雅克·蒂茨Jacques Tits

雅克·蒂茨出生在布鲁塞尔南郊的Uccle。他的父母是Léon 蒂茨，一位教授，和Lousia André。蒂茨就读于Uccle的Athénée，然后在布鲁塞尔自由大学学习。他在布鲁塞尔的论文导师是Paul Libois，蒂茨于1950年毕业，获得博士学位，提交了他的学位论文Généralisation des groupes projectifs basés sur la notion de transitivité Ⓣ（基于传递性概念的射影群的推广）。从1948年到1956年，他由比利时国家科学研究基金资助。

蒂茨的第一批论文，继他为博士论文所做的工作之后，是关于三重传递群的推广。他在1949年发表了一篇分为两部分的论文Généralisations des groupes projectifs Ⓣ（射影群的推广），关于这个主题，推广了一维射影变换群。所证明的结果包括在三重传递群中刻画射影群。在Groupes triplement transitifs et généralisations Ⓣ（三重传递群及推广）（1950）中，蒂茨继续研究$n$重传递群的推广，定义了一个几乎$n$重传递群。这推广了平面的直射变换群，它是几乎四重传递的。在Sur les groupes triplement transitifs continus; généralisation d'un théorème de Kerékjártó Ⓣ（关于三重传递连续群；Kerékjártó定理的推广）（1951）中，蒂茨利用他早期的结果，在三重传递群中刻画射影群，研究拓扑空间的三重传递变换群。

蒂茨于1956年9月8日与历史学家Marie-Jeanne Dieuaide结婚。从1956年到1962年，他是布鲁塞尔大学的助理。他于1962年晋升为教授，并在布鲁塞尔担任此职两年，然后于1964年接受波恩大学的教授职位。在蒂茨在布鲁塞尔的博士生中，我们提到Francis Buekenhout，他于1965年获得博士学位。1973年，蒂茨接受了法兰西公学院的群论讲席。担任此职后不久，他于1974年入籍法国。蒂茨一直担任此讲席直到2000年退休。

蒂茨引入的大量重要数学发展，远非此处能详细涵盖。也许他工作中最重要的部分是引入建筑，这在Ronan的[3]中被置于背景中。我们给出他的总结：-

本文是一篇论述群论的发展如何导致各类单群的发现，以及这些单群又如何导致建筑理论的文章。故事梗概如下。埃瓦里斯特·伽罗瓦首次在专业意义上使用了“群”这个术语，并发现了最早的单群。Jordan在其1870年出版的著名著作Traité des substitutions et des équations algébriquesⓉ（《置换与代数方程论》）中，推广了埃瓦里斯特·伽罗瓦的工作，并将群论置于坚实的基础之上。此时群被当作置换群来处理，但群论的其他方面很快也发展起来。索菲斯·李于1870年作为研究生访问巴黎，随后创立了连续变换群理论。威廉·基灵独立地得到了这类群，并于1888年利用半单复索菲斯·李代数（族A到G）发现了单索菲斯·李群的分类。埃利·嘉当于1894年改进了这一分类，修正了证明中的一些错误，现在它被称为威廉·基灵-埃利·嘉当分类。经典族（A到D）很快引出了实数或复数以外的域上的群，伦纳德·尤金·迪克森于1901年发表了一项全面的研究。后来他处理了$E_{6}$和G，但其他方面的进展直到第二次世界大战之后才出现。蒂茨正在研究这个问题，谢瓦莱也在研究，后者当时是一位更资深的数学家。谢瓦莱于1955年取得了成功，他的论文很快引出了Steinberg、蒂茨、Suzuki和Ree的变体。在此期间，蒂茨逐渐发展了建筑理论，他1974年的著作“Buildings of spherical type and finite BN-pairs”产生了一套完整的理论，此后得到了许多应用。……我们提到蒂茨关于建筑的一些早期工作，并讨论他上述关于球型建筑的著作的内容。最后……提到了后来由蒂茨提出的另一种建筑研究方法，并在结尾回到利用建筑理论构造李型例外群的问题。

通过大量其他重要角色，蒂茨在数学生活中发挥了重要作用。例如，他于1980年至1999年担任I.H.E.S.数学出版物的主编。他于1978年和1994年两次担任颁发菲尔兹奖的委员会成员。他还于1985年担任颁发巴尔赞奖的委员会成员。

