in generalized recursion theory and descriptive set theory: Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society 79 (1955), pp. 312–340; On the forms of the predicates in the theory of constructive ordinals (second paper), American Journal of Mathematics 77 (1955), pp. 405–428; and Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society 61 (1955), pp. 193–213. 1983 Shiing-Shen Chern for the cumulative influence of his total mathematical work, high level of research over a period of time, particular influence on the development of the field of differential geometry, and influence on mathematics through Ph.D. students. 1982 Lars Ahlfors for his expository work in Complex analysis (McGraw–Hill Book Company, New York, 1953), and in Lectures on quasiconformal mappings (D. Van Nostrand Co., Inc., New York, 1966) and Conformal invariants (McGraw–Hill Book Company, New York, 1973). 1982 Tsit Yuen Lam for his expository work in his book Algebraic theory of quadratic forms (1973), and four of his papers: K_0 and K_1-an introduction to algebraic K-theory (1975), Ten lectures on quadratic forms over fields (1977), Serre's conjecture (1978), and The theory of ordered fields (1980). 1982 John W. Milnor for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics (2) 64 (1956), pp. 399–405. 1982 Fritz John for the cumulative influence of his total mathematical work, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students. 1981 Oscar Zariski for his work in algebraic geometry, especially his fundamental contributions to the algebraic foundations of this subject. 1981 Eberhard Hopf for three papers of fundamental and lasting importance: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Mathematisch-Naturwissenschaftliche Klasse, volume 95 (1943), pp. 3–22; A mathematical example displaying features of turbulence, Communications on Applied Mathematics, volume 1 (1948), pp. 303–322; and The partial differential equation u_t + uu_x = u_{xx}, Communications on Pure and Applied Mathematics, volume 3 (1950), pp. 201–230. 1981 Nelson Dunford, Jacob T. Schwartz for their expository book, Linear operators, Part I, General theory, 1958; Part II, Spectral theory, 1963; Part III, Spectral operators, 1971, Interscience Publishers, New York. 1980 André Weil for the total effect of his work on the general course of twentieth century mathematics, especially in the many areas in