奥托·黑塞Otto Hesse

奥托·黑塞的父亲是Johann Gottlieb 黑塞，他是一位商人和酿酒师。黑塞的母亲是Anna Karoline Reiter（1788-1865）。马克斯·玻恩在柯尼斯堡，黑塞在这座著名的城市长大，就读于老城文理中学。他的父亲于1829年去世，当时他正在读高中。他于1832年获得毕业证书，随后进入柯尼斯堡大学。

在大学里，黑塞学习数学和自然科学，他的讲师包括卡尔·古斯塔夫·雅各布·雅可比、弗里德里希·威廉·贝塞尔、卡尔·诺伊曼和F J Richelot。如果不是因为卡尔·古斯塔夫·雅各布·雅可比的启发式教学，黑塞可能会选择专攻数学以外的科学科目。然而，黑塞于1837年毕业，获得了在中学教授数学、物理和化学的资格，随后他在柯尼斯堡的Kneiphof文理中学担任了一年的试用教师。1838年夏天，他游历了德国和意大利，继续深造。在秋季学期开始前回到柯尼斯堡后，他在那里的一所职业学校担任了物理和化学教师。

黑塞 继续在 卡尔·古斯塔夫·雅各布·雅可比 的指导下攻读博士学位，并于 1840 年在提交学位论文 De octo punctis intersectionis trium superficium secundi ordinis Ⓣ（二阶曲线的八个交点）后从柯尼斯堡获得学位。1841 年，他向柯尼斯堡提交了 教授资格论文（Habilitation） 学位论文，并被任命为 privatdozent。此时他辞去了职业学校教职。同年，黑塞 与 Marie Sophie Emilie Dulk 结婚，她是 Friedrich Philipp Dulk（1788-1852）的女儿，后者是柯尼斯堡的化学教授；他们育有一子五女。

1845 年，黑塞 被提升为柯尼斯堡的编外教授，并在那里度过了他最多产的岁月，将大部分工作发表在 奥古斯都·利奥波德·克雷勒 的杂志上。许多著名数学家都在 黑塞 的指导下攻读博士学位。这些博士包括柯尼斯堡的 古斯塔夫·基尔霍夫 和 卡尔·诺伊曼，但他也在那里为其他几位后来成为杰出数学家的学生授课，包括 Siegfried Aronhold、阿尔弗雷德·克莱布什 和 鲁道夫·利普希茨。1855 年，黑塞 被任命为哈勒的正式教授，但他只担任此职一年，因为当他被提供海德堡的讲席以接替 Ferdinand Schweins 时，他急切地接受并加入他在那里的前学生 古斯塔夫·基尔霍夫 和本生。1856 年，他就任海德堡的职位，从 9 月 9 日正式上任，并一直留在那里直到 1868 年，他在新的慕尼黑理工学院任职。我们提到了在 黑塞 指导下在柯尼斯堡攻读博士学位的著名学生。在海德堡鲁普雷希特-卡尔大学，他拥有更长的著名研究学生名单，包括 阿道夫·迈尔、恩斯特·施勒德、海因里希·马丁·韦伯、奥劳斯·亨里齐 和 马克斯·诺特。

黑塞的主要工作是在代数函数论和不变量论的发展方面。Haas在[1]中写道：-

然而，只有将他的成就与同时代人的成就紧密联系起来，才能对其作出评价。黑塞最初关于二次曲线与二次曲面理论的工作，其灵感和起点得益于卡尔·古斯塔夫·雅各布·雅可比关于二次型线性变换的研究。在证明中（再次受到卡尔·古斯塔夫·雅各布·雅可比的影响），他使用了新近发展起来的行列式，这使他的表述达到了此前未曾有过的优雅。

事实上，黑塞在1842年的一篇论文中，在研究三次曲线和二次曲线时引入了“Hessian行列式”。此后，这一概念在代数几何中得到了广泛应用。

一些近期研究提示，黑塞所做的远不止改进卡尔·古斯塔夫·雅各布·雅可比某些结果的表述。例如，在[3]中，Fraser提出了一个有力的论证，认为黑塞的工作比许多人所以为的更为根本。他考察了卡尔·古斯塔夫·雅各布·雅可比1837年关于变分法的结果以及黑塞在1857年的重新表述：-

卡尔·古斯塔夫·雅各布·雅可比的结果在当前关于变分法的教材中并不太显眼，黑塞的结果更是如此。两人都对这一领域作出了重大贡献，而这一领域或许可以被视为泛函分析的门槛。在莱昂哈德·欧拉和约瑟夫·拉格朗日取得重要突破之后，研究第二变分是自然的一步。卡尔·古斯塔夫·雅各布·雅可比在1837年提出了第二变分理论，总体上得到了很好的接受。1857年，黑塞发表了该理论的另一种表述，这一表述一直被视为仅仅是对卡尔·古斯塔夫·雅各布·雅可比结果的改进性阐述。［[3]的］作者对这些观点提出了挑战，认为黑塞实际上给出了该理论的一种不同表述。的确，黑塞从变分法的算法进路转向强调其分析特征。这正是极值场方法所采用的研究路线，而该方法体现了19世纪后期变分法的进展。黑塞或许可以被视为这些发展的先驱。

