弗里德里希·希策布鲁赫Friedrich Hirzebruch

弗里德里希·希策布鲁赫是Fritz Hirzebruch博士和Martha Holtschmidt的儿子。Fritz Hirzebruch是哈姆一所中学的校长，他也在那里教数学。上学时，希策布鲁赫喜欢数学，并通过与父亲交谈和阅读父亲的数学书来学习更高级的课题。1937年，希策布鲁赫满十岁，当时他被要求加入德意志少年团（Deutsches Jungvolk），这是希特勒青年团的一个分支，面向10至14岁的男孩。1943年，15岁的他随全班被征召为“助手”，在他家乡的一个防空阵地服役。

1945年3月，希策布鲁赫在盟军即将渡过莱茵河时被征入德国军队。到4月，盟军已控制了德国大部分地区，希策布鲁赫被俘。他于1945年7月1日获释。1945年11月，他获准进入明斯特威斯特法伦威廉大学，学习数学、物理和数理逻辑。大学的大部分建筑已被轰炸严重损坏，所以起初他不得不在家里进行大部分学习。然而，条件很快得到改善，几乎正常的大学生活恢复了。

他于1950年在明斯特大学获得博士学位，学位论文为Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen Ⓣ（论四维波恩哈德·黎曼曲面上两个复变量的多值解析函数），此前师从Heinrich Behnke。1949年至1950年间，他还在苏黎世的联邦理工学院与海因茨·霍普夫一起学习代数拓扑学和代数几何。这一时期对他准备学位论文非常重要。

1951年至1952年在埃尔朗根大学担任科学助理之后，他于1952年至1954年在美国普林斯顿高等爱德华·斯图迪研究所与阿曼德·博雷尔、小平邦彦和唐纳德·斯宾塞一起研究层论、向量丛、示性类和勒内·托姆配边等课题。他于1952年与Ingeborg Spitzley结婚；他们有两个女儿和一个儿子。

他的第一篇论文发表于1951年，题为Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten Ⓣ（《论一类单连通复流形》）。S S Cairns 的评论开头如下：-

两个复流形如果能够彼此拓扑地且解析地映射，则称为解析等价的。本文对确定一类拓扑等价流形中所有解析等价类这一一般问题作出了贡献。

希策布鲁赫于1953年发表了一篇与他的学位论文同名的论文，同年还发表了另外三篇论文Über die quaternionalen projektiven Räume Ⓣ（《论四元数射影空间》）；On Steenrod's reduced powers, the index of inertia, and the Todd genus和Übertragung einiger Sätze aus der Theorie der algebraischen Flächen auf komplexe Mannigfaltigkeiten von zwei komplexen Dimensionen Ⓣ（《将代数曲面理论的一些结果转移到二维复流形》）。他最著名的结果之一，现称为希策布鲁赫-波恩哈德·黎曼-Roch定理，出现在他1954年的论文Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties中。同年他发表了Some problems on differentiable and complex manifolds。在提交他的教授资格论文（Habilitation）学位论文（与1956年出版的著作相同）之后，他于1954-55年在明斯特大学担任Dozent。此后，他被任命为普林斯顿大学的助理教授，并在那里度过了1955-56年。

希策布鲁赫关于波恩哈德·黎曼-Roch定理的令人印象深刻的工作，完整地阐述在他1956年出版的一本书中，书名为Neue topologische Methoden in der algebraischen Geometrie Ⓣ（《代数几何中的新拓扑方法》）。唐纳德·斯宾塞写道：-

这本专著阐述了围绕任意维代数流形的基本黎曼-Roch定理的主要思想和概念，该定理的证明最近由作者找到，并在此首次以完整形式呈现。代数流形被理解为一种紧复流形，它可以无奇点地复解析嵌入到适当选取维数的复射影空间中。

1956年，他成为莱茵希策布鲁赫-威廉·波恩大学的正式教授，担任此职位直到1993年退休。在此期间，他于1962年至1964年担任数学与自然科学学院院长。

希策布鲁赫所著的书籍清单，无论在数量、阐述的质量，还是对数学研究方向的影响上，都令人印象深刻。我们给出一些细节，关于那些继上文提到的他1956年的杰作之后的著作，那本书被翻译为Topological methods in algebraic geometry，并在多年间出现了几个更新版本。Garben-und Cohomologietheorie Ⓣ（层与上同调）（1957），与G Scheja合著，阐述了层的上同调理论。1962年，希策布鲁赫在布兰迪斯和乔治·伯克利举办了一系列研讨会。这些研讨会的笔记由S S Koh记录，文本由W D Neumann修订和更新，最终以Differentiable manifolds and quadratic forms为书名，于1971年作为上述三位数学家的联合著作出版。1974年，希策布鲁赫与D Zagier合作出版了The Atiyah-Singer theorem and elementary number theory。值得引用Harvey Cohn评论的前几句话：-

