Jesse Douglas (3 July 1897- 7 Oct 1965) was a
mathematician. Born in New York, he was the son Louis Douglas and Sarah Kommel. Jesse developed a love for mathematics while he was studying at
high school in New York. After graduation from high school, he entered
the City College of New York and won the Belden Medal for excellence in
mathematics in his first year at the College. In winning this medal, he
became the youngest person ever to receive it. After an outstanding
undergraduate career he graduated with honours in mathematics in 1916
and entered Columbia University.
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. He died in New York.
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