赫尔曼·阿曼杜斯·施瓦茨Hermann Schwarz

赫尔曼·阿曼杜斯·施瓦茨的父亲是一位建筑师。雅各布·赫尔曼在多特蒙德的文理中学学习，他最喜欢的科目是化学。离开学校时，他打算攻读化学学位，并为此目的进入了Gewerbeinstitut，后来称为柏林工业大学。

施瓦茨 在柏林开始学习化学，但不久之后，恩斯特·爱德华·库默尔 和 卡尔·魏尔斯特拉斯 就影响他转到了数学。第一位影响他最终研究方向的是 Karl Pohlke。通过他，施瓦茨 对几何学产生了兴趣。施瓦茨 在 1861 年听了 卡尔·魏尔斯特拉斯 关于 The integral calculus 的讲座，施瓦茨 在这些讲座上做的笔记至今仍然存在。他对几何学的兴趣很快就与 卡尔·魏尔斯特拉斯 的分析思想结合了起来。正如 Bölling 在 [3] 中所写：-

……来自几何考虑的想法被[施瓦茨]翻译成了分析的语言。

他继续在柏林学习，由 卡尔·魏尔斯特拉斯 指导，直到 1864 年获得博士学位。他的学位论文由 恩斯特·爱德华·库默尔 审查。

在柏林期间，施瓦茨研究极小曲面（面积最小的曲面），这是变分法的一个典型问题。约瑟夫·普拉托于1866年发表了一篇关于该主题的著名论文，同年卡尔·魏尔斯特拉斯在极小曲面理论与解析函数理论之间建立了桥梁。施瓦茨在1865年做出了重要贡献，他发现现在被称为施瓦茨极小曲面的曲面。这个极小曲面的边界由正四面体的四条棱组成。

施瓦茨继续在柏林学习以取得教师资格，并于1867年完成。那一年，他被任命为哈勒大学的Privatdozent。1869年，他被任命为苏黎世联邦理工学院的数学教授，然后，在1875年，他接受了哥廷根大学数学讲席的任命。

也许令人惊讶的是，在 施瓦茨 于 1892 年接替 卡尔·魏尔斯特拉斯 接受柏林的一个教授职位之后，德国数学最杰出的大学——毫无疑问曾是柏林——的优势开始向哥廷根转移。这有几个原因。首先，施瓦茨 在搬迁后未能保持其数学研究的产出。路德维希·比贝尔巴赫 在 [2] 中说得相当好，他写道 施瓦茨 于 1892 年退休回到柏林。情况如此，本不应让做出任命的人完全感到意外，因为 施瓦茨 已在 1890 年，也就是两年前，出版了他的全集。Boerner 在 [1] 中写道：-

……教学职责和对[施瓦茨的]许多学生的关心占据了他太多时间，以至于他此后发表的东西很少。一个促成因素可能是他倾向于以同样的彻底性处理重要和琐碎的事情，这一特点在他的数学论文中也很明显。

我们不应给人这样的印象：柏林从德国数学领先大学降为第二所大学，唯一原因是 施瓦茨。另一个影响是 菲利克斯·克莱因，他在哥廷根的充满活力的领导使哥廷根繁荣起来，而柏林则付出了代价，在那里 费迪南德·格奥尔格·弗罗贝尼乌斯 和 施瓦茨 无法提供同样富有灵感的做法。也许哥廷根超越柏林的最终标志出现在 1902 年，当时 费迪南德·格奥尔格·弗罗贝尼乌斯 和 施瓦茨 选择 大卫·希尔伯特 来继任因 拉扎勒斯·福克斯 去世而空缺的柏林讲席。大卫·希尔伯特 拒绝了这一邀请，宁愿留在哥廷根。柏林讲席随后由 弗里德里希·赫曼·肖特基 填补，但像他之前的 施瓦茨 一样，他是在数学研究的黄金时期已经过去之后才搬到柏林的。

施瓦茨在柏林继续任教直到1918年。我们稍后将描述他的一些非常出色的数学成就，但首先我们注意到他在数学之外还有几项兴趣，尽管他的婚姻是数学的，因为他娶了恩斯特·爱德华·库默尔的女儿。在数学之外，他是当地志愿消防队的队长，更令人惊讶的是，他协助当地火车站的站长关闭火车车门。