这本书的观点具有如此潜在的革命性，以至于只有通过判断未来的发展，才能充分评价这本书，并充分评估这一观点。基本上，拓扑学在数论中的作用已经从诸如p进理论这样的局部方法，进步到诸如同调类的相交数这样的全局方法。

1981年，希策布鲁赫的五次讲座和Gerard van der Geer的八次讲座被两位作者合并成笔记，题为Lectures on Hilbert modular surfaces。他们特别发展了足够的代数曲面理论，以允许研究大卫·希尔伯特模曲面。与希策布鲁赫作为合著者的进一步著作有Zahlen（1983）、Geradenkonfigurationen und Algebraische Flächen Ⓣ（线配置与代数曲面）（1987）和Manifolds and modular forms（1992）。

希策布鲁赫还通过自1957年以来组织著名的Arbeitstagung会议，对德国的数学生活产生了重要影响。他组织的前30次Arbeitstagung构成了“第一系列”，而“第二系列”始于1993年。在此必须提到与迈克尔·阿蒂亚的合作，这一点非常重要。在前30次Arbeitstagung期间，迈克尔·阿蒂亚作了32次演讲，经常作为第一位演讲者。从1959年到1970年，迈克尔·阿蒂亚和希策布鲁赫发表了九篇联合论文。最初的论文受到亚历山大·格罗滕迪克在1957年第一次Arbeitstagung期间系列讲座的启发，这些讲座推广了希策布鲁赫的波恩哈德·黎曼-Roch定理。亚历山大·格罗滕迪克的思想导致了K-理论的发展。

希策布鲁赫于1980年在波恩创立了Max Planck Society for the Advancement of Science的数学研究所。该研究所拥有少量固定人员，几乎所有成员都是来自世界各地的数学家，在此度过一段固定时间。整个理念的基础是为问题和思想的交流提供最适宜的环境。图书馆、行政部门和计算机组提升了研究环境。希策布鲁赫于1980年至1995年担任该研究所所长。

希策布鲁赫于1961-62年以及1990年再次担任德国数学会主席。他还于1990年至1994年担任欧洲数学会主席。他是1998年在柏林举行的国际数学家大会的名誉主席。

希策布鲁赫作为名誉主席在1998年国际数学家大会上所作致辞的一部分见THIS LINK。

他的杰出贡献带来了最令人瞩目的荣誉清单。他当选为功勋勋章科学与艺术勋章、German Academy of Sciences Leopoldina、Heidelberg Academy、美因茨科学院、Royal Netherlands Academy of Sciences、北莱茵-威斯特法伦科学院、National Academy of Sciences（美国）、巴伐利亚科学院、Finnish Academy of Science and Letters、Russian Academy of Sciences、Paris Academy of Sciences、Göttingen Academy of Sciences、American Academy of Arts and Sciences、Ukrainian Academy of Sciences、萨克森科学院、Berlin Academy of Science、Royal Society of London、Royal Irish Academy、Polish Academy of Sciences、欧洲科学院、欧洲艺术与科学院、Austrian Academy of Sciences和爱丁堡皇家学会的成员。

此外，希策布鲁赫获得了华威大学、哥廷根大学、牛津大学、伍珀塔尔大学、圣母大学、都柏林大学、雅典大学、波茨坦大学、康斯坦茨大学、柏林洪堡大学、巴伊兰大学、奥斯陆大学以及芝加哥伊利诺伊大学、奥格斯堡大学和布加勒斯特的罗马尼亚科学院的名誉博士学位。他获得了许多奖项：瑞士联邦理工学院银质奖章（1950年）、沃尔夫数学奖（1988年）、Russian Academy of Sciences颁发的罗巴切夫斯基奖（1989年）、Japanese Mathematical Society颁发的关孝和-高木奖（1996年）、利奥波尔迪纳科滕尼乌斯金质奖章（1997年）、Russian Academy of Sciences颁发的罗蒙诺索夫金质奖章（1997年）、阿尔伯特·爱因斯坦奖章（1999年）、Polish Academy of Sciences颁发的斯特凡·巴拿赫奖章（2000年）、克虏伯科学奖（2000年）、柏林-勃兰登堡科学院颁发的赫尔曼·冯·亥姆霍兹奖章（2002年）、German Mathematical Society颁发的格奥尔格·康托尔奖章（2004年）。