蒂茨已经获得并将继续获得许多荣誉。其中我们提到L Empain跨学院科学奖（1955年）、皇家比利时科学院的Wettrems奖（1958年）、比利时政府十年数学奖（1965年）、French Academy of Sciences大奖（1976年）、沃尔夫数学奖（1993年）以及德国数学会的康托尔奖章（1996年）。他当选为许多科学院和学会的成员，包括德国利奥波第那科学院（1977年）、Royal Netherlands Academy of Sciences（1988年）、欧洲科学院创始成员（1988年）、Royal Belgium Academy of Science（1991年）、American Academy of Arts and Sciences（1992年）、美国国家科学院（1992年）和伦敦数学会（1993年）。他获得了乌得勒支大学（1970年）、根特大学（1979年）、波恩大学（1988年）和鲁汶大学（1992年）的荣誉博士学位。他被授予荣誉军团骑士勋章（1995年）和国家功绩军官勋章（2001年）。

2000年退休后，蒂茨成为鲁汶大学Vallée-夏尔-让·德拉瓦莱·普桑讲席的第一位持有者。他于2001年10月18日作了就职演讲Immeubles : une approche géométrique des groupes algébriques simples et des groupes de Kac-Moody。随后他就以下主题进行了三个系列的讲座

(1) Généralités sur les nombres $p$-adiques. Groupes algébriques simples sur les corps $p$-adiquesⓉ（p进数概述。p进域上的单代数群）；
(2) Schémas en groupes à fibre générique simple sur les anneaux d'entiersⓉ（整数环上整群的简单一般纤维图）；
(3) Réseaux invariants dans les espaces de représentations. Applications algébriquesⓉ（表示空间中的不变格。代数应用）。

2008年，Norwegian Academy of Science and Letters将尼尔斯·阿贝尔奖授予约翰·格里格斯·汤普森和蒂茨：-

……以表彰他们在代数方面的深刻成就，特别是为塑造现代群论所做的贡献。

新闻稿对蒂茨的贡献作了如下总结：-

蒂茨 创立了一种极具影响力的新观点，将群视为几何对象。他引入了如今被称为 蒂茨 建筑的结构，它以几何术语编码了线性群的代数结构。建筑理论是一个核心的统一原理，其应用范围惊人，例如用于代数群和李群以及有限单群的分类，用于Kac-Moody群（理论物理学家所用），用于组合几何（计算机科学所用），以及用于负曲率空间中的刚性现象研究。蒂茨 的几何方法在研究和实现包括Monster在内的散在群中至关重要。他还确立了著名的“蒂茨 二择一”：每个有限生成的线性群要么是几乎可解的，要么包含两个生成元的自由群的一个副本。这一结果激发了众多变体和应用。约翰·格里格斯·汤普森 和 蒂茨 的成就具有非凡的深度和影响力。它们相互补充，共同构成了现代群论的支柱。

Jacques Tits was born in Uccle, on the southern outskirts of Brussels. His parents were Léon Tits, who was a professor, and Lousia André. Jacques attended the Athénée of Uccle and then studied at the Free University of Brussels. His thesis advisor in Brussels was Paul Libois, and Tits graduated with his doctorate in 1950 having submitted his dissertation Généralisation des groupes projectifs basés sur la notion de transitivité Ⓣ . From 1948 to 1956 he was funded by the Belgium Fonds National de la Recherche Scientifique.

Tits' first papers, following the work he had undertaken for his doctoral dissertation, were on generalisations of triply transitive groups. He published a two part paper Généralisations des groupes projectifs Ⓣ in 1949 on this topic generalising the group of one-dimensional projective transformations. Among the results proved were characterisations of projective groups among triply transitive groups. In Groupes triplement transitifs et généralisations Ⓣ (1950), Tits went on to look at generalisations of $n$-tuply transitive groups, defining an almost $n$-tuply transitive group. This generalises the group of collineations of the plane which is almost quadruply transitive. In Sur les groupes triplement transitifs continus; généralisation d'un théorème de Kerékjártó Ⓣ (1951) Tits looked at triply transitive groups of transformations of a topological space using his earlier results which characterised the projective groups among triply transitive groups.

Tits married Marie-Jeanne Dieuaide, a historian, on 8 September 1956. From 1956 to 1962 he was an assistant at the University of Brussels. He was promoted to professor in 1962 and remained in this role at Brussels for two years before accepting a professorship at the University of Bonn in 1964. Among Tits' doctoral students in Brussels we mention Francis Buekenhout who was awarded his doctorate in 1965. In 1973 Tits accepted the Chair of Group Theory at the Collège de France. Shortly after taking up this post, he became a naturalised French subject in 1974. Tits held this chair until he retired in 2000.

