洛朗·拉福格Laurent Lafforgue

洛朗·拉福格出生于巴黎南郊的安东尼。他于1972年进入安东尼的朱尔·费里小学，1977年转入同一城镇的勒内·笛卡儿中学。在一次采访[4]中，他谈到了自己的求学经历：-

我非常喜欢我从小就读的学校。我的祖父母十二岁就开始工作，但他们一直非常尊重学校，并将这种尊重传给了他们的子女和孙辈。

他于1984年获得中学毕业会考文凭（优异成绩），并在综合竞赛中获得数学一等奖。此时他已经赢得了国际声誉，因为他在1984年国际数学奥林匹克竞赛中获得银牌，并在1985年的竞赛中再次获得银牌。在巴黎路易大帝中学度过1984-86两年准备大学学习后，他于1986年进入巴黎高等师范学院。他于1988年获得Agrégation de Mathématiques学位。随后，他在Christophe Soulé的指导下开始研究代数几何和Arakelov理论。

拉福格于1990年在法国国家科学研究中心（CNRS）成为chargé de recherche ，并在奥赛的巴黎南大学算术与代数几何团队工作。1991-92年间，他在科埃基当的法国国家军事学院圣西尔军校服兵役。这一年之后，他回到奥赛巴黎南大学的职位，并于1994年以D-stukas de Drinfeld为题、由Gérard Laumon指导完成学位论文，获得博士学位。由于这项杰出工作，他获得了法兰西公学院的佩科奖，并受邀讲授佩科课程。他继续在法国国家科学研究中心工作，并于1998年受邀在柏林国际数学家大会上作“群与李代数”分会报告。同年，他获得了CNRS铜奖。

2000年，拉福格被提升为法国国家科学研究中心directeur de recherche ，在巴黎南大学数学系工作。此后不久，他被任命为法国比尔斯-伊维特高等科学研究所的教授。事实上，2000年对拉福格来说在另一方面也很重要，因为在5月24日，在法兰西公学院的巴黎千年会议上，他获得了2000年克莱研究奖。在会议上，安德鲁·怀尔斯宣布将该奖项授予拉福格，Lavinia Clay向他颁发了弗格森雕塑。在[9]中，给出了拉福格的成就及其背景的详细信息：-

拉福格为比先前已知更广泛的一类情况建立了罗伯特·朗兰兹对应。这些对应将算术性质与某些称为自守表示的特殊群表示的解析性质联系起来。它由罗伯特·朗兰兹在1960年代末提出。在秩1的情况下，这个猜想不过是埃米尔·阿廷的现在传统的“类域论”。在秩2和数域的情况下，这个猜想的第一个重大确认是皮埃尔·德利涅对拉马努金猜想的证明，以及罗伯特·朗兰兹本人对埃米尔·阿廷猜想的证明，除了一种情况。在七十年代初，弗拉基米尔·德林费尔德在更一般的代数背景下攻克这些猜想。为此，他构造了类似于模曲线的簇，并展示了秩2中罗伯特·朗兰兹猜想的某些情况。然后，由于这些簇无法达到所有想要的表示，弗拉基米尔·德林费尔德引入了“chtoucas”，这一步使他能够证明秩2中的罗伯特·朗兰兹猜想。在克服了巨大的技术困难之后，这使得一般情况变得可行。拉福格解决这个问题的关键贡献是构造某些模簇的紧化。这个证明是里程碑式的，是六年多集中努力的结果。

拉福格很快因其卓越的数学成就而获得了更多重要奖项。他于2001年获得了科学院颁发的雅克·埃尔布朗奖，随后在第二年，他获得了被认为是数学家最高荣誉的奖项，即约翰·查尔斯·菲尔兹奖章。他于2002年8月20日在中国北京举行的国际数学家大会开幕式上获得了该奖章。Michael Rapoport写道[6]（或[5]）：-

拉福格因其证明了正特征函数域上一般线性群$GL_{r}$的罗伯特·朗兰兹对应而获得了菲尔兹奖。他对这一问题的处理方法遵循了弗拉基米尔·德林费尔德二十五年前在证明$GL_{2}$时引入的基本策略。弗拉基米尔·德林费尔德的证明已经极其困难。拉福格的证明是一项真正的壮举，占用了数百页高度浓缩的推理。通过他的成就，拉福格证明了自己是一位具有非凡力量和毅力的数学家。

Allyn Jackson写道[7]：-

拉福格通过证明函数域的全局罗伯特·朗兰兹对应，在罗伯特·朗兰兹纲领中取得了巨大进展。罗伯特·朗兰兹纲领由罗伯特·朗兰兹在20世纪60年代提出，提出了一张连接埃瓦里斯特·伽罗瓦表示和自守形式的关系网。罗伯特·朗兰兹纲领的影响力多年来不断增长，每一次新进展都被誉为重要成就。罗伯特·朗兰兹纲领的根源在于数论中最深刻的结果之一——二次互反律，该定律由卡尔·弗里德里希·高斯于1801年首次证明。这一定律允许人们描述，对于任何正整数d，使得同余式$x^{2} = d mod p$有解的素数p。尽管这一定律有许多证明（卡尔·弗里德里希·高斯本人就给出了六个不同的证明），它仍然是数论中最神秘的事实之一。对二次互反律推广的探索刺激了19世纪数论的大量研究。埃米尔·阿廷在20世纪20年代的里程碑式工作产生了当时已知的最一般的互反律。罗伯特·朗兰兹纲领背后的最初动机之一就是提供对互反律的完整理解。