根据引文，1988年沃尔夫基金会奖授予了希策布鲁赫：-

……因其结合拓扑学、代数与微分几何以及代数数论的杰出工作；以及因其对数学合作与研究的促进。

引文随后更全面地叙述了希策布鲁赫导致获奖的工作：-

在过去的三十五年里，希策布鲁赫教授的名字一直与拓扑学、代数几何和整体微分几何领域的著名成果联系在一起，这些成果都标志着重要理论的开端，并对现代数学的发展产生了巨大影响。希策布鲁赫的成就包括

1. 可微流形的符号定理的发现，以及代数簇的波恩哈德·黎曼-Roch定理的表述与证明，

2. 可微流形示性类的整性定理，

3. 复齐性流形的比例定理，以及与阿曼德·博雷尔合作的紧索菲斯·李群的齐性空间示性类的一般理论，

4. 复K-理论及其谱序列和各种几何应用（与迈克尔·阿蒂亚合作），

5. 通过4-流形理论对理查德·戴德金互反定理的‘拓扑’证明，以及微分拓扑与代数数论之间的其他引人入胜的关系

6. 对大卫·希尔伯特模形式与模曲面及其与类数关系的系统研究。

许多数学家扩展和推广了希策布鲁赫的思想。他本人一直对优美的特殊情形和具体问题感兴趣，他通过创造结合了非凡的几何、代数和算术直觉的新方法来解决问题。此外，通过他精彩的演讲和写作，通过“波恩工作坊”（每年最高水平的国际会议），以及通过他在科学组织中的奉献工作，他极大地促进了世界范围内的研究合作。

German Mathematical Society于2004年9月在海德堡举行的学会会议上向希策布鲁赫颁发了格奥尔格·康托尔奖章。颁奖词如下：-

为表彰其卓越成就，German Mathematical Society将格奥尔格·康托尔奖章授予希策布鲁赫教授博士。通过这一荣誉，DMV表彰一位享有世界声誉的数学家，其开创性工作极大地推动了数学的发展。他的思想和发现——特别是与波恩哈德·黎曼-Roch定理、示性类和K理论相关的——促成了20世纪下半叶数学最重要的发展之一。他为德国数学的国际融合以及东德和西德数学家融入一个共同组织所做的贡献超过任何人。

Friedrich Hirzebruch was the son of Dr Fritz Hirzebruch and Martha Holtschmidt. Fritz Hirzebruch was the headmaster of a secondary school in Hamm, and he also taught mathematics there. When he was at school, Friedrich enjoyed mathematics and learnt more advanced topics by talking to his father and reading his father's mathematics books. In 1937 Friedrich reached the age of ten and at that time he was required to join the Deutsches Jungvolk (German Youth), a subdivision of the Hitler youth for boys aged 10 to 14. In 1943 at the age of 15 he was drafted with his whole school class as "helper" into an anti aircraft position in his hometown.

In March 1945 Hirzebruch was drafted into the German army at the time when Allied forces were on the point of crossing the Rhine. By April, Allied forces had taken control of much of Germany and Hirzebruch was taken prisoner. He was released on 1st July 1945. He was allowed to enter the Westfälische Wilhelms University of Münster in November 1945 where he studied mathematics, physics and mathematical logic. Much of the University had been severely damaged by bombing, so at first he had to do most of his studying at home. Quickly, however, conditions improved and nearly normal university life returned.

He received his Ph.D. from the University of Münster in 1950 for his thesis Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen Ⓣ after studying under Heinrich Behnke. He also studied algebraic topology and algebraic geometry with Heinz Hopf at the Eidgenössische Technische Hochschule in Zürich from 1949 to 1950. This period was very important for the preparation of his thesis.

After serving as a Scientific Assistant at the University of Erlangen during 1951-52, he spent the two years 1952-54 at the Institute for Advanced Study in Princeton in the United States working with Armand Borel, Kunihiko Kodaira, and D C Spencer on topics such as sheaf theory, vector bundles, characteristic classes and Thom cobordism. He married Ingeborg Spitzley in 1952; they had two daughters and one son.

