让·勒雷Jean Leray

让·勒雷的父亲是Francis Leray，是一位教授，母亲是Baptistine Pineau。勒雷先在南特读中学，然后转到雷恩的中学，最后在巴黎高等师范学院完成学业并获得博士学位。在巴黎，他研究流体动力学。他于1932年10月20日与Marguerite Trumier结婚。他们育有三个孩子：勒雷-Claude、Françoise和Denis。

1933年，尤利乌什·绍德尔凭借洛克菲勒奖学金抵达巴黎，与雅克·阿达马合作。这促成了勒雷与尤利乌什·绍德尔之间的合作，他们的共同工作产生了一篇论文Topologie et équations fonctionelles Ⓣ（拓扑学与函数方程），发表在Annales scientifiques de l'École Normale Supérieure上。这篇1934年关于拓扑学与偏微分方程的论文极为重要：-

在这篇论文中，定义了如今所称的勒雷-尤利乌什·绍德尔度（一个同伦不变量）。随后，这一度被用于一种巧妙的方法，以证明复杂偏微分方程解的存在性。

在1934年与尤利乌什·绍德尔合作发表论文之后，勒雷于次年发表了一篇关于代数拓扑学的论文，研究斯特凡·巴拿赫空间的拓扑学。随后他重新转向分析方面的工作，特别是研究由流体力学产生的微分方程。他研究了三維克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程初值问题的解。他不仅考察了解的存在性和唯一性，还证明了解仅在有限时间内保持光滑，此后便出现湍流解。在建立这一理论的过程中，勒雷引入了许多泛函分析的思想，这些思想如今已成为标准工具。

1936年，勒雷被任命为南锡理学院教授。第二次世界大战于1939年爆发，勒雷担任军官。他于1940年被俘，被送往奥地利的一个战俘营，在那里一直待到1945年战争结束。在战俘营期间，勒雷和一些同被俘的人组织了一所“战俘大学”，勒雷成为其校长。由于不希望德国人知道他是一位流体动力学专家，因为他担心一旦被发现，就会被迫为他们从事战争工作，勒雷声称自己是一位拓扑学家。在被关押于战俘营的那些年里，他只研究拓扑学问题。

尽管勒雷已经从事过一些拓扑学工作，但在不阅读拓扑学文献的情况下研究这一课题对他来说并不容易。他通过当时在苏黎世的海因茨·霍普夫获得了一些论文，但勒雷的大部分工作是在与该学科已发生的发展相独立的情况下完成的。1945年获释后，勒雷发表了一部三部分的著作Algebraic topology taught in captivity。

回到巴黎后，勒雷继续研究拓扑学问题，并于1947年成为法兰西公学院教授。对于勒雷 [2]：-

代数拓扑不仅应研究空间的拓扑，即附着于空间、在同态下不变的代数对象，还应研究表示的拓扑（连续映射），即连续映射的类似性质的拓扑不变量。

沿着这条路线，他发表了论文，引入了层以及连续映射的谱序列。

20世纪50年代，勒雷在若干领域工作。他研究了时间依赖的双曲型偏微分方程，也开始研究奥古斯丁·路易·柯西问题。特别地，他于1956年发表了一篇关于变系数方程的奥古斯丁·路易·柯西问题的论文。1957年，他解释了自己在这一领域工作的目标：-

我们提议在复情形下整体地研究线性奥古斯丁·路易·柯西问题，然后在实双曲情形下研究，假设给定数据是解析的。

他能够将常线性解析微分方程理论中的结果推广，从而得到偏微分方程的类似结果。勒雷关于奥古斯丁·路易·柯西问题的工作引导他研究留数理论。1959年，他[2]：-

……在复流形上发展了一般留数理论，并将其应用于研究由求解奥古斯丁·路易·柯西问题产生的、依赖于具体积分的参数。

在5中，Ekeland如下总结了勒雷的成就：-

勒雷之所以远远超越他的时代，是因为他非凡的技术能力和几何洞察力。在他手中，偏微分方程的能量估计与代数拓扑的思想（如不动点定理）以一种极具原创性的方式结合起来，从而攻克了最困难的问题。他是第一个采纳现代观点的人，即函数不是两组变量之间的复杂关系，而是某个无限维空间中的一个点……可以说勒雷是第一位现代分析学家。

勒雷获得了许多荣誉。他自1953年起是科学院的成员，并于1965年当选为美国的国家科学院院士。次年，他当选为苏联科学院院士。他还是比利时皇家科学院的成员、Royal Society of London的会士，以及米兰、波士顿、Göttingen、Turin、巴勒莫、华沙和Lincei等科学院的成员。1967年，他被授予芝加哥大学荣誉博士学位。授奖词写道：-