拉福格证明的$GL_{n}$的全局罗伯特·朗兰兹对应提供了对函数域互反律的完整理解。拉福格为任何给定的函数域建立了其埃瓦里斯特·伽罗瓦群的表示与该域相关的自守形式之间的精确联系。他建立在1990年约翰·查尔斯·菲尔兹奖章得主弗拉基米尔·德林费尔德的工作之上，后者在20世纪70年代证明了$GL_{2}$的全局罗伯特·朗兰兹对应。

2003年，拉福格成为荣誉军团骑士，并当选为巴黎Académie des Sciences院士。同样在2003年，他的重要著作Chirurgie des grassmanniennes（Grassmann流形上的割补）由美国数学会出版。然而，他生命中的一个新篇章即将开始，即他对教育问题的热情投入[4]：-

我对教育这一主题的具体兴趣始于几年前，当时我签署了一份请愿书，为希腊语和拉丁语作为学术科目辩护，因为它们正处于严重危险之中。这一戏剧性的状况令我震惊，而只有少数教师对此予以谴责，于是我开始了更深入的探究，阅读了不同意识形态取向的人所写的书，他们共同之处在于对工作的认真态度，以及对学校和年轻人未来的热情。这些阅读深深地震撼了我——拉丁语和希腊语只是冰山一角！在法国，甚至连法语本身的教学也岌岌可危。新式法国学校与我仅仅二十五年前所了解的学校已毫无共同之处。

2004年5月15日，拉福格在一次为支持中学拉丁语和希腊语教学而组织的会议上发表了A mathematician and the classics演讲。从那时起，他便高度投身于教育问题。2005年11月8日星期二，教育高级委员会（Haut Conseil de l'Education）成立，取代了其他几个教育机构。其职责包括界定所有法国儿童在16岁前需要掌握的知识和技能的内容。它还负责制定教师大学学院的规范。法兰西共和国总统Jacques Chirac任命拉福格为教育高级委员会成员，该委员会于2005年11月17日星期四举行了第一次会议。次日，拉福格被要求从委员会辞职，因为他质疑了听取所谓"experts of the Ministry of National Education"建议的必要性。在致教育高级委员会主席的一封信中，拉福格曾问道[1]：——

[委员会]是希望“将那些其政策已导致我们学校当前灾难的专家们委托以制定未来政策的任务”，还是会有益地决心“与当前所有教育官僚彻底决裂”，并“致力于制定政府可以用来将我们的教育体系从彻底而确定的毁灭中拯救出来的政策建议”？

我们应当以提供一些关于拉福格教育观点的细节来结束这篇传记。我们鼓励读者详细阅读文章[11]，因为它对这些观点作了极好的阐述。我们引用该文章中的一段话，其中他指出了法国学校的问题：——

学生，所有学生，都是学校被摧毁的首要受害者。这种摧毁是过去几十年历届政府教育政策的结果。这不是教师的责任，因为他们本身就是受害者：首先，他们被阻止正确教学，因为颁布的国家课程越来越混乱、不连贯且内容空洞；其次，因为他们的学生多年来积累的知识缺口使教学条件变得越来越困难，并使他们暴露于青少年日益增多的无礼和暴力事件之中，这些青少年从未被教导过基本的知识、工作的习惯或自我控制，而这些对于他们学业的进步是不可或缺的；最后，因为年轻一代的教师已经受到了一个已经退化的教育计划的伤害，以至于他们自己的理解不如他们的前辈确定，并且，除了一些性格温和的人之外，已被教师学院大肆散布的荒谬培训弄得迷失了方向。

最后，引用同一篇文章，让我们看看拉福格认为数学在学校应如何教授：——

让我们以四种运算为例：加法、减法、乘法和除法。近几十年来，它们的教学被大大推迟和忽视，以至于大多数初中生不知道乘法表，许多高中生不会做两个分数的加法。这些运算被忽视是因为计算器的出现，以及一种信念，即机器进行的运算可以与人的精神进行的运算相同。就结果而言，这是一回事——假设一个人计算正确且没有按错键，并且保留一点：同样有许多场合计算器不能替代心算：我最近收到一位祖父的来信，他的孙女在一家市场做了几个小时的售货员后被解雇，因为她不会找零。但最重要的是，一个人编程让计算器执行某些运算，它只知道那些被编程的运算。而学生习得并掌握的同样的运算则成为他精神的养料，赋予他力量，被他消化，成为他自己的东西，扩大并唤醒他的数学能力和力量。对数字的熟悉，同样对几何对象的熟悉，使得赋予他的生命一点一点地进入数学世界。

Laurent Lafforgue was born in Antony, on the southern outskirts of Paris. He entered the primary school Jules Ferry in Antony in 1972, moving to the Lycée Descartes in the same town in 1977. In an interview [4] he spoke of his own schooling:-

I'm very fond of the school I attended since my earliest years. My grandparents began working when they were twelve, but they always respected school very much and passed on this respect to their children and grandchildren.