His first paper, published in 1951, was Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten Ⓣ. S S Cairns begins his review as follows:-

Two complex manifolds are analytically equivalent if they can be topologically and analytically mapped onto one another. This paper makes a contribution to the general problem of determining all analytic equivalence classes among a class of topologically equivalent manifolds.

Hirzebruch published a paper with the same title as his doctoral thesis in 1953, also publishing three further papers Über die quaternionalen projektiven Räume Ⓣ; On Steenrod's reduced powers, the index of inertia, and the Todd genus and Übertragung einiger Sätze aus der Theorie der algebraischen Flächen auf komplexe Mannigfaltigkeiten von zwei komplexen Dimensionen Ⓣ in the same year. One of his most famous results, now named the Hirzebruch-Riemann-Roch theorem, appeared in his 1954 paper Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties. He published Some problems on differentiable and complex manifolds in the same year. After submitting his habilitation thesis (identical with the book pubished in 1956) he spent the year 1954-55 as a Dozent at the University of Münster. Following this he was appointed as an Assistant Professor at Princeton University where he spent the year 1955-56.

Hirzebruch's impressive work on the Riemann-Roch theorem was fully set out in a book he published in 1956 entitled Neue topologische Methoden in der algebraischen Geometrie Ⓣ. D C Spencer writes:-

This monograph is an exposition of the main ideas and concepts centring around the fundamental Riemann-Roch theorem for algebraic manifolds of arbitrary dimension, a theorem whose proof has recently been found by the author and which is presented here in complete form for the first time. An algebraic manifold is understood to mean a compact complex manifold which can be imbedded complex-analytically without singularities in a complex projective space of suitably chosen dimension.

In 1956 he became a full professor at the Rheinischen Friedrich-Wilhelms University of Bonn, holding this position until he retired in 1993. During this time he served as Dean of the Faculty of Mathematics and Natural Sciences from 1962 to 1964.

The list of books written by Hirzebruch is impressive both for the number, the quality of the exposition, and for the influence on the direction of mathematical research. We give some details of those which followed his 1956 masterpiece mentioned above which was translated as Topological methods in algebraic geometry and appeared in several updated editions over many years. Garben-und Cohomologietheorie Ⓣ (1957), written with G Scheja, sets out the cohomology theory of sheaves. In 1962 Hirzebruch gave a series of seminars at Brandeis and Berkeley. Notes from these were taken by S S Koh and the text revised and updated by W D Neumann for the book Differentiable manifolds and quadratic forms which was published as a joint work by the three mentioned mathematicians in 1971. In 1974 Hirzebruch, jointly with D Zagier, published The Atiyah-Singer theorem and elementary number theory. It is well worth reproducing the first few sentences of a review by Harvey Cohn:-

The point of view of this book is so potentially revolutionary that the book can be adequately reviewed and the point of view can be adequately assessed only by judging future developments. Basically, the role of topology in number theory has progressed beyond the local methods such as p-adic theory to global methods such as intersection numbers of homology classes.

In 1981 a series of five lectures by Hirzebruch and eight lectures by Gerard van der Geer were combined into notes by the two authors entitled Lectures on Hilbert modular surfaces. In particular they develop enough of the theory of algebraic surfaces to allow an investigation of Hilbert modular surfaces. Further works with Hirzebruch as a coauthor are Zahlen (1983), Geradenkonfigurationen und Algebraische Flächen Ⓣ (1987), and Manifolds and modular forms (1992).

Hirzebruch also had an important influence on the mathematical life in Germany by organizing the famous Arbeitstagung meetings since 1957. The first 30 Arbeitstagungen which he organised form the "First Series" with a "Second Series" beginning in 1993. It is very important to mention at this point the cooperation with Michael Atiyah. During the first 30 Arbeitstagungen Atiyah lectured 32 times, often as the first speaker. From 1959 to 1970 Atiyah and Hirzebruch published nine joint papers. The first papers were stimulated by the series of lectures by A Grothendieck during the first Arbeitstagung 1957 generalizing Hirzebruch's Riemann-Roch theorem. Grothendieck's ideas led to the development of K-theory.

Hirzebruch founded the Institut für Mathematik of the Max Planck Society for the Advancement of Science in Bonn in 1980. The Institute has a small permanent staff, almost all members being mathematicians from around the world spending a fixed period there. The whole concept is based on providing the most suitable environment for the exchange of problems and ideas. The research environment is enhanced by the library, the administration and the computer group. Hirzebruch served as director of the Institute from 1980 to 1995.