一位具有洞察力和原创性的数学家，其发明革新了偏微分方程和代数拓扑。

他于1938年获得Malaxa奖，1971年获得Feltrinelli奖，1979年获得沃尔夫奖，1988年获得M V Lomonosov金质奖章。他还被授予荣誉军团司令勋章。

我们应当以对勒雷授课风格的一些评论来结束这部传记[5]：-

他是一位举止温和、衣着整洁、留着灰色小胡子的人，他眯着眼睛看着听众，很快就失去了他们的注意；但他在恭敬的沉默中继续在黑板上书写，确信数学就在那里供所有人观看，无需进一步解释。

Jean Leray's father was Francis Leray, who was a professor, and his mother was Baptistine Pineau. Jean attended the Lycée at Nantes, then moving to the Lycée at Rennes before completing his education at the École Normale Supérieure where he was awarded his doctorate. In Paris he worked on hydrodynamics. He married Marguerite Trumier on 20 October 1932. They had three children, Jean-Claude, Françoise, and Denis.

In 1933 Juliusz Schauder arrived in Paris on a Rockefeller scholarship to work with Hadamard. This led to a collaboration between Leray and Schauder and their joint work led to a paper Topologie et équations fonctionelles Ⓣ published in the Annales scientifiques de l'École Normale Supérieure. This 1934 paper on topology and partial differential equations is of major importance:-

In this paper what is now known as Leray-Schauder degree (a homotopy invariant) is defined. This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.

After his 1934 paper with Schauder, Leray published a paper on algebraic topology in the following year on the topology of Banach spaces. He then returned to work on analysis, in particular studying differential equations arising from hydrodynamics. He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations. He examined not only the existence and uniqueness of solutions but he showed that the solutions remained smooth for only a finite time after which turbulent solutions arise. In producing this theory Leray introduced many ideas of functional analysis which have today become standard tools.

In 1936 Leray was appointed Professor at the Faculty of Science at Nancy. World War II began in 1939 and Leray served as an army officer. He was captured in 1940 and sent to a prisoner of war camp in Austria where he remained until the end of the war in 1945. While at the camp Leray and some of his fellow captives organised a "université en captivité" and Leray became its rector. Not wishing the Germans to know that he was an expert in hydrodynamics, since he feared that if they found out he would be forced to undertake war work for them, Leray claimed to be a topologist. He worked only on topological problems for the years he was held captive in the camp.

Although he had undertaken some topological work it was not easy for Leray to work on the topic without reading topological literature. He was able to obtain some papers through Hopf who was at this time in Zürich but much of Leray's work was done independently of the developments which had taken place in the subject. After his release in 1945 Leray published a three part work Algebraic topology taught in captivity.

Leray continued to work on topological questions after his return to Paris where he became professor at the Collège de France in 1947. For Leray [2]:-

... algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps.

Following this line he published papers which introduced sheaves, and the spectral sequence of a continuous map.

In the 1950s Leray worked in a number of areas. He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem. In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956. In 1957 he explained the aims of his work in this area:-

We propose to study globally the linear Cauchy problem in the complex case, then in the real hyperbolic case, assuming that the given data is analytic.

He was able to generalise results in the theory of ordinary linear analytic differential equations to obtain similar results for partial differential equations. Leray's work on the Cauchy problem led him to study residues theory. In 1959 he [2]:-

... developed a general residue theory on complex manifolds and applied it to the investigation of concrete integrals depending on parameters arising from solving the Cauchy problem.

In [5] Ekeland sums up Leray's achievements as follows:-

Leray was so far ahead of his time because of his tremendous technical capability and geometrical insight. In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems. He was the first to adopt the modern point of view, whereby a function is not a complicated relation between two sets of variables, but a point in some infinite dimensional space ... Leray [can be said to have been] the first modern analyst.

Leray received many honours. He was a member of the Academy of Sciences from 1953 and he was elected to the National Academy of Sciences in the United States in 1965. The following year he was elected to the USSR Academy of Sciences. He was also a member of the Royal Academy of Belgium, a fellow of the Royal Society of London, and a member of the academies of Milan, Boston, Göttingen, Turin, Palermo, Warsaw, and Lincei. In 1967 he was awarded an honorary doctorate from the University of Chicago. The citation stated:-

Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology.

He was awarded the Malaxa prize in 1938, the Feltrinelli prize in 1971, the Wolf prize in 1979 and the M V Lomonosov Gold Medal in 1988. He was also made Commandeur de la Légion d'Honneur.

We should end this biography with some comments on Leray's lecturing style [5]:-

He was a mild mannered, dapper man with a grey moustache, who squinted at his audience and lost it rather quickly; but he continued to write on the blackboard amidst a respectful silence, confident that the mathematics were there for all to see and needed no further explanation.