http://dx.doi.org/10.1090/pspum/028.1 MATHEMATICAL DEVELOPMENTS ARISING FROM HILBERT PROBLEMS PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XXVIII, Part 1 MATHEMATICAL DEVELOPMENTS ARISING FROM HILBERT PROBLEMS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1976 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT NORTHERN ILLINOIS UNIVERSITY DEKALB, ILLINOIS MAY 1974 EDITED BY FELIX E. BROWDER Prepared by the American Mathematical Society with partial support from National Science Foundation grant GP-41994 Library of Congress Cataloging in Publication Data |Tp Symposium in Pure Mathematics, Northern Illinois University, 197^. Mathematical developments arising from Hilbert problems 0 (Proceedings of symposia in pure mathematics ; v. 28) Bibliography: p. 1. Mathematics--Congresses. I. Browder, Felix E. II. American Mathematical Society. III. Title. IV. Title: Hilbert problems. V. Series. QA1.S897 197^ 510 76-2CA-37 ISBN 0-8218-11*28-1 AMS (MOS) subject classifications (1970). Primary 00-02. Copyright © 1976 by the American Mathematical Society Reprinted 1977 Printed by the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. CONTENTS Introduction vii Photographs of the speakers x Hilbert's original article 1 Problems of present day mathematics 35 Hilbert's first problem: the continuum hypothesis 81 DONALD A. MARTIN What have we learnt from Hilbert's second problem? 93 G. KREISEL Problem IV: Desarguesian spaces 131 HERBERT BUSEMANN Hilbert's fifth problem and related problems on transformation groups 142 C.T.YANG Hilbert's sixth problem: mathematical treatment of the axioms of physics... 147 A. S. WlGHTMAN Hilbert's seventh problem: on the Gel'fond-Baker method and its applica­ tions 241 R. TIJDEMAN Hilbert's 8th problem: an analogue 269 E. BOMBIERI An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields (Hilbert's problem 8) 275 NICHOLAS M. KATZ Problems concerning prime numbers (Hilbert's problem 8) 307 HUGH L. MONTGOMERY Part 2 Problem 9: the general reciprocity law 311 J.TATE Hilbert's tenth problem. Diophantine equations: positive aspects of a nega­ tive solution 323 MARTIN DAVIS, YURI MATIJASEVIC and JULIA ROBINSON Hilbert's eleventh problem: the arithmetic theory of quadratic forms 379 O. T. O'MEARA CONTENTS Some contemporary problems with origins in the Jugendtraum (Hilbert's problem 12) 401 R. P. LANGLANDS The 13-th problem of Hilbert 419 G. G. LORENTZ Hilbert's fourteenth problem—the finite generation of subrings such as rings of invariants 431 DAVID MUM FORD Problem 15. Rigorous foundation of Schubert's enumerative calculus 445 STEVEN L. KLEIMAN Hilbert's seventeenth problem and related problems on definite forms 483 ALBRECHT PFISTER Hilbert's problem 18: on crystalographic groups, fundamental domains, and on sphere packing 491 J.MILNOR The solvability of boundary value problems (Hilbert's problem 19) 507 JAMES SERRIN Variational problems and elliptic equations (Hilbert's problem 20) 525 ENRICO BOMBIERI An overview of Deligne's work on Hilbert's twenty-first problem 537 NICHOLAS M. KATZ On Hilbert's 22nd problem 559 LIPMAN BERS Hilbert's twenty-third problem: extensions of the calculus of variations 611 GUIDO STAMPACCHIA Introduction In May 1974, the American Mathematical Society sponsored a special Sym­ posium on the mathematical consequences of the Hilbert problems which was held at Northern Illinois University in De Kalb, Illinois. The present volume con­ tains the proceedings of that symposium and includes papers corresponding to all the invited addresses with one exception, that of Dr. J. R. Conway of Cam­ bridge University, England. It contains as well the address of Professor G. Stamp- acchia that could not be delivered at the Symposium because of health problems. The volume includes a number of other features to which we direct the attention of the reader. Thus, we have included photographs of the speakers (by the courtesy of Paul Halmos), and a translation of the text of the Hilbert Problems as published in the Bulletin of the American Mathematical Society of 1903. The papers are published in the order of the problems to which they are affiliated, and not in the alphabetical order of their authors. We should remark explicitly that the central concern of this Symposium was not the explicit solutions (so far as they have been given) of Hilbert's prob­ lems nor commentary upon their details. It was rather an attempt to focus up­ on those areas of importance in contemporary mathematical research which can be seen as descended in some way from the ideas and tendencies put forward by Hilbert in his speech at the International Congress of Mathematicians in Paris in 1900. The connection was direct and unequivocal in a number of cases, and a good deal more ambiguous in others. The