施瓦茨研究的一个重要领域是共形映射。1870年，他做出了与波恩哈德·黎曼映射定理相关的工作。尽管波恩哈德·黎曼已经给出了定理的证明，即平面上任何单连通区域都可以共形映射到圆盘上，但他的证明涉及使用约翰·彼得·古斯塔夫·勒热纳·狄利克雷问题。卡尔·魏尔斯特拉斯已经表明约翰·彼得·古斯塔夫·勒热纳·狄利克雷对此的解决方案并不严格，详情见[10]。施瓦茨给出了一种将多边形区域共形映射到圆的方法。然后，通过用多边形逼近任意单连通区域，他能够给出波恩哈德·黎曼映射定理的严格证明。施瓦茨还给出了求解约翰·彼得·古斯塔夫·勒热纳·狄利克雷问题的交替方法，这很快成为标准技术。施瓦茨工作的这一方面在[10]中有详细考察。

他最重要的工作是为卡尔·魏尔斯特拉斯70岁生日所作的纪念文集。施瓦茨回答了给定极小曲面是否真的产生极小面积的问题。这项工作中的一个想法，即他使用逐次逼近构造了一个函数，引导埃米尔·皮卡给出了微分方程解的存在性证明。它还包含了现在称为“施瓦茨不等式”的积分不等式，详情见[9]。

施瓦茨竟然提出了现在称为奥古斯丁·路易·柯西-施瓦茨不等式（或奥古斯丁·路易·柯西-维克托·布尼亚科夫斯基-施瓦茨不等式）的一般结果的一个特例，这并不令人惊讶，因为他大部分工作的特点是研究相当具体和狭窄的问题，但使用具有极大普遍性的方法来解决它们，这些方法此后得到了广泛应用。他发现这样的一般方法，充分说明了他伟大的直觉，这也许基于对几何的深刻感受。

例如，奥古斯丁·路易·柯西-施瓦茨不等式在维克托·布尼亚科夫斯基、奥古斯丁·路易·柯西、赫尔曼·格拉斯曼、冯·诺伊曼和赫尔曼·外尔等数学家的著作中以算术、几何和函数论的形式出现。今天通常呈现该不等式的形式：

$< x + my, x + my > ≥ 0$

及其标准现代证明似乎最早由赫尔曼·外尔于1918年给出。

在回答卡尔·弗里德里希·高斯的超几何级数何时为代数函数这一问题时，施瓦茨像他多次做过的那样，发展出了一种将导向更一般结果的方法。正是在这项工作中，他定义了一个以圆弧为边的三角形到单位圆盘的共形映射，现在被称为“施瓦茨函数”。这个函数是自守函数的一个早期例子，在这项工作中，施瓦茨正在考察一些想法，这些想法引导菲利克斯·克莱因和儒勒·昂利·庞加莱发展了自守函数理论。

让我们以引用Bölling [3]关于施瓦茨性格的论述作为结束。他写道：-

施瓦茨深受卡尔·魏尔斯特拉斯的影响。从他们的通信中可以发现，施瓦茨常常以精确到最细微之处的态度称呼他的老师，有时几乎显得胆怯。施瓦茨的举止被描述为天真、戏剧化、粗鲁。尽管给人自信的印象，他实际上相当缺乏安全感，此外，在事务方面也不高效。

Hermann Schwarz's father was an architect. Hermann studied at the Gymnasium in Dortmund where his favourite subject was chemistry. When he left school he intended to take a degree in chemistry and he entered the Gewerbeinstitut, later called the Technical University of Berlin, with this aim.

Schwarz began his study of chemistry at Berlin but it was not long before Kummer and Weierstrass had influenced him to change to mathematics. The first of his teachers to influence the direction that his research would eventually take was Karl Pohlke. Through him Schwarz became interested in geometry. Schwarz attended Weierstrass's lectures on The integral calculus in 1861 and the notes that Schwarz took at these lectures still exist. His interest in geometry was soon combined with Weierstrass's ideas of analysis. As Bölling writes in [3]:-

... ideas coming from geometrical considerations were translated [by Schwarz] into the language of analysis.

He continued to study in Berlin, being supervised by Weierstrass, until 1864 when he was awarded his doctorate. His doctoral dissertation was examined by Kummer.

While in Berlin, Schwarz worked on minimal surfaces (surfaces of least area), a characteristic problem of the calculus of variations. Plateau published a famous memoir on the topic in 1866 and in the same year Weierstrass established a bridge between the theory of minimal surfaces and the theory of analytic functions. Schwarz had made an important contribution in 1865 when he discovered what is now known as the Schwarz minimal surface. This minimal surface has a boundary consisting of four edges of a regular tetrahedron.

