In May 1974, the American Mathematical Society sponsored a special symposium on the mathematical consequences of the Hilbert problems, held at Northern Illinois University, DeKalb, Illinois. The central concern of the symposium was to focus upon 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 Organizing Committee's 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. The Committee consisted of P. R. Bateman (secretary), F. E. Browder (chairman), R. C. Buck, D. Lewis, and D. Zelinsky. The volume contains the proceedings of that symposium and includes papers corresponding to all the invited addresses with one exception.It contains as well the address of Professor B. Stanpacchia that could not be delivered at the symposium because of health problems. The volume includes 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 filiated, and not in the alphabetical order of their authors. An additional unusual feature of the 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 Dieudonne through correspondence with a number of mathematicians throughout the world. The resulting problems, as well as others obtained by the editor, appear in the form in which they were suggested.

Table of Contents:
Part 1: Hilbert's first problem: The continuum hypothesis by D. A. Martin What have we learnt from Hilbert's second problem? by G. Kreisel Problem IV: Desarguesian spaces by H. Busemann Hilbert's fifth problem and related problems on transformation groups by C. T. Yang Hilbert's sixth problem: Mathematical treatment of the axioms of physics by A. S. Wightman Hilbert's seventh problem: On the Gelfond-Baker method and its applications by R. Tijdeman Hilbert's 8th problem: An analogue by E. Bombieri An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields (Hilbert's problem 8) by N. M. Katz Problems concerning prime numbers (Hilbert's problem 8) by H. L. Montgomery Part 2: Problem 9: The general reciprocity law by J. Tate Hilbert's tenth problem. Diophantine equations: Positive aspects of a negative solution by M. Davis, Y. Matijasevic, and J. Robinson Hilbert's eleventh problem: The arithmetic theory of quadratic forms by O. T. O'Meara Some contemporary problems with origins in the Jugendtraum (Hilbert's problem 12) by R. P. Langlands The 13-th problem of Hilbert by G. G. Lorentz Hilbert's fourteenth problem--the finite generation of subrings such as rings of invariants by D. Mumford Problem 15. Rigorous foundation of Schubert's enumerative calculus by S. L. Kleiman Hilbert's seventeenth problem and related problems on definite forms by A. Pfister Hilbert's problem 18: On crystalographic groups, fundamental domains, and on sphere packing by J. Milnor The solvability of boundary value problems (Hilbert's problem 19) by J. Serrin Variational problems and elliptic equations (Hilbert's problem 20) by E. Bombieri An overview of Deligne's work on Hilbert's twenty-first problem by N. M. Katz Hilbert's twenty-third problem: Extensions of the calculus of variations by G. Stampacchia.