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数学家生平——Laurent Moise Schwartz

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发表于 2023-11-19 11:47 | 显示全部楼层 |阅读模式
数学家生平——Laurent Moise Schwartz

来源:数学往事

作者:Mathematical Memories


简介


Laurent Moise Schwartz(1915.3.5-2002.7.4)

Laurent Schwartz 是一位法国数学家。他开创了分布理论,为诸如狄拉克函数之类的对象赋予了明确的含义。1950 年,他因在分布理论方面的工作而获得菲尔兹奖。

生平

Laurent Schwartz 来自一个阿尔萨斯血统的犹太家庭,具有深厚的科学背景:他的父亲(Anselme Schwartz)是法国人,不喜欢在德国生活,所以十四岁时他离开家乡去了巴黎,在那里他成为了一名外科医生,1907年,他娶了Claire Debré(1888-1972),她的家族有很多著名的人物: 例如,Claire 的哥哥 Robert Debré(他为联合国儿童基金会的创建做出了杰出贡献)是一位著名的儿科医生,Robert 的儿子 Michel Debré(1912-1996)在 1959 年成为法国总理,这个大家庭里也有顶尖的数学家,Jacques Hadamard 娶了 Claire Debré 母亲的妹妹。Laurent 是他父母三个儿子中最大的一个,他有两个兄弟 Daniel(1917 年出生)和 Bertrand(1919 年出生)。

Schwartz 十一岁时患上了小儿麻痹症。尽管他在几个月内康复了,但这种疾病使得他终生虚弱。1926 年 9 月,当他还在小儿麻痹症的康复阶段时,他的父母在 Autouillet 买了一栋乡间别墅。那是一座大房子,有壮丽的花园,四周是草地和田野,孩子们可以在其中玩耍。Schwartz 年轻时,全家每个周末都会在 Autouillet 度过,但平日里住在巴黎。在巴黎就读中学时,Schwartz 在数学以及希腊语和拉丁语等古典语言方面都表现出色。他面临着一个艰难的选择,尤其是在他在全国拉丁语“concours général”中排名第一,在翻译方面排名第四的成绩之后。他必须在最后的学年学习哲学和人文学科以准备大学学习古典文学或学习数学和哲学之间做出选择。他的母亲,一直在鼓励儿子学习方面发挥重要作用,她向哥哥 Robert Debré 寻求建议。他的医学专长是照顾孩子,他给出了他的专业意见,即 Schwartz 应该学习数学。再加上他在中学的老师们的一直建议,Schwartz 决定放弃拉丁语,同时学习数学和哲学,并同时获得两个科目的学士学位。在 lycée Janson de Sailly 的最后几年,他学习了数学和哲学课程。他爱上了几何,由一位鼓舞人心的老师教授,但对教学不太好的哲学课程感到失望。同样在扬森德塞利中学,他爱上了比他高一班的玛丽-海伦·列维。Marie-Hélène 是 向她的兄弟罗伯特·德布雷寻求建议。他的医学专长是照顾孩子,他给出了他的专业意见,即施瓦茨应该学习数学。再加上他在中学的老师们的相同意见,施瓦茨决定放弃拉丁语,同时学习数学和哲学,并同时获得两个科目的学士学位。在 lycée Janson de Sailly 的最后几年,他学习了数学和哲学课程。他爱上了由一位鼓舞人心的老师教授的几何,但对教学质量差得多的哲学课程感到失望。同样在 lycée Janson de Sailly 他爱上了比他高一班的 Marie-Hélène Levy , 她是 Paul Lévy 的女儿。

1934 年,Schwartz 进入巴黎高等师范学院,师从世界上一些顶尖的数学家。1935 年,他与同样学习数学的 Marie-Hélène 订婚。与她的父亲 Paul Lévy 友好的关系,对这个年轻人产生了重要的数学影响。Lévy 的主要研究兴趣是概率论和泛函分析,让这个年轻人热爱这些话题,这将成为他一生的主要研究兴趣。大学通常是学生参与政治的时候,事实上,Schwartz 对“left-wing beliefs”很活跃。他此时的政治活动在[1]中有所描述:

The intellectual ferment of these years was paralleled by political engagement. Though from a traditionally right-wing background, he was a strong supporter of Leon Blum's Popular Front Government until he became disillusioned by its failure to support the Spanish Republicans. Similarly, his sympathies for communism were soon dampened by Stalin's show trials, though he then spent ten years as a Trotskyite, up to 1947. He claimed never to regret this, even though it almost prevented him travelling to America to receive the Fields Medal.