The large and important mathematical developments introduced by Tits are far too numerous to cover here in any detail. Perhaps the most important part of his work was the introduction of buildings and this is put into context by Ronan in [3]. We give his summary:-

This paper is an essay on how the development of group theory led to the discovery of various families of simple groups, and how these in turn led to the theory of buildings. In outline the story is this. Galois first used the term 'group' in the technical sense, and found the first simple groups. Jordan, in his famous Traité des substitutions et des équations algébriques Ⓣ, published in 1870, promoted Galois' work and put the theory of groups on a firm foundation. At this time groups were treated as groups of permutations, but other aspects of group theory were soon on the way. Lie visited Paris in 1870 as a graduate student, and went on to create the theory of continuous transformation groups. Killing came to such groups independently, and in 1888 found the classification of the simple Lie groups, using semisimple complex Lie algebras (families A through G). Cartan refined this classification in 1894, correcting some errors in the proofs, and it is now known as the Killing-Cartan classification. The classical families (A through D) soon led to groups over fields other than the real or complex numbers, and a comprehensive study was published by Dickson in 1901. Later he dealt with $E_{6}$ and G, but progress on the others did not occur until after the Second World War. Tits was working on the problem, as was Chevalley, who was a more established mathematician at that time. Chevalley succeeded in 1955, and his paper was soon followed by variations due to Steinberg, Tits, Suzuki, and Ree. During this time Tits was gradually developing the theory of buildings, and his book "Buildings of spherical type and finite BN-pairs" in 1974 produced a fully-fledged theory that has since found many uses. ... we mention some of Tits' early work on buildings, and we discuss the contents of his above-mentioned book concerning buildings of spherical type. Finally ... a later approach to buildings, also due to Tits, is mentioned, and we return at the end to the construction of the exceptional groups of Lie type using building theory.

Through a large number of other important roles, Tits played a major part in mathematical life. For example he was editor-in-chief for mathematical publications at I.H.E.S. from 1980 to 1999. He served on the committee awarding the Fields medals in 1978 and again in 1994. He also served on the committee awarding the Balzan Prize in 1985.

Tits has received, and continues to receive, many honours. Among these we mention the Prix scientifique Interfacultataire L Empain (1955), the Wettrems Prize of the Royal Belgium Academy of Science (1958), the Prix décennal de mathématique from the Belgium government (1965), the Grand Prix of the French Academy of Sciences (1976), the Wolf Prize in Mathematics (1993), and the Cantor Medal from the German Mathematical Society (1996). He was elected to many academies and societies including the German Academy of Sciences Leopoldina (1977), the Royal Netherlands Academy of Sciences (1988), founder member of Academia Europaea (1988), the Royal Belgium Academy of Science (1991), the American Academy of Arts and Sciences (1992), the National Academy of Sciences of the United States (1992), and London Mathematical Society (1993). He has been awarded honorary doctorates from the universities of Utrecht (1970), Ghent (1979), Bonn (1988) and Leuwen (1992). He was made Chevalier de la Légion d'Honneur (1995) and Officier de l'Ordre National du Mérite (2001).

After retiring in 2000, Tits became the first holder of the Vallée-Poussin Chair from the University of Louvain. He gave his inaugural lecture Immeubles : une approche géométrique des groupes algébriques simples et des groupes de Kac-Moody on 18 October 2001. He followed this with three series of lectures on the following topics

(1) Généralités sur les nombres $p$-adiques. Groupes algébriques simples sur les corps $p$-adiques Ⓣ;
(2) Schémas en groupes à fibre générique simple sur les anneaux d'entiers Ⓣ;
(3) Réseaux invariants dans les espaces de représentations. Applications algébriques Ⓣ.

In 2008 the Norwegian Academy of Science and Letters awarded the Abel Prize to John Griggs Thompson and Jacques Tits:-

... for their profound achievements in algebra and in particular for shaping modern group theory.

The Press Release gives the following summary of Tits's contributions:-

Tits created a new and highly influential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits's geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated "Tits alternative": every finitely generated linear group is either virtually solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications. The achievements of John Thompson and of Jacques Tits are of extraordinary depth and influence. They complement each other and together form the backbone of modern group theory.