The Vision, Insight, and Influence of Oswald Veblen Steve Batterson hen Oswald Veblen assumed the proof of the Jordan Curve Theorem. It was Veblen presidency of the American Math- who made Poincaré’s brilliant algebraic topol- ematical Society in 1923, there ogy concepts accessible to the field’s pioneers. was little support for mathemati- Among the students supervised by Veblen were cal research in the United States. R. L. Moore, Alonzo Church, and J. H. C. Whitehead, WUniversity teaching loads varied from nine hours who went on to become leaders in the development to fifteen hours per week and higher. The birth of of their respective specialities. the National Science Foundation was over a quarter I first came to appreciate Veblen’s significance century in the future. Research when I began research for a book on the origins grants and postdoctoral positions of the Institute for Advanced Study. In discuss- (with little or no teaching) did not ing my project with others it became evident that exist for mathematicians. Veblen’s contributions were largely unknown to As a professor at Princeton the mathematicians of today. Oswald was fre- University, Veblen himself was a quently confused with his more famous uncle, member of a premier mathemat- economist Thorstein Veblen. The objective of this ics department. Yet just four of article is to review Oswald Veblen’s overall influ- Veblen’s colleagues were actively ence as mathematician, mentor, and advocate. I engaged in research, and, among am grateful to Michele Benzi and Albert Lewis for them, James Alexander, Einar their suggestions. Hille, and Joseph Wedderburn worked with no office space. Hille, Mathematical Ascendance as an instructor, was assigned Oswald Veblen was of Norwegian descent. His teaching duties that included two mother Kirsti Hougen was born in Norway while sections of trigonometry. Instruc- his father, Andrew, and Uncle Thorstein were tors at Columbia and Michigan first generation Americans from Wisconsin. The taught twelve and sixteen hours Oswald Veblen, 1924. Hougen and Veblen families settled on nearby per week respectively [1], [2], [3]. farms in Minnesota. Kirsti and Andrew married Veblen took up the task of improving the circumstances for research math- in 1877 [4], [5]. ematicians. His success in this endeavor was On June 25, 1880, Oswald Veblen was born in but one facet of a career that had an enormous Decorah, Iowa, where his father was teaching at impact on mathematics and the profession. For Luther College. Shortly afterward Andrew began example, due entirely to Veblen’s initiative the graduate study at Johns Hopkins. In 1883 the fam- young scholars Kurt Gödel and John von Neumann ily returned to Iowa for Andrew to teach physics were recruited for academic opportunities in the and mathematics at the University of Iowa. There United States. As a mathematician Veblen was Oswald was educated from elementary school no slouch. His work included the first rigorous through an undergraduate degree at the university in 1898. Two years later he received a second B.A. Steve Batterson is associate professor of mathematics and from Harvard. computer science at Emory University. His email address In 1900 Veblen began graduate school at the is [email protected]. University of Chicago. He arrived without any 606 NOTICES OF THE AMS VOLUME 54, NUMBER 5 financial support, “but with a hope that something recommended Veblen. Veblen might turn up.” It did. Another student with a fel- and Eisenhart were joined by lowship dropped out during his first quarter, and Gilbert Bliss and J. W. Young as Veblen was awarded the US$320 stipend. [4/2/38, the first mathematics precep- 5/2/38 Oswald Veblen to John Howe, from Oswald tors [1]. Veblen Papers, Manuscript Division, Library of The selections were superb. Congress] Financial considerations aside, Chicago Each preceptor was both a rising was a wise choice for graduate study in mathemat- scholar and leader. Veblen and ics. The university, begun just eight years earlier, Eisenhart would work with Fine in was in the midst of a historic period in the edu- engineering Princeton’s advance- cation of American mathematicians. Among the ment. Bliss and Young remained rising generation of leaders to receive Ph.D.s there for three years before moving at this time were Leonard Dickson (1896), Gilbert on. Bliss returned to Chicago and Bliss (1900), Veblen (1903), R. L. Moore (1905), and together with Dickson became G. D. Birkhoff (1907). Each would later serve a term the department’s second genera- E. H. Moore, 1902. as president of the AMS. tion of leaders. Young eventually The head of the Chicago department was E. H. settled at Dartmouth where he Moore who was one of the most highly regarded initiated a transformation toward mathematicians in the country. Among Moore’s research. interests was David Hilbert’s recent work on the At Princeton Veblen began