格列戈里·马尔古利斯Grigory Margulis

格列戈里·马尔古利斯在莫斯科高中接受教育，1962年毕业。同年，他开始在莫斯科大学攻读本科，并于1967年获得第一个学位。马尔古利斯留在莫斯科大学攻读研究生。

他展现出作为数学家的巨大潜力，他获得的第一个重要奖项是在研究生期间，于1968年获得了莫斯科数学会颁发的青年数学家奖。马尔古利斯于1970年完成研究生学业，并因论文On some problems in the theory of U-systems获得科学副博士学位。

在获得科学副博士学位（相当于英国或美国的博士学位）后，马尔古利斯开始在工作信息传输问题研究所工作。1970年至1974年，他在那里担任初级科研人员，之后晋升为高级科研人员。他担任此职位直到1986年再次晋升，这次晋升为首席科研人员。

1978年，马尔古利斯获得了国际荣誉，在赫尔辛基国际数学家大会上被授予菲尔兹奖。然而，这对马尔古利斯来说并不是一个愉快的时刻，因为苏联当局不允许他前往赫尔辛基领取奖章。雅克·蒂茨在发表演讲[7]时，对马尔古利斯无法出席表示悲伤：-

……我不得不表达我深深的失望——毫无疑问，在座的许多人也有同感——马尔古利斯缺席了这次仪式。鉴于赫尔辛基这座城市的象征意义，我确实有理由希望，我终于有机会见到一位我只通过其工作了解的数学家，我对他怀有最大的敬意和钦佩。

也许雅克·蒂茨关于“象征意义”的评论应该解释一下。他在赫尔辛基的芬兰大厦发表了演讲，那里本应是马尔古利斯领取奖章的地方，也是1975年8月1日签署《赫尔辛基协议》的地方。这项重大协议是在第一次欧洲安全与合作会议结束时签署的。《赫尔辛基协议》由所有欧洲国家（不包括阿尔巴尼亚）以及美国和加拿大签署，旨在通过接受当时的欧洲边界来缓和冷战紧张局势。

雅克·蒂茨在[7]中谈到了马尔古利斯在组合学、微分几何、遍历理论、动力系统和李群的离散子群方面的工作范围。约翰·查尔斯·菲尔兹奖章的授予主要是为了他在后一主题上的工作：-

儒勒·昂利·庞加莱已经在思考描述李群G中所有有限余体积的离散子群的可能性。$G = PSL_{2}(\mathbb{R})$中这类子群的繁多使人起初怀疑这种可能性。然而，$PSL_{2}(\mathbb{R})$在很长一段时间内是唯一已知包含非算术有限余体积离散子群的单李群，而1965年由Makarov和Vinberg发现的进一步例子只涉及少数其他李群，从而增加了阿特勒·塞尔伯格和Pyatetski-Shapiro的猜想的可信度，即“对于大多数半单李群”，有限余体积的离散子群必然是算术的。马尔古利斯最引人注目的成就是完全解决了这个问题，特别是证明了所讨论的猜想。

马尔古利斯很快得以离开苏联阵营，并于1979年在波恩大学度过了三个月。1988年至1991年间，马尔古利斯多次访问波恩的马克斯·普朗克研究所、高等研究院和法兰西学院、哈佛大学以及普林斯顿高等研究院。自1991年起，他在耶鲁大学担任讲席。

亚历山大·奥本海姆猜想于1929年提出，涉及不定无理二次型在整数点处的值。早期工作基于沃伊捷赫·亚尔尼克和阿诺德·华菲斯的结果。20世纪40年代，哈罗德·达文波特和汉斯·海尔布龙通过证明特殊情形做出了贡献，1946年乔治·奈维尔·沃森扩展了他们的结果，表明该猜想在更多特殊情形下成立。马尔古利斯于1986年证明了完整猜想，并在[3]中对导致这一解决的工作给出了精彩的综述。在那里，马尔古利斯解释道：-

处理这一猜想及相关猜想（和定理）的不同方法涉及解析数论、李群与代数群理论、遍历理论、表示论、约化理论、数的几何以及其他一些主题。

马尔古利斯因其工作获得了许多荣誉。除了菲尔兹奖章外，他还被授予法兰西公学院奖章（1991年），并于同年当选为美国艺术与科学院的名誉成员。1995年，他获得了洪堡奖，1996年，他荣幸地当选为塔塔基础研究所的成员。

