VEBLEN, OSWALD

born June 24, 1880, Decorah, Iowa, U.S. died Aug. 10, 1960, Brooklin, Maine U.S. mathematician who made important contributions to differential geometry and early topology. Many of his contributions found application to atomic physics and relativity. Veblen taught mathematics at Princeton University (190532) and was appointed professor at the Institute for Advanced Study, Princeton, N.J., in 1932. He became professor emeritus in 1950. Veblen's work on the axiom systems of projective geometry came about because of his great interest in the foundations of geometry. John Wesley Young collaborated with him on the first volume of Projective Geometry, 2 vol. (191018; 1965). The first book to systematically cover the basic ideas of topology was Veblen's Analysis Situs (1922), his most influential work and for many years the best available topology text. He also laid the foundations of the Princeton school of topological research. Soon after the discovery of general relativity, Veblen turned to differential geometry and took a leading part in the development of generalized affine and projective geometry. His work, The Invariants of Quadratic Differential Forms (1927), is distinguished by precise and systematic treatment of the basic properties of Riemann geometry. In collaboration with his brilliant student J.H.C. Whitehead, Veblen extended the knowledge of the Riemann metric for more general cases in The Foundations of Differential Geometry (1932). This work provided a clear definition of a differentiable manifold (in the large) and inspired others to refine the definition, the refined definition being basic to modern geometrical research. Veblen's belief that the foundations of geometry must be studied both as a branch of physics and as a branch of mathematics quite naturally led him to the study of relativity and the search for a geometric structure to form a field theory unifying gravitation and electromagnetism. With respect to the Kaluza-Klein field theory, which involved field equations in five-dimensional space, he provided the first physical interpretation of the fifth coordinate. By regarding the coordinate as a gauge variable, he was able to interpret the theory as one involving four-dimensional spacetime. In connection with this contribution, Veblen provided a new treatment of spinors (expressions used to represent electron spin), which is summarized in Projektive Relativittstheorie (1933; Projective Relativity Theory). Veblen played a key role in the formation of the school of mathematics at the Institute for Advanced Study. This role, combined with his tremendous influence in encouraging and developing young mathematicians, represents a contribution equal to that of his mathematical innovations.