黑塞另一个被证明特别有影响的结果，是他在1866年研究射影几何时给出的“转移原理”。Hawkins在有趣的论文[4]中研究了这一原理如何影响了数学的许多不同领域。Wilhelm Meyer在1883年给出了黑塞转移原理的一般形式，而这一形式又被埃利·嘉当在1913年用来构造一个复半单李代数的所有不可约表示。

黑塞 的工作也受到 雅各布·施泰纳 的影响，特别是他在代数变换的几何解释方面所做的工作。尤里乌斯·普吕克 和 让-维克托·彭赛列 也做出了重大贡献，黑塞 在此基础上发展。他的学生 Aronhold 表明 黑塞 在这里的一些结果是最佳的。黑塞 研究了一些 阿瑟·凯莱 也在研究的课题，两人同时发表了齐次型理论。

Haas在[1]中写到黑塞作为教师的贡献：-

黑塞的教学也很有影响。在长期担任讲师期间，他不断表现出对数学的热情，他关于解析几何的教科书必须放在这一背景下看待。黑塞在这些书中使用的线性方程和平面方程的特殊形式，在该学科的所有现代教科书中都被称为黑塞的线性方程和平面方程的标准形式。

黑塞在海德堡期间写的两本教科书是Vorlesungen über analytische Geometrie des Raumes: insbesondere über Oberflächen zweiter Ordnung Ⓣ（空间解析几何讲义：特别是关于二次曲面）（1861）和Vorlesungen über analytische Geometrie der geraden Linie, des Punktes und des Kreises in der Ebene Ⓣ（平面上直线、点和圆的解析几何讲义）（1865）。我们还要提到他的重要论文Sieben Vorlesungen aus der analytischen Geometrie der Kegelschnitte Ⓣ（圆锥曲线解析几何七讲），该文于1874年发表在Zeitschrift für Mathematik und Physik上。

许多科学院授予黑塞院士称号，包括柏林科学院和哥廷根科学院（Königliche Gesellschaft der Wissenschaften）于1856年，以及慕尼黑的巴伐利亚科学院（Königlich Bayerischen Akademie der Wissenschaften）于1869年。1871年，他成为伦敦数学会的荣誉外籍院士。1872年，他还荣获柏林科学院的雅各布·施泰纳奖。

黑塞在慕尼黑因肝脏问题去世，但应其要求葬于海德堡，因为他一直觉得那座城市是他的第二故乡。他的全集由巴伐利亚科学院于1897年出版，前言由瓦尔特·冯·戴克、S Gundelfinger、雅各布·吕罗特和马克斯·诺特撰写。这本731页的书由Chelsea Publishing Company于1972年重印（这就是[2]这本书）。

Otto Hesse's father was Johann Gottlieb Hesse who was a merchant and brewer. Otto's mother was Anna Karoline Reiter (1788-1865). Born in Königsberg, Otto Hesse grew up in the famous city where he attended the Old City Gymnasium. His father died in 1829 while he was at the high school. He graduated with his leaving certificate in 1832 and then entered the University of Königsberg.

At the university Hesse studied mathematics and natural sciences where his lectureres included Jacobi, Bessel, Carl Neumann, and F J Richelot. If it had not been for the inspired teachings of Jacobi, Hesse may have chosen to specialise in a science subject other than mathematics. However Hesse graduated in 1837 with a qualification which allowed him to teach mathematics, physics and chemistry in secondary schools, and then he spent a year as a probationary teacher at the Kneiphof Gymnasium in Königsberg. In the summer of 1838 he travelled through Germany and Italy, furthering his education. Back in Königsberg by the start of the autumn school term he took up a post teaching physics and chemistry at a trade school there.

Hesse had continued to study for his doctorate under Jacobi's supervision and he was awarded the degree from Königsberg in 1840 after submitting his thesis De octo punctis intersectionis trium superficium secundi ordinis Ⓣ. In 1841 he submitted his habilitation thesis to Königsberg and was appointed as a privatdozent. At this point he resigned his teaching position at the trade school. In the same year Hesse married Marie Sophie Emilie Dulk, the daughter of Friedrich Philipp Dulk (1788-1852) who was the Professor of Chemistry at Königsberg; they had one son and five daughters.

