAL SOCIETY REPRESENTATION THEORY A N E LECTRONIC J OURNAL OF THE AMS

MANAGING Most recent articles EDITOR On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected David Vogan covering group of Sl(2, R) Massachusetts Bertram Kostant Institute of Technology On square-integrable representations of classical p-adic groups II EDITORIAL Chris Jantzen BOARD MEMBERS Large Schubert varieties Michel Brion and Patrick Polo A. Knapp Harish-Chandra modules for quantum symmetric pairs SUNY at Stony Brook Gail Letzter J. Lepowsky Symmetric polynomials and Uq(sl2) Rutgers University Naihuan Jing Verifying Kottwitz' conjecture by computer G. Lusztig Bill Casselman Massachusetts Institute of Technology Stable nilpotent orbital integrals on real reductive Lie algebras Robert E. Kottwitz D. Milicic Involutions in Weyl groups University of Utah Robert E. Kottwitz B. Speh Differential operators on some nilpotent orbits Cornell University T. 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 representation theory. It seeks to maintain a high standard for exposition as well as for mathematical content. Papers are posted in several different electronic formats on e-MATH via the World Wide Web.