foundations of geometry. Veblen attended Moore’s to collaborate with Young on fall 1901 seminar covering this topic. Hilbert had projective geometry. Their work employed the undefined terms of point, line, and culminated in a classic text on plane, to devise a scheme of 20 axioms. Questions the subject. The first volume soon arose over the independence of these axioms. appeared in 1910, with a second In the seminar Moore identified and sharpened the volume slated for the following independence deficiencies. Veblen was inspired year. The scheduling was opti- to go further in his 1903 thesis. Creating a frame- mistic. The authors were sepa- work based on just the two undefined elements of rated, and Young was moving point and order, Veblen proposed twelve axioms from the chair duties at Kansas to for Euclidean geometry. He showed the axioms the the headship at Dartmouth. were independent and that they were satisfied by Veblen, who had taken on other Henry Fine, 1911. a system that was essentially unique [6]. projects as well, finished the sec- Veblen remained at Chicago for two notable ond volume on his own in 1917. postdoctoral years. During this period he polished Some indication of the enduring his thesis for publication, proved a major theorem, value of these books is given by began a collaboration, and joined his advisor E. H. a mid-2006 ISI Web of Knowledge Moore in directing the thesis research of a newly citation search, which recorded arrived student. The student was R. L. Moore and twenty-seven hits among articles the collaborator a young Scottish algebraist, Joseph published since 2000. Wedderburn. The theorem was the long accepted The year 1905 marks the be- Jordan Curve Theorem for which previous proofs ginning of Princeton’s ascen- were unsatisfactory. Veblen was well prepared to dance to world class mathematics go out on his own. standing. Veblen’s fingerprints It was at this time, in 1905, that Princeton Uni- immediately appear all over the versity President Woodrow Wilson embarked on a university’s successful efforts to program to upgrade the university. A key element identify and recruit promising of his plan was the creation of a large number of young talent. Prior to the com- junior faculty positions to reduce class size and pletion of his first year, Veblen G. D. Birkhoff, 1913. individualize undergraduate education. These wrote to R. L. Moore to sound out “preceptors” were to teach traditional classes as his interest in a position. Moore well as provide one-on-one supervision to upper became the fifth preceptor, before going to North- class students. Selection of the mathematics pre- western University and then to the University of ceptors was in the hands of mathematician and Pennsylvania and the University of Texas. With archives. AMS Dean of the Faculty Henry Fine. the departures of Moore, Bliss, and Young, two Fine recognized an opportunity to change the outstanding mathematicians came to Princeton direction of a department that, aside from himself as replacements in 1909. Both had overlapped and the young instructor Luther Eisenhart, had with Veblen at Chicago. They were his collabora- no commitment to research. When Fine contacted tor Wedderburn and G.D. Birkhoff, a later E. H. E. H. Moore to solicit nominations, Moore strongly Moore student. the from are page this on Photographs MAY 2007 NOTICES OF THE AMS 607 Birkhoff was just twenty-five when his appoint- of mathematical research. According to longtime ment began at Princeton. Frequent walks there with Princeton mathematician Albert Tucker, Veblen the four-year-older Veblen cemented a lifelong was sufficiently concerned over the subject’s personal and professional friendship. It would standing that he advised Alexander to obtain his have been fascinating to listen in on these conver- Ph.D. in a more fashionable area [7]. In 1915 Alex- sations, for the interaction undoubtedly stimulated ander published two important papers. His thesis, subsequent developments at this historic time under T. H. Gronwall, was on univalent functions. in the emergence of mathematical research in In another article Alexander established that the America. Birkhoff and Veblen were pursuing dif- homology of a 3-dimensional polyhedron was in- ferent programs of Poincaré. With Birkhoff it was dependent of its triangulation. celestial mechanics that would lead to his seminal Veblen’s standing as a mathematician grew proof of the existence of fixed points for certain steadily at Princeton. An offer from Yale contrib- area preserving maps of the annulus. Veblen be- uted to his 1910 promotion to the rank of full came interested in Poincaré’s conceptualization of professor. At that time, one of the most prestigious algebraic topology (known then as analysis situs) recognitions for an American mathematician was which, to this point, was largely undeveloped. an invitation to deliver the Colloquium Lectures.