Schwartz 在完成了出色的本科生涯后,他于 1937 年从 Agrégation de Mathématiques 毕业。在这个阶段,他决定他将服兵役而不是拖延。他被分配到 DCA( Défense Contre Aéronefs - 飞机防御),但由于身体虚弱且缺乏灵活性,他发现自己无法执行诸如拆卸和组装机枪之类的任务。他首先被派往巴兰古,然后被派往比斯卡罗斯,尽管他因二战不得不延长兵役,但他从未参加过战斗,并于 1940 年 8  月退伍。在此期间,1938 年,他与 Marie-Hélène 结婚。事实上,他们原本计划在 1935 年 12 月结婚,但 Marie-Hélène 感染了肺结核,被送往上萨瓦省帕西的疗养院。十八个月的被迫分开,在此期间他们只能通过信件通信,这对他们俩来说都是极其困难的。1940 年 Schwartz 退伍后,他和妻子一起去了图卢兹(Toulouse),他的父母在德国入侵和法国沦陷后搬到了那里。

在图卢兹,他代表巴黎高等师范学院进行口头演讲时遇到了亨利·嘉当(Henri Cartan)。事实上,Marie-Hélène 也借此机会与 Henri Cartan 交谈,因为她想继续她的数学学习。Cartan 建议他在克莱蒙费朗(Clermont-Ferrand)攻读博士学位,二战开始时德国军队入侵法国后,斯特拉斯堡大学(University of Strasbourg)就搬到了克莱蒙费朗。Schwartz 成为 Caisse National des Sciences(后来成为 CNRS )的成员,该组织一直资助他到 1942 年底。在这种支持结束后,他从 ARS( Aid à la Recherche Scientifique )获得了资金,该基金支持他到战争结束。Schwartz 从常驻巴黎的 Georges Valiron 那里得到了数学建议。他在本科学习复变函数课程时就认识了 Valiron 。

Schwartz 的论文 Etude des sommes d'exponentielles (T) 于 1943 年提交给斯特拉斯堡大学,其中包含以下关于 Valiron 所给予帮助的确认:

I want to especially thank Georges Valiron who not only gave me much advice, but also, through the correspondence he kindly entered into with me, helped me to overcome many difficulties.(我要特别感谢 Georges Valiron ,他不仅给了我很多建议,而且通过与我友好的通信,帮助我克服了许多困难。

Valiron、Charles Ehresmann 和 Andre Roussel 都是论文的指导老师. Schwartz 也在他的论文中写道:

I would also like to express my gratitude to Jacques Hadamard and Paul Lévy who have guided and enriched my mathematical development.

在战争期间,他的政治活动、托洛茨基主义信仰和犹太背景使他处于各种微妙的境地。他采用了一个假身份,称自己为 Laurent-Marie Sélimartin ,只有凭借技巧和好运气,他才逃脱了侦查。然而,他虚弱的身体状况意味着他无法协助抵抗运动。自从他在巴黎高等师范学院读书以来,他一直是托洛茨基党的坚定支持者,但他的感情在战争期间开始发生变化[4] :

Trotskyism gave me, during my years at the ENS, a remarkable education, clearly more advanced and sophisticated than that of most youngsters of my age. But by the extremism and sectarianism of its ideas, and by its stereotyped language, it neutralised me during the occupation. My judgment remains extremely severe on my own actions as well as those of the majority of the Trotskyist party during that period.

1943 年 3 月,Schwartz 的儿子 Marc-André 出生;这只会增加他们的危险。Marc-André 后来成为作家和诗人,但过着悲惨的生活——见下文。1944-1945 年,Schwartz 在格勒诺布尔的理学院任教,然后搬到南希,在 Jean Delsarte 和 Jean Dieudonné 的推荐下,他成为了理学院的教授。正是在他职业生涯的这一时期,他创作了关于分布理论的著名著作。我们在下面更详细地描述了这个想法,但在这一点上,我们引用了 Schwartz 自己在 1944 年提出这个想法的描述[4]:

In my youth I used to have insomnias lasting several hours and never took sleeping pills. I remained in my bed, the light off and without writing, did mathematics. My inventive energy was redoubled and I advanced rapidly without tiring. I felt entirely free, without any of the brakes imposed by my daily life and writing. After some hours ... especially if an unexpected difficulty came up ... I would stop and sleep until morning. I would be tired but happy for the whole of the following day. ... On this particular night I felt sure of myself and filled with a sense of exaltation. I lost no time in rushing to explain everything to Henri Cartan who ... lived next door. He was enthusiastic: "There you are. You've just resolved all the difficulties of differentiation. Now we'll never again have functions without derivatives".(在我年轻的时候,我经常失眠几个小时,从不服用安眠药。我躺在床上,关着灯,不写字,做数学。我的创造力倍增,我不累地快速前进。我感到完全自由,没有任何日常生活和写作的阻碍。几个小时后…尤其是遇到意外困难时…我会停下来睡到早上。我会很累,但第二天一整天都很开心 …… 在这个特别的夜晚,我对自己充满了信心,充满了兴奋。我立即向住在隔壁的 Henri Cartan 解释了这一切。他热情地说:“There you are. You've just resolved all the difficulties of differentiation. Now we'll never again have functions without derivatives。”。