马尔古利斯还被授予俄罗斯科学院的罗巴切夫斯基国际奖，并当选为美国国家科学院。2005年他被授予沃尔夫数学奖：-

……因其对代数的巨大贡献，特别是对半单李群中格理论，以及将其显著应用于遍历理论、表示论、数论、组合学和测度论。

American Mathematical Society的Notices2005年5月部分的一篇文章解释了导致该奖项的工作：-

Gregory Margulis工作的核心是他对阿特勒·塞尔伯格-Piatetskii-Shapiro猜想的证明，该猜想断言高秩Lie群中的格是算术的，这个问题可追溯到儒勒·昂利·庞加莱。这是通过一项非凡的壮举实现的，其中围绕遍历定理的非交换版本的概率思想与p进分析以及代数几何思想相结合，表明先前由马尔古利斯等人建立的“刚性”现象可以表述为（“超刚性”）从而蕴含算术性。这项工作在代数方法和分析方法上都展现出惊人的技术精湛和独创性。这项工作随后重塑了流形上一般群作用的遍历理论。

在第二项壮举中，马尔古利斯解决了1929年的亚历山大·奥本海姆猜想，该猜想断言，三个以上变元的不定无理非退化二次型在整数点处的取值集合在$\mathbb{R}^{n}$中稠密。这已被（Rhagunathan）归约为关于齐性空间上单幂流的猜想，并由马尔古利斯证明。这一方法将此前仅在解析数论中研究的一类问题转化到了这一遍历设定中。

第三个戏剧性的突破出现在马尔古利斯表明大卫·卡日丹的“性质T”（已知对刚性格成立）可用于单个算术格构造，以解决两个看似无关的问题。一个是解决Rusiewicz提出的关于球面和欧几里得空间上的有限可加测度的问题。另一个是首次显式构造有界度数的无限族扩展图，这是高效通信网络设计中的一个实际应用问题。马尔古利斯的工作特点是非凡的深度、技术力量、对来自数学不同领域的思想和方法的创造性综合，以及其最终形式的宏大建筑统一性。尽管他的工作涉及深刻的未解决问题，但他的解决方案建立在具有广泛而持久应用的新概念和方法论框架中。他是过去半个世纪的数学巨人之一。

2008年，Pure and Applied Mathematics Quarterly出版了一期特刊以纪念马尔古利斯。引言中写道：-

马尔古利斯是一位极具深度和独创性的数学家。除了他在高秩半单Lie群不可约格的超刚性和算术性方面的著名结果，以及解决关于无理不定二次型在整数点取值的亚历山大·奥本海姆猜想之外，他还开创了许多其他研究方向，并解决了各种著名的未解决问题。

最后，我们引用雅克·蒂茨 [7]来结束这篇传记：-

马尔古利斯完全或几乎完全解决了李群离散子群理论中的若干重要问题，这些问题根植于遥远的过去，其意义远远超出了该理论本身。可以毫不夸张地说，他多次通过解决当时看来完全无法企及的问题，令专家们感到困惑。他之所以能做到这一点，是因为他掌握了多种多样的技巧，并以非凡的技艺和独创性加以运用。他发明的新的、最强有力的方法，除了为其所创的目的之外，已经有了其他重要应用；考虑到它们的普遍性，我毫不怀疑它们在未来还会有更多应用。

Gregori Margulis was educated at Moscow High School, graduating in 1962. In that year he began his undergraduate studies at Moscow University and he was awarded his first degree in 1967. Margulis remained at Moscow University for his postgraduate studies.

He showed great potential as a mathematician and the first important award which he won was during his time as a postgraduate student when he received the young mathematicians prize from the Moscow Mathematical Society in 1968. Margulis completed his graduate studies in 1970 and he was awarded the degree of Candidate of Science for a thesis On some problems in the theory of U-systems.

After being awarded the Candidate of Science degree (the equivalent of a British or American Ph.D.), Margulis began to work in the Institute for Problems in Information Transmission. He was a Junior scientific worker there from 1970 to 1974 when he was promoted to Senior scientific worker. He held this post until 1986 when he was promoted again, this time to Leading scientific worker.

International honour was given to Margulis in 1978 when he was awarded a Fields Medal at the International Congress at Helsinki. However it was not a happy occasion for Margulis who was not permitted by the Soviet authorities to travel to Helsinki to receive the Medal. Tits, delivering the address [7] spoke of his sadness that Margulis could not be present:-

... I cannot but express my deep disappointment - no doubt shared by many people here - in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration.