Schwarz continued studying in Berlin for his teacher's training qualification which he completed by 1867. In that year he was appointed as a Privatdozent to the University of Halle. In 1869 he was appointed as professor of mathematics at the Eidgenössische Technische Hochschule in Zürich then, in 1875, he accepted appointment to the chair of mathematics at Göttingen University.

Perhaps surprisingly after Schwarz succeeded Weierstrass accepting a professorship in Berlin in 1892, the balance in favour of the most eminent university in Germany for mathematics, which had undoubtedly been Berlin, began to shift towards Göttingen. There were several reasons for this. Firstly Schwarz failed to keep up his output of mathematical research after his move. Bieberbach in [2] put it rather well when he wrote that Schwarz retired to Berlin in 1892. That this was the case should not have come as a complete surprise to those making the appointment for Schwarz had published his Complete Works in 1890, two years earlier. Boerner writes in [1] that:-

... teaching duties and concern for [Schwarz's] many students took so much of his time that he published very little more. A contributing element may have been his propensity for handling both the important and the trivial with the same thoroughness, a trait also evident in his mathematical papers.

We should not give the impression that the only reason for Berlin moving down from being the leading German university for mathematics to become its second university was due to Schwarz. The other effect was Klein whose dynamic leadership in Göttingen made it prosper at the expense of Berlin where Frobenius and Schwarz could not provide the same inspired approach. Perhaps the final sign that Göttingen had overtaken Berlin came in 1902 when Frobenius and Schwarz chose Hilbert to succeed to the Berlin chair which had become vacant on the death of Fuchs. Hilbert turned down the offer, preferring to remain at Göttingen. The Berlin chair was then filled by Schottky but, like Schwarz before him, he had moved to Berlin after his best days for mathematical research were behind him.

Schwarz continued teaching at Berlin until 1918. We shall describe some of his very fine mathematical achievements in a moment, but first we note that he had several interests outside mathematics, although his marriage was a mathematical one since he married Kummer's daughter. Outside mathematics he was the captain of the local Voluntary Fire Brigade and, more surprisingly, he assisted the stationmaster at the local railway station by closing the doors of the trains.

One important area which Schwarz worked on was that of conformal mappings. In 1870 he produced work related to the Riemann mapping theorem. Although Riemann had given a proof of the theorem that any simply connected region of the plane can be mapped conformally onto a disc, his proof involved using the Dirichlet problem. Weierstrass had shown that Dirichlet's solution to this was not rigorous, see [10] for details. Schwarz's gave a method to conformally map polygonal regions to the circle. Then, by approximating an arbitrary simply connected region by polygons he was able to give a rigorous proof of the Riemann mapping theorem. Schwarz also gave the alternating method for solving the Dirichlet problem which soon became a standard technique. This aspect of Schwarz's work is examined in detail in [10].

His most important work is a Festschrift for Weierstrass's 70th birthday. Schwarz answered the question of whether a given minimal surface really yields a minimal area. An idea from this work, in which he constructed a function using successive approximations, led Émile Picard to his existence proof for solutions of differential equations. It also contains the inequality for integrals now known as the 'Schwarz inequality', see [9] for details.

The fact that Schwarz should have come up with a special case of the general result now known as the Cauchy-Schwarz inequality (or the Cauchy-Bunyakovsky-Schwarz inequality) is not surprising for much of his work is characterised by looking at rather specific and narrow problems but solving them using methods of great generality which have since found widespread applications. That he found such general methods says much for his great intuition which was perhaps based on a deep feeling for geometry.

For example the Cauchy-Schwarz inequality appears in arithmetic, geometric and function-theoretic formulations in works of mathematicians such as Bunyakovsky, Cauchy, Grassmann, von Neumann and Weyl. The form in which the inequality is usually presented today:

$< x + my, x + my > ≥ 0$

with its standard modern proof seems to have been first given by Weyl in 1918.

In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results. It was in this work that he defined a conformal mapping of a triangle with arcs of circles as sides onto the unit disc which is now known as the 'Schwarz function'. This function is an early example of an automorphic function and in this work Schwarz was looking at ideas which led Klein and Poincaré to develop the theory of automorphic functions.

Let us end with quoting Bölling [3] on Schwarz's character. He writes:-

Schwarz was deeply influenced by Weierstrass. From their correspondence one finds that Schwarz addressed his teacher often with an accuracy going down to the last detail, sometimes almost timidly. Schwarz's demeanour has been described as naive, dramatic, coarse. In spite of giving the impression of self-confidence, he was, in fact, rather insecure and besides, not efficient in business matters.