Perhaps it is worth at this stage explaining the difference between Columbia University and Columbia College. In 1754 King's College was founded in New York but following the American War of Independence, this name was regarded as inappropriate and the College was renamed Columbia College. In 1912 Columbia College became Columbia University but the university retained the name Columbia College for the undergraduate liberal arts school for men. It was therefore Columbia University that Douglas entered in 1916 to undertake research under the supervision of Edward Kasner. He took part in Kasner's seminar on differential geometry and it was there that Douglas developed a love of geometry. Also at the seminar he first met the Plateau Problem for which he would become famous. He submitted his doctoral thesis On Certain Two-Point Properties of General Families of Curves; The Geometry of Variations in 1920. In the following year he published the main results of his doctoral thesis in the Transactions of the American Mathematical Society.
Douglas continued to undertake research in differential geometry while teaching at Columbia College from 1920 to 1926. His publications from this period are Normal congruences and quadruply infinite systems of curves in space (1924), and A criterion for the conformal equivalence of a Riemann space to a Euclidean space (1925). Then he was awarded a National Research Fellowship and, from 1926 to 1930, he visited Princeton (1926-27), Harvard (1927), Chicago (1928), Paris (1928-30), and Göttingen (1930). It was during this period that he worked out a complete solution to the Plateau problem which had been posed by Lagrange in 1760 and then had been studied by leading mathematicians such as Riemann, Weierstrass and Schwarz. The problem is to prove the existence of a surface of minimal area bounded by a given contour. Before Douglas's solution only special cases of the problem had been solved. In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28). Douglas presented full details of his solution in Solution of the problem of Plateau in the Transactions of the American Mathematical Society in 1931. For this fine achievement he was awarded the Fields Medal at the International Congress of Mathematicians at Oslo in 1936.
In 1930 Douglas was appointed as an assistant professor of mathematics at the Massachusetts Institute of Technology. He was promoted to associate professor in 1934 and spent the year 1934-35 as a research fellow at the Institute for Advanced Study at Princeton. He left the Massachusetts Institute of Technology in 1937, spending another year as a research fellow at the Institute for Advanced Study at Princeton in 1938-39. He received Guggenheim Foundation Fellowships in 1940 and 1941, then was appointed to Brooklyn College and from 1942 to 1954 he taught there and at Columbia University. Douglas married Jessie Nayler on 30 June 1940; they had one son Lewis Philip Douglas.
After giving a complete solution to the Plateau Problem, Douglas went on to study generalisations of it. He published One-sided minimal surfaces with a given boundary (1932) and A Jordan space curve no arc of which can form part of a contour which bounds a finite area (1934). In 1943 Douglas was awarded the Bôcher Prize by the American Mathematical Society for his memoirs on the Plateau Problem. In particular the award was for three papers all published in 1939: Green's function and the problem of Plateau and The most general form of the problem of Plateau published in the American Journal of Mathematics and Solution of the inverse problem of the calculus of variations published in the Proceedings of the National Academy of Sciences. In the first of these Douglas looked at the following form of the problem: Given an aggregate G of k non-intersecting Jordan curves in n-space, to find a minimal surface bounded by G and having a prescribed genus h and a prescribed orientability character (one-sided or two-sided). In the second paper the following problem is studied by Douglas: Given a Riemann surface (or semi-surface) R with boundary C, and given in n-space a topological image G of C, to prove the existence of a minimal surface topologically equivalent to R and bounded by G. The third paper does not give the compete proof for the solution of the inverse problem of the calculus of variations but is an announcement of the result.
These three papers were, amazingly, not the only ones which Douglas published in 1939. He also published The analytic prolongation of a minimal surface across a straight line which gives a generalisation of some earlier results on minimal surfaces with a simpler proof, The higher topological form of Plateau's problem which compares the methods which Douglas used in the first two of his papers which won the Bôcher Prize, and Minimal surfaces of higher topological structure. Another five papers by Douglas appeared in 1940: Theorems in the inverse problem of the calculus of variations; Geometry of polygons in the complex plane; On linear polygon transformations; A converse theorem concerning the diametral locus of an algebraic curve and A new special form of the linear element of a surface. The second and third of these papers generalise the following elementary geometrical theorem: If on each side of any triangle as base an isosceles triangle with 120° as vertex-angle is constructed (always outward or always inward), then the vertices of these isosceles triangles form an equilateral triangle.
In 1942 Douglas published a non-technical survey of the theory of integration. His starting point was the quadrature of a circle and of a segment of a parabola by Archimedes. He then gave the definitions of the Riemann, Riemann-Stieltjes and Lebesgue integrals, and presented their properties. In the 47 page text, Douglas also mentions Fourier series and transforms, Denjoy integrals and the double integrals of Riemann and of Lebesgue. Douglas also worked group theory and, in 1951, studied groups with 2 generators x, y such that every element can be expressed in the form xnym, where n, m are integers. He published a number of papers on this topic entitled On finite groups with two independent generators. He also presented a series of papers On the basis theorem for finite abelian groups.
His wife, Jessie Douglas died in 1955, the year in which Douglas was appointed professor of mathematics at the City College of New York. He remained in that post for the final ten years of his life, living in Butler Hall, 88 Morningside Drive in New York.
Article by: J J O'Connor and E F Robertson