Perhaps Tits's comment about 'symbolic meaning' should be explained. He delivered the address in the Finlandia Hall in Helsinki where Margulis should have received the Medal and where the Helsinki Accords had been signed on 1 August 1975. This major agreement was signed at the end of the first Conference on Security and Cooperation in Europe. The Helsinki Accords, signed by all the countries of Europe (excluding Albania) and by the United States and Canada, were designed to reduce the cold war tension by accepting the European boundaries as they then were.

Tits talks in [7] about the range of Margulis's work in combinatorics, differential geometry, ergodic theory, dynamical systems and discrete subgroups of Lie groups. The award of the Fields Medal was mainly for his work on this latter topic:-

Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups in $G = PSL_{2}(\mathbb{R})$ makes one at first doubt of any such possibility. However, $PSL_{2}(\mathbb{R})$ was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that "for most semisimple Lie groups" discrete subgroups of finite covolume are necessarily arithmetic. Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question.

Margulis was soon able to leave the Soviet bloc and, in 1979, he was able to spend three months at the University of Bonn. Between 1988 and 1991 Margulis made a number of visits to the Max Planck Institute in Bonn, to the Institut des Hautes Études and to the Collège de France, to Harvard and to the Institute for Advanced study in Princeton. From 1991 he has held a chair at Yale University.

The Oppenheim conjecture was made in 1929 and concerns values of indefinite irrational quadratic forms at integer points. Early work was based on results of Jarnik and Walfisz. In the 1940s Davenport and Heilbronn contributed by proving special cases and in 1946 Watson extended their results showing the conjecture to be true for further special cases. Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [3]. There Margulis explains that:-

The different approaches to this and related conjectures (and theorems) involve analytic number theory, the theory of Lie groups and algebraic groups, ergodic theory, representation theory, reduction theory, geometry of numbers and some other topics.

Margulis has received many honours for his work. In addition to the Fields Medal he has been awarded the Medal of the Collège de France (1991) and in the same year he was elected an honorary member of the American Academy of Arts and Science. In 1995 he received the Humboldt Prize and in 1996 he was honoured by election as a member of the Tata Institute of fundamental research.

Margulis has also been awarded the Lobachevsky International Prize of the Russian Academy of Sciences and has been elected to the United States National Academy of Sciences. In 2005 he was awarded the Wolf Prize for Mathematics:-

... for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics and measure theory.

An article in the May 2005 part of the Notices of the American Mathematical Society explains the work which led to the award:-

At the centre of the work of Gregory Margulis lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincaré. This was achieved by a remarkable tour de force, in which probabilistic ideas revolving around a noncommutative version of the ergodic theorem were combined with p-adic analysis and with algebraic geometric ideas showing that "rigidity" phenomena, earlier established by Margulis and others, could be formulated in such a way ("super-rigidity") as to imply arithmeticity. This work displays stunning technical virtuosity and originality, with both algebraic and analytic methods. The work has subsequently reshaped the ergodic theory of general group actions on manifolds.

In a second tour de force, Margulis solved the 1929 Oppenheim Conjecture, stating that the set of values at integer points of an indefinite irrational nondegenerate quadratic form in more than three variables is dense in $\mathbb{R}^{n}$. This had been reduced (by Rhagunathan) to a conjecture about unipotent flows on homogeneous spaces, proved by Margulis. This method transformed to this ergodic setting a family of questions till then investigated only in analytic number theory.

A third dramatic breakthrough came when Margulis showed that Kazhdan's "Property T" (known to hold for rigid lattices) could be used in a single arithmetic lattice construction to solve two apparently unrelated problems. One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces. The other was the first explicit construction of infinite families of expander graphs of bounded degree, a problem of practical application in the design of efficient communication networks. Margulis's work is characterized by extraordinary depth, technical power, creative synthesis of ideas and methods from different areas of mathematics, and a grand architectural unity of its final form. Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks of broad and enduring application. He is one of the mathematical giants of the last half century.

In 2008 the Pure and Applied Mathematics Quarterly produced a Special Issue in honour of Margulis. The Introduction states:-

Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems.

Finally we end this biography by quoting from Tits [7]:-

Margulis has completely or almost completely solved a number of important problems in the theory of discrete subgroups of Lie groups, problems whose roots lie deep in the past and whose relevance goes far beyond that theory itself. It is not exaggerated to say that, on several occasions, he has bewildered the experts by solving questions which appeared to be completely out of reach at the time. He managed that through his mastery of a great variety of techniques used with extraordinary resources of skill and ingenuity. The new and most powerful methods he has invented have already had other important applications besides those for which they were created and, considering their generality, I have no doubt that they will have many more in the future.