杰西·道格拉斯Jesse Douglas

杰西·道格拉斯的父母是Louis Douglas和Sarah Kommel。道格拉斯在纽约上高中时培养了对数学的热爱。高中毕业后，他进入纽约城市学院，并在大学第一年赢得了贝尔登数学卓越奖章。赢得这枚奖章使他成为有史以来获得该奖章的最年轻的人。在出色的本科生涯之后，他于1916年以数学荣誉学位毕业，并进入哥伦比亚大学。

也许在这一点上值得解释一下哥伦比亚大学和哥伦比亚学院的区别。1754年，国王学院在纽约成立，但在美国独立战争之后，这个名字被认为不合适，学院更名为哥伦比亚学院。1912年，哥伦比亚学院成为哥伦比亚大学，但大学保留了哥伦比亚学院这个名字作为男子本科文理学院。因此，道格拉斯于1916年进入的是哥伦比亚大学，在爱德华·卡斯勒的指导下进行研究。他参加了爱德华·卡斯勒关于微分几何的讨论班，正是在那里，道格拉斯培养了对几何的热爱。也是在讨论班上，他第一次遇到了后来使他成名的约瑟夫·普拉托问题。他于1920年提交了博士学位论文On Certain Two-Point Properties of General Families of Curves; The Geometry of Variations。次年，他在Transactions of the American Mathematical Society上发表了其博士学位论文的主要结果。

道格拉斯从1920年到1926年在哥伦比亚学院任教期间，继续从事微分几何研究。他这一时期的出版物有Normal congruences and quadruply infinite systems of curves in space（1924年）和A criterion for the conformal equivalence of a Riemann space to a Euclidean space（1925年）。随后他获得了国家研究奖学金，并于1926年至1930年间访问了普林斯顿（1926-27年）、哈佛（1927年）、芝加哥（1928年）、巴黎（1928-30年）和哥廷根（1930年）。正是在这一时期，他完整解决了一个由约瑟夫·拉格朗日在1760年提出、随后被波恩哈德·黎曼、卡尔·魏尔斯特拉斯和赫尔曼·阿曼杜斯·施瓦茨等顶尖数学家研究过的约瑟夫·普拉托问题。该问题是证明由给定周线界定的极小面积曲面的存在性。在道格拉斯的解答之前，该问题只有一些特殊情形得到了解决。从1927年起，在一系列论文中，道格拉斯朝着完整解答迈进：Extremals and transversality of the general 变分法 problem of the first order in space（1927年）、The general geometry of paths（1927-28年）和A method of numerical solution of the problem of Plateau（1927-28年）。道格拉斯于1931年在Transactions of the American Mathematical Society上以Solution of the problem of Plateau给出了其解答的全部细节。由于这一杰出成就，他于1936年在奥斯陆举行的国际数学家大会上被授予菲尔兹奖。

1930年，道格拉斯被任命为麻省理工学院数学助理教授。1934年晋升为副教授，并在1934-35年作为研究员在普林斯顿高等爱德华·斯图迪研究所工作。他于1937年离开麻省理工学院，1938-39年再次作为研究员在普林斯顿高等斯图迪研究所工作一年。他在1940年和1941年获得古根海姆基金会奖学金，随后被任命到布鲁克林学院，从1942年到1954年，他在那里和哥伦比亚大学任教。道格拉斯于1940年6月30日与Jessie Nayler结婚；他们有一个儿子Lewis Philip Douglas。

在完整解决了约瑟夫·普拉托问题之后，道格拉斯继续研究其推广。他发表了One-sided minimal surfaces with a given boundary（1932年）和A Jordan space curve no arc of which can form part of a contour which bounds a finite area（1934年）。1943年，道格拉斯因关于约瑟夫·普拉托问题的回忆录而被美国数学会授予马克希莫·博谢奖。特别是，该奖授予三篇均发表于1939年的论文：Green's function and the problem of Plateau和The most general form of the problem of Plateau发表在American Journal of Mathematics上，Solution of the inverse problem of the calculus of variations发表在Proceedings of the National Academy of Sciences上。在第一篇中，道格拉斯考察了问题的如下形式：给定$n$-空间中$k$条不相交的卡米耶·若尔当曲线的一个集合$G$，求一个以$G$为边界、具有指定亏格$h$和指定可定向性（单侧或双侧）的最小曲面。在第二篇论文中，道格拉斯研究了如下问题：给定一个以$C$为边界的波恩哈德·黎曼曲面（或半曲面）$R$，并在$n$-空间中给定$C$的一个拓扑像$G$，证明存在一个与$R$拓扑等价且以$G$为边界的最小曲面。第三篇论文没有给出变分法逆问题解答的完整证明，而是宣布了结果。

令人惊讶的是，这三篇论文并不是道格拉斯在1939年发表的唯一论文。他还发表了The analytic prolongation of a minimal surface across a straight line，其中给出了关于最小曲面的一些早期结果的推广，并给出了更简单的证明；The higher topological form of Plateau's problem比较了道格拉斯在其获得马克希莫·博谢奖的前两篇论文中使用的方法；以及Minimal surfaces of higher topological structure。道格拉斯在1940年又发表了五篇论文：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和A new special form of the linear element of a surface。其中第二篇和第三篇论文推广了如下初等几何定理：如果在任意三角形的每一边上，以该边为底构造一个顶角为120°的等腰三角形（总是向外或总是向内），那么这些等腰三角形的顶点构成一个等边三角形。

1942年，道格拉斯发表了一篇关于积分论的非技术性综述。他的出发点是阿基米德对圆和抛物线弓形的求积。然后他给出了波恩哈德·黎曼积分、波恩哈德·黎曼-汤姆斯·斯蒂尔吉斯积分和昂利·勒贝格积分的定义，并介绍了它们的性质。在这篇47页的文本中，道格拉斯还提到了Fourier series和变换、阿尔诺·当茹瓦积分以及波恩哈德·黎曼和昂利·勒贝格的二重积分。道格拉斯还研究了group theory，并在1951年研究了具有2个生成元$x, y$的群，使得每个元素都可以表示为$x^{n}y^{m}$的形式，其中$n, m$是整数。他发表了多篇关于这一主题的论文，题为On finite groups with two independent generators。他还发表了一系列论文On the basis theorem for finite abelian groups。

他的妻子Jessie Douglas于1955年去世，同年道格拉斯被任命为纽约市立大学数学教授。他在这一职位上度过了生命的最后十年，住在纽约晨边大道88号的巴特勒大楼。

Jesse Douglas's parents were 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 $x^{n}y^{m}$, 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.