Hirzebruch was President of the German Mathematical Society in 1961-62 and again in 1990. He was also President of the European Mathematical Society from 1990 to 1994. He was Honorary President of the International Congress of Mathematicians held in Berlin in 1998.

Part of Hirzebruch's address to the 1998 ICM address as Honorary President is at THIS LINK.

His outstanding contributions have led to the most impressive list of honours. He was elected to the Orden pour le mérite für Wissenschaft und Künste, the German Academy of Sciences Leopoldina, the Heidelberg Academy, the Mainz Academy, the Royal Netherlands Academy of Sciences, the Nordrheinwestfalen Academy, the National Academy of Sciences (United States), the Bayerische Akademie der Wissenschaften, the Finnish Academy of Science and Letters, the Russian Academy of Sciences, the Paris Academy of Sciences, the Göttingen Academy of Sciences, the American Academy of Arts and Sciences, the Ukrainian Academy of Sciences, the Sächsische Akademie, the Berlin Academy of Science, the Royal Society of London, the Royal Irish Academy, the Polish Academy of Sciences, Academia Europaea, the European Academy of Arts and Sciences, the Austrian Academy of Sciences and the Royal Society of Edinburgh.

In addition Hirzebruch has been awarded honorary doctorates from the University of Warwick, Göttingen, Oxford, Wuppertal, Notre Dame, Dublin, Athens, Potsdam, Konstanz, Humboldt-Berlin, Bar-Ilan, Oslo, and the University of Illinois at Chicago, Augsburg and the Romanian Academy of Sciences in Bucharest. He has been awarded many prizes: the Silver Medal from the Swiss Federal Institute of Technology (1950), the Wolf Prize for Mathematics (1988), the Lobachevsky Prize from the Russian Academy of Sciences (1989), the Seki-Takakazu Prize of the Japanese Mathematical Society (1996), the Cothenius Gold Medal Leopoldina (1997), the Lomonosov Gold Medal of the Russian Academy of Sciences (1997), the Albert Einstein Medal (1999), the Stefan Banach Medal of the Polish Academy of Sciences (2000), the Krupp-Wissenschaftspreis (2000), the Helmholtz Medal of the Berlin-Brandenburg Academy of Sciences 2002, the Georg Cantor Medal of the German Mathematical Society (2004).

According to the citation the 1988 Wolf Foundation Prize was awarded to Hirzebruch:-

... for outstanding work combining topology, algebraic and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research.

The citation then gave a fuller account of Hirzebruch's work which led to the award:-

For the past three and a half decades, the name of Professor Friedrich Hirzebruch has been connected with famous results in the areas of topology, algebraic geometry, and global differential geometry, results which all mark the beginning of important theories and which have had an enormous influence on the development of modern mathematics. Hirzebruch's achievements include

1. the discovery of the signature theorem for differentiable manifolds and the formulation and proof of the Riemann-Roch theorem for algebraic varieties,

2. the integrality theorem for characteristic classes of differentiable manifolds,

3. the proportionality theorem for complex homogeneous manifolds and (with Armand Borel) the general theory of characteristic classes of homogeneous spaces of compact Lie groups,

4. complex K-theory and its spectral sequence and various geometrical applications (with M F Atiyah),

5. the 'topological' proof of the Dedekind reciprocity theorem through 4-manifold theory and other fascinating relations between differential topology and algebraic number theory

6. the systematic study of Hilbert modular-forms and-surfaces and their relation to class numbers.

Many mathematicians have expanded and generalized Hirzebruch's ideas. He himself has always been interested in the beautiful particular case and concrete problem, which he solves by creating new methods that combine unusual geometric, algebraic, and arithmetic intuition. Moreover, through his brilliant lecturing and writing, through the "Arbeitstagung Bonn" (yearly international meetings at the highest level), and through his dedicated work in scientific organizations he has greatly stimulated world-wide cooperation in research.

The German Mathematical Society presented Hirzebruch with their Georg Cantor Medal during a meeting of the Society in Heidelberg in September 2004. The citation reads:-

In recognition of his remarkable achievements the German Mathematical Society bestows the Georg Cantor Medal on Prof Dr Friedrich Hirzebruch. With this distinction the DMV honors a mathematician of worldwide reputation whose path-breaking works have substantially furthered mathematics. His ideas and discoveries - particularly in connection with Riemann-Roch theorems, characteristic classes, and K-theory - have contributed to the instigation of one of the most important developments in mathematics in the second half of the 20th century. He has contributed more than anyone else to the international integration of German mathematics and to the absorption of East and West German mathematicians into a common organization.