Organizing Committee of the Sym­ posium consisted of P. R. Bateman (Secretary), F. E. Browder (Chairman), R. C. Buck, D. Lewis, and D. Zelinsky. Its basic objective was to obtain as broad a representation of significant mathematical research as possible within the general constraint of relevance to the Hilbert problems. An additional unusual feature of the present volume is the article entitled Problems of present day mathematics which appears immediately after the text of Hilbert's article. The development of this material was initiated by Jean Dieudonrie through correspondence with a number of mathematicians through­ out the world. The resulting problems as well as others obtained by the Editor, appear in the form in which they were suggested by the mathematicians whose vii vm INTRODUCTION names are attached to them. The editing process as carried on by Professor Dieudonne or the present Editor consisted simply in selection of problems in cases where suggested problems overlapped seriously in their content. The general reaction of the participants to the Symposium was one of great enthusiasm and approval, both by speakers and audience, except for one point on which comment is appropriate. This concerned the provisional text of the collection of new problems as distributed at the Symposium and discussed at a special hour session. Criticism was put forward of the method by which the problems had been collected and of the scarcity of problems in a number of important directions, particularly in analysis and geometry. The Editor also received correspondence criticizing the texts of several of the draft problems, particularly in so far as they impinged on mathematical physics, logic, and com­ puter science. In the form in which the problems appear in the present volume, I believe that some of these criticisms have been met, particularly in terms of new sets of problems in various areas of geometry and analysis, as well as a new set of problems in mathematical physics. Despite efforts in the direction of logic and computer science, it turned out to be impracticable to obtain problems on the right level in these directions. Two sets of problems, those by Hugh Mont­ gomery on the theory of prime numbers and those by Arthur Wightman on mathematical physics, were transposed from their submitted papers to the Prob­ lem Collection, and in the case of Montgomery's paper, this was the whole of the submitted manuscript. We believe that the final result meets most of the substantive criticisms which have been made. It does not meet the criticism that these problems were obtained by consulting a self-selecting elite. That criticism could have been made with much greater force against the problems presented by Hilbert (see the discussion of the circumstances of Hilbert's address by Con­ stance Reid in her biography of Hilbert). In either case, the significance of the problems depends upon their intrinsic merit, not their origin. The significance of this set of problems can only be judged in the future, in terms of their consequences and the role they may play in focusing the pro­ cess of mathematical discovery. We should not believe that anyone, even Hil­ bert, could or can foresee the mysteries of the future in terms of new discoveries and new turns of interest. Despite the great role of the Hilbert problems as re­ corded in the present volume, we should not fail to note that there is no hint in them of such decisive developments in the following decade as the develop­ ment of topology, both combinatorial and set-theoretic, or of functional analysis (as in the theory of linear integral equations to which Hilbert himself was to devote most of his own efforts). Nor, despite the scope of the Hilbert problems and the breadth of the present Symposium, should we ignore the fact that a rather different Symposium of equal importance could be organized around the INTRODUCTION IX interests and achievements of Hilbert's great contemporary, Henri Poincare. There is obviously an enormous difference between the collection of prob­ lems which is given below as originating from a number of leading mathemati­ cians in different areas of mathematical research and the synthesis of problems made by Hilbert as the product of a single mind. There is no common statement of philosophy or attitude such as Hilbert's article begins with, nor a common conclusion. The whole thrust of the Symposium and of the new problems leads however to a re-endorsement of an essential part of Hilbert's own conclusion: "The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of to-day is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up onto separate branches, whose representatives scarce­ ly understand one another and whose connection becomes ever more loose.