Following his graduation from school, van Kampen entered the University of Leiden. After graduating with a first degree he continued to undertake research in mathematics. In 1927 he travelled to Göttingen where he met van der Waerden who was undertaking research there for his habilitation. He also met Aleksandrov in Göttingen, for around this period Aleksandrov spent every summer there. Discussions with these two mathematicians proved important for van Kampen's research and he began to study ways of making a satisfactory topological definition of a variety. Back at the University of Leiden, his research was supervised by Willem van der Woude, himself a student of Pieter Schoute. Van Kampen submitted his thesis Die kombinatorische Topologie und die Dualitätssatze to the University of Leiden and was awarded his doctorate in 1929.
Before the award of his doctorate, van Kampen had spent the summer months of 1928 at the University of Hamburg where he worked with Artin. Van Kampen's first published work was an example of a knot which appeared in the Hamburger Abhandlungen in 1928. The knot provided a counterexample to a result which Artin had claimed to be true in 1925. At this stage in his career van Kampen was approached by Johns Hopkins University in the United States and offered a position but he was still too young to enter the United States so decided to work in Europe before taking up the offer from Johns Hopkins. He accepted an appointment as an assistant to Schouten, who was professor of mathematics at the University of Delft. Schouten worked all his life on tensor analysis and although this seems quite far removed from the topics that van Kampen had been undertaking research on, nevertheless he collaborated with Schouten on three papers on tensor analysis, published in 1930, 1931 and 1933. However he continued to work on topological ideas, in particular embedding complexes in Euclidean space.
In 1931 van Kampen left Europe and travelled to the United States to take up the position which he had been offered at Johns Hopkins University in Baltimore, Maryland. There he met Zariski who had taught at Johns Hopkins University as a Johnston Scholar from 1927 until 1929 when he had joined the Faculty. Zariski had been working on the fundamental group of the complement of an algebraic curve, and he had found generators and relations for the fundamental group but was unable to show that he had found sufficient relations to give a presentation for the group. Van Kampen solved the problem, showing that Zariski's relations were sufficient, and the result is now known as the Zariski-van Kampen theorem.
Van Kampen spent the year 1933 at Princeton University where J W Alexander, A Einstein, M Morse, O Veblen, von Neumann, and H Weyl were working at the newly founded Institute for Advanced Study. Shortly before this Pontryagin, who had been working on problems in topology and algebra, had been studying duality. He had proved that compact abelian groups are dual to discrete abelian groups, and von Neumann was interested in extending this result. Van Kampen became interested in Pontryagin's duality and wrote sixteen papers on this topic, including an excellent survey article published in 1935.
In 1935 van Kampen began to work in a different area of mathematics when he became interested in the work which Wintner was undertaking. Wintner had worked at Johns Hopkins since 1930, the year before van Kampen arrived, and his interests were in almost-periodic functions and differential equations. Van Kampen had become involved with the study of almost-periodic functions when he had visited Princeton, and von Neumann had explained to him how Pontryagin's duality results could be extended using almost-periodic functions. Van Kampen published a paper on almost-periodic functions in the Journal of the London Mathematical Society in 1937, having published his first joint paper with Wintner On the canonical transformations of Hamiltonian systems in the American Journal of Mathematics in the previous year.
By the late 1930s van Kampen was complaining of headaches in letters which he wrote to his family back in Belgium. At first doctors thought that the problem was coming from his neck and he had physiotherapy to try to cure the pain. However the headaches worsened and cancer was diagnosed, originating from a birth mark near his left ear. In April 1941 van Kampen entered hospital and doctors operated to remove the malignant growth. At first the operation was thought to have been successful and van Kampen resumed teaching in the autumn of 1941. However the headaches soon returned even more severe than before, and he lost the hearing in his left ear. With his health rapidly deteriorating, van Kampen entered hospital again in December 1941 and another operation was carried out in January 1942. It was not successful and van Kampen lapsed into unconsciousness on 10 February and died on the following day.
During his final illness van Kampen's papers were still appearing in print. Seven papers appeared in 1939, four on statistics and three on almost periodic functions. Wintner was a joint author of four of these papers, Kac of three, and two had van Kampen as sole author. Five papers appeared in 1940, one of them a major article over 30 pages in length in the American Journal of Mathematics with the title Infinite product measures and infinite convolutions. Also in 1940 he published a paper written jointly with Erdős, Kac and Wintner: Ramanujan sums and almost periodic functions. Three more papers appeared in print in 1941 and at the time of his death he had 54 papers in print, a remarkable achievement over a period of 12 years. Five years after van Kampen's death Wintner published a paper On the asymptotic distribution of geodesics on surfaces of revolution which he made a joint work with van Kampen since it contained ideas on which the two had been working.
Article by: J J O'Connor and E F Robertson