Oswald Veblen made important contributions to projective and differential geometry, and topology.
Veblen attended school in Iowa City before entering the University of Iowa in 1894 receiving his A.B. in 1898. After a year spent as a laboratory assistant, Veblen spent a year at Harvard University before going to the University of Chicago to undertake research. Archibald writes in [4]:-
He received the major part of his mathematical training at the University of Chicago from that inspiring trio Bolza, Maschke, and Eliakim Moore. Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors. His often quoted dissertation under Eliakim Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899).
Veblen's doctoral dissertation was entitled A System of Axioms for Geometry and he was awarded his doctorate from the University of Chicago in 1903. He taught mathematics at Princeton University from 1905 to 1932. In the academic year 1928-29 he taught at Oxford as part of an exchange with G H Hardy. In 1932 he helped organise the Institute for Advanced Study in Princeton and he became a professor there in 1932.
Veblen's first work on topology appeared just before he arrived in Princeton and Veblen went on to establish Princeton as one of the leading centres in the World for topology research.
His interest in the foundations of geometry led to his work on the axiom systems of projective geometry. Together with John Wesley Young he published Projective geometry (1910-18). The introduction to this work justifies the study of foundations:-
Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view to some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to its foundations. We believe, moreover, that the abstract treatment is particularly desirable in projective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Since it is most natural to derive the geometrical disciplines associated with the names of the names of Euclid, Descartes, Lobachevsky etc. from projective geometry than to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry.
Veblen's Analysis Situs (1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology.
Soon after Einstein's theory of general relativity appeared Veblen turned his attention to differential geometry. This work led to important applications in relativity theory, and much of his work also found application in atomic physics. His work The invariants of quadratic differential forms (1927) is a systematic treatment of Riemann geometry while his work, written jointly with his student Henry Whitehead, The foundations of differential geometry (1933) gives the first definition of a differentiable manifold.
In Projective relativity theory (1933) he gave a new treatment of spinors, used to represent electron spin.
Veblen was an active member of the American Mathematical Society, serving the Society as Vice-President in 1915 and President in 1923-24. He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology.
He was honoured with memberships of other societies around the World. For example he was a member of the London Mathematical Society, serving on the council in 1928 when he was replacing Hardy at Oxford. Oxford further honoured him with an Honorary D.Sc. in 1929, while in the same year he was honoured by the University of Oslo on the occasion of the centenary celebrations for Abel.
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