Schwartz 还在政治上活跃,作为托洛茨基主义者参加了 1945 年的法国选举,在未能当选后,他在 1946 年南希再次参选(也没有成功)。在南希期间,他教了一批杰出的学生,包括 Bernard Malgrange , Jacques-Louis Lions , François Bruhat , and Alexander Grothendieck 等。此外,在这些年中,他成为数学界的国际明星,1947 年 10 月应哈拉尔·玻尔的邀请在哥本哈根演讲,1949 年在温哥华举行的第一届加拿大数学大会上,他是四位主要演讲人之一;施瓦兹的女儿 Claudine 出生于 1947 年,她后来也成为了一名数学家,是文献[42]的作者。(Claudine 于 1971 年嫁给了数学家 Raoul Robert)1953 年,Schwartz 的妻子 Marie-Hélène Schwartz 因其论文《 Formules apparentées à celles de Gauss-Bonnet et Nevanlinna-Ahlfors pour certaines applications d'une variété à n dimensions dans une autre(T) 被巴黎大学授予博士学位。Valiron 是她论文导师,十年前他是她丈夫论文的指导教师。

1953 年,Schwartz 回到巴黎,并担任教授直到 1959 年。1959 年至 1980 ,他在巴黎的埃科尔理工学院任教。随后,他在法国巴黎七大待了三年,1983 年退休。下面我们略为介绍一下他卓越的数学贡献,但在我们介绍这些之前,我们先介绍他在巴黎职业生涯中参与的一些政治活动。

1956 年,他是法国反对苏联入侵匈牙利抗议活动的领导人之一。然后在接下来的一年里,他卷入了一件与他个人关系更密切的事情,阿尔及利亚的“奥丁事件”[1]:-

Audin, a mathematician and communist based in Algiers, was writing his thesis under Schwartz's supervision. But in June 1957 the 25-year-old father of three and opponent of French rule in Algeria was abducted by paratroopers, tortured and killed. Schwartz was tireless in his calls for justice, and organised a presentation of the young man's thesis in his absence.

Vocal in his opposition to the French campaign, he signed the famous "Declaration des 121" in favour of military insubordination. The riposte of Pierre Messmer, the Minister for the French Army(and, by the same token, of the École), was to strip him of his position at the Polytechnique, for reasons of "common sense and honour". To which Schwartz replied that since the Army commanded by Messmer had sanctioned torture and promoted torturers, such remarks were absurd.

After a brief exile in New York, he regained his post two years later ...

Schwartz 在 20 世纪 40 年代末对数学做出的杰出贡献是他在分布理论方面的工作。在之前,我们描述了他想出这个 idea 的那个晚上。他展示这些想法的第一份出版物是 Généralisation de la concept de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques(T) ,出版于 1948 年。分布理论是对微分学和积分学的极大扩展。Heaviside 和 Dirac  已经把微积分推广到特定的应用中。然而,这些和其他类似的形式计算方法并不是建立在抽象和严格的数学基础上的。Schwartz 的分布理论的发展为这种类型的方法奠定了坚实的基础,并极大地扩展了它们的应用范围,为许多领域的应用提供了强大的工具。

1949 年,他在温哥华所做的讲座成为了 Schwartz 两卷专著 Théorie des distribution(1950, 1951)的基础。Irving Segal 在一篇评论中写道:

This is a generally clear, carefully organized, and detailed account of the basic aspects of the theory of "distributions'' due to the author, and described by him in earlier publications ... This theory provides a convenient formalism for many common situations in theoretical and applied analysis, but its greatest significance may be in connection with partial differential equations, particularly those of hyperbolic type, where its adaptability to local problems gives it an advantage over Hilbert space(and other primarily global)techniques.(这是作者对“分布”理论的基本方面的一个总体上清晰、精心组织和详细的描述,他在早期的出版物中描述过 …… 这一理论为理论和应用分析中的许多常见情况提供了一种方便的形式,但它最大的意义可能是与偏微分方程有关,特别是那些双曲型方程,它对局部问题的适用性使其优于希尔伯特空间(和其他主要的)技术。