In 1845 Hesse was promoted to extraordinary professor at Königsberg and spent his most productive years there publishing most of his work in Crelle's Journal. Many famous mathematicians did their doctoral studies under Hesse's supervision. These doctoral include Gustav Kirchhoff and Carl Neumann at Königsberg, but he also lectured to several other students there who would go on to become exceptional mathematicians including Siegfried Aronhold, Alfred Clebsch and Rudolph Lipschitz. In 1855 Hesse was appointed as an ordinary professor at Halle, but he only held this post for one year since when he was offered the chair at Heidelberg, to succeed Ferdinand Schweins, he was eager to accept and join his former students Kirchhoff and Bunsen there. In 1856 he took up the appointment in Heidelberg, officially taking up the position from 9 September, and remained there until 1868 when he took up a post in the new Munich Polytechnic. We mentioned the famous students who undertook doctoral studies under Hesse's supervision at Königsberg. At the Ruprecht-Karls University of Heidelberg he had an even longer list of famous research students including Adolph Mayer, Ernst Schröder, Heinrich Weber, Olaus Henrici, and Max Noether.

Hesse's main work was in the development of the theory of algebraic functions and the theory of invariants. Haas writes in [1]:-

His achievements can be evaluated, however, only in close connection with those of his contemporaries. Hesse was indebted to Jacobi's investigations on the linear transformation of quadratic forms for the inspiration and starting point of his initial works on the theory of quadratic curves and planes. For proof (again influenced by Jacobi) he used the newly developed determinants which allowed his presentation to reach an elegance not previously attained.

In fact Hesse introduced the 'Hessian determinant' in a paper in 1842 during an investigation of cubic and quadratic curves. Subsequently this concept has been widely applied in algebraic geometry.

Some recent research has suggested that Hesse did much more than improve the presentation of certain results by Jacobi. For example in [3] Fraser presents a strong argument that Hesse's work was more fundamental than many have considered. He examines Jacobi's 1837 result on the calculus of variations and Hesse's reformulation in 1857:-

Jacobi's result does not have much visibility in current texts on the calculus of variations and even less so Hesse's. Both have major contributions to this field, which might be considered the doorstep of functional analysis. After the important breakthrough of Euler and Lagrange, it was a natural step to study the second variation. Jacobi in 1837 proposed a theory of the second variation which was generally well received. In 1857 Hesse published another presentation of the theory, which has been regarded merely as an improved exposition of Jacobi's results. The author [of [3]] challenges these views, attributing to Hesse an effectively distinct presentation of the theory. Indeed, Hesse shifts from an algorithmic approach to the calculus of variations to an emphasis on its analytical character. This was the line of research adopted in the method of fields of extremals, which characterizes progress in the calculus of variations in the late nineteenth century. Hesse might be considered a precursor of these developments.

Another result by Hesse which has proved particularly influential is the 'principle of transfer' which he gave in 1866 during his work on projective geometry. How this influenced many different areas of mathematics is studied by Hawkins in the interesting paper [4]. Wilhelm Meyer gave a general form of Hesse's principle of transfer in 1883 which in turn was used by Cartan in 1913 to construct all irreducible representations of a complex semisimple Lie algebra.

Hesse's work was also influenced by Steiner, particularly work he did on the geometrical interpretation of algebraic transformations. Plücker and Poncelet had also made major contributions which Hesse built on. His student Aronhold showed that some of Hesse's results here were best possible. Hesse worked on some topics that Cayley was also working on and both produced a theory of homogeneous forms which they published at the same time.

Haas writes in [1] of Hesse's contributions as a teacher:-

Hesse's teaching was also influential. In his long years as a lecturer, he continually showed his enthusiasm for mathematics, and his textbooks on analytical geometry must be seen in this context. the special forms of linear equation and of planar equation that Hesse used in these books are called Hesse's normal form of the linear equation and of the planar equation in all modern textbooks on the discipline.

The two textbooks which Hesse wrote during his years in Heidelberg are Vorlesungen über analytische Geometrie des Raumes: insbesondere über Oberflächen zweiter Ordnung Ⓣ (1861) and Vorlesungen über analytische Geometrie der geraden Linie, des Punktes und des Kreises in der Ebene Ⓣ (1865). Let us also mention his important paper Sieben Vorlesungen aus der analytischen Geometrie der Kegelschnitte Ⓣ which appeared in Zeitschrift für Mathematik und Physik in 1874.

Many academies honoured Hesse with membership including the Berlin Academy of Sciences and the Göttingen Academy of Sciences (Königliche Gesellschaft der Wissenschaften) in 1856 and the Bavarian Academy of Sciences (Königlich Bayerischen Akademie der Wissenschaften) in Munich in 1869. In 1871 he became an honorary foreign member of the London Mathematical Society. He was also honoured with the award of the Steiner prize of the Berlin Academy of Sciences in 1872.

Hesse died in Munich from a liver problem but was buried in Heidelberg at his request since he always felt that city to be his second home. His complete works was published by the Bavarian Academy of Sciences in 1897 with a foreword by Walther von Dyck, S Gundelfinger, Jacob Lüroth and Max Noether. The 731 page book was reprinted by the Chelsea Publishing Company in 1972 (this is the book [2]).