在《大英百科全书( Encyclopaedia Britannica)》中关于“分析”的文章中,François Treves 将 Schwartz 的工作描述如下:

... Schwartz's idea(in 1947)was to give a unified interpretation of all the generalized functions that had infiltrated analysis as(continuous)linear functionals on the space Cç of infinitely differentiable functions vanishing outside compact sets. He provided a systematic and rigorous description, entirely based on abstract functional analysis and on duality. It is noteworthy that such an approach had a precedent, in the presentation by André Weil of the integration of locally compact groups ... Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated. This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables. Schwartz's ideas can be applied to many other spaces of test-functions beside C, as he himself and others have shown ...(…… Schwartz 的想法(1947年)是将所有渗透到分析中的广义函数统一解释为在紧支集外消失的无穷可微函数空间上的(连续)线性泛函。他提供了一个系统和严格的描述,完全基于抽象的泛函分析和对偶观点。值得注意的是,这种方法在 André Weil 关于局部紧群积分的介绍中有先例 …… 由于分布理论中的可微性要求,测试函数的空间及其对偶稍微复杂一些。这导致了对拓扑向量空间的广泛研究,超出了 Hilbert 和 Banach 空间这两个熟悉的类,这些研究反过来又为某些分析领域提供了有用的新见解,例如偏微分方程或多个复变量的函数。Schwartz 的想法可以应用于 C 以外的许多其他测试函数空间,正如他自己和其他人所展示的那样 ……

1950 年 8 月 30 日,Harald Bohr 在哈佛举行的国际数学家大会上向 Schwartz 颁发了菲尔兹奖,以表彰他在分布理论方面的贡献。Harald Bohr 把施瓦茨1948年的论文描述为:

... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ...( ... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ...(…… 这肯定会成为我们这个时代的经典数学论文之一。我想引用他论文的每一位读者,和我一样,在看到该理论所引导的微积分整体结构的美妙和谐,以及理解其应用的进步对更高分析的许多部分的重要性时,都会留下相当多令人愉快的兴奋,如谱理论、势理论、事实上,线性偏微分方程的整个理论 ……

除了菲尔兹奖章外,Schwartz 还获得了一系列奖章和荣誉。他分别于 1955 年、1964 年和 1972 年获得巴黎科学院的奖项,1972 年当选为该院院士。他曾获得许多大学的荣誉博士学位,包括洪堡(1960 年)、布鲁塞尔(1962 年)、隆德(1981 年)、特拉维夫(1981 年),蒙特利尔(1985 年)和雅典(1993 年)。

Schwartz 后来在随机微分学方面的工作在他的综述文章[45]中有描述,另见[44]。后来的政治运动包括反对美国参与越南战争、苏联入侵阿富汗和俄罗斯对车臣的战争。然而,他的政治活动导致了一场家庭悲剧[21]:

His son Marc-André committed suicide in 1971, the result of the life-long trauma following his kidnapping at the hands of French nationalists seeking revenge on his father for his commitment to anti-colonialism and support of the Algerians seeking independence.(他的儿子 Marc-André 于 1971 年自杀。他的儿子因反对殖民主义和支持阿尔及利亚独立的父亲而被法国民族主义者绑架,这是他一生的创伤。

由于 Schwartz 如此热衷于数学和政治,人们可能会认为他没有时间发展其他爱好。然而,这是完全错误的,因为他是一个狂热的蝴蝶收藏家,拥有超过 2 万个标本。

在 Schwartz 写的其他几本书中,我们提到 Méthodes mathématiques pour les 科学物理(T)(1961)。它由 George Temple 评论:

Those who have been privileged to see the notes of the lectures which Professor Schwartz delivered in British Columbia a few years ago will know that he is a master of clear, precise exposition which can be readily adapted to the needs of mathematical physicists. The title of this volume has clearly been chosen with some care, to indicate that it treats of mathematical methods involved in modern mathematical physics, and that it is not necessarily an introductory text-book for the mathematical physicist. The book does give, in fact, a splendid introduction to a number of basic topics in mathematical physics, treated with a degree of rigour and abstraction which may well surprise the physics student at British universities. ... The whole book is written in a concise and lucid style which we have learnt to associate with the name of Professor Schwartz.

让我们引用 Schwartz 的两句话作为结束,第一句是关于政治的,第二句是关于数学的:

I have always thought that morality in politics was something essential, just like feelings and affinities.(我一直认为政治中的道德是必不可少的,就像感情和亲和力一样。

To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.(要在数学中有所发现, 就要克服循规蹈矩。如果你不具有颠覆性,你就无法前进。

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 楼主| 发表于 2023-11-19 11:47 | 显示全部楼层
参考文献

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