He received his Baccalauréat (with distinction) in 1984, and received the First Prize in mathematics in the Concours général. By this time he was already achieving international fame for he had won a silver medal at the International Mathematical Olympiad competition in 1984 and he won a second silver medal at the competition in 1985. After spending the two years 1984-86 at the lycée Louis-le-Grand in Paris preparing for his university studies, he entered the École Normale Supérieure in Paris in 1986. He was awarded his Agrégation de Mathématiques in 1988. He then began research in algebraic geometry and the theory of Arakelov under the direction of Christophe Soulé.

Lafforgue became chargé de recherche at the Centre National de la Recherche Scientifique (CNRS) in 1990 and worked in the Arithmetic and Algebraic Geometry team at the Université Paris-Sud at Orsay. During 1991-92 he undertook military service at the École Spéciale Militaire De Saint-cyr, the French national military academy at Coëtquida. After this year he returned to his position at Université Paris-Sud at Orsay where he received his doctorate in 1994 for his dissertation D-stukas de Drinfeld written with Gérard Laumon as his thesis advisor. For this exceptional piece of work he was awarded the Peccot Prize from the Collège de France and was invited to give the Cours Peccot. He continued to work at the Centre National de la Recherche Scientifique and in 1998 he was invited to address the "Groups and Lie algebras" section at the International Congress of Mathematicians in Berlin. In the same year he was awarded the Bronze Medal of the CNRS.

In 2000 Lafforgue was promoted to directeur de recherche of the CNRS working in the Mathematics Department of the Université Paris-Sud. Shortly after this he was named as Professor at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. In fact the year 2000 was significant for Lafforgue in another way too, for on 24 May, at the Paris Millennium Meeting at the Collège de France, he received the 2000 Clay Research Award. At the meeting, Andrew Wiles announced the award to Lafforgue and Lavinia Clay presented him with the Ferguson sculpture. In [9] details of Lafforgue's achievements and their background are given:-

Laurent Lafforgue established the Langlands Correspondences for a much wider class of cases than previously known. These correspondences connect arithmetic properties to analytic properties of some special group representations called automorphic representations. It was formulated by Robert Langlands at the end of the 1960's. In rank 1, this conjecture is nothing other than the now traditional "class field theory" of Emil Artin. In rank 2 and for number fields, the first great confirmations of this conjecture were the proof of the conjecture of Ramanujan by Pierre Deligne and the proof by Langlands himself of the conjecture of Artin except for a case. At the beginning of the seventies, Vladimir Drinfeld attacked the conjectures in a more general algebraic context. For that purpose, he built varieties similar to modular curves and showed certain cases of the conjecture of Langlands in rank 2. Then, as these varieties did not make it possible to reach all desired representations, Drinfeld introduced the "chtoucas", a step which enabled him to prove the conjecture of Langlands in rank 2. This turned out to make the general case accessible, after formidable technical difficulties were surmounted. The crucial contribution by Laurent Lafforgue to solve this question is the construction of compactifications of certain varieties of modules. The proof, which is monumental, is the result of more than six years of concentrated efforts.

Lafforgue soon received further major prizes for his remarkable mathematical achievements. He received the Jacques Herbrand Prize from the Academy of Sciences in 2001 and then, in the following year, he received what is considered the greatest honour for any mathematician, namely a Fields Medal. He received the Medal at the opening ceremonies of the International Congress of Mathematicians in Beijing, China on 20 August 2002. Michael Rapoport writes [6] (or [5]):-

Laurent Lafforgue was awarded the Fields Medal for his proof of the Langlands correspondence for the general linear groups $GL_{r}$ over function fields of positive characteristic. His approach to this problem follows the basic strategy introduced twenty-five years ago by V Drinfeld in his proof for $GL_{2}$. Already Drinfeld's proof is extremely difficult. Lafforgue's proof is a real tour de force, taking up as it does several hundred pages of highly condensed reasoning. By his achievement Lafforgue has proved himself a mathematician of remarkable strength and perseverance.

Allyn Jackson writes [7]:-

Laurent Lafforgue has made an enormous advance in the Langlands Program by proving the global Langlands correspondence for function fields. The Langlands Program, formulated by Robert Langlands in the 1960s, proposes a web of relationships connecting Galois representations and automorphic forms. The influence of the Langlands Program has grown over the years, with each new advance hailed as an important achievement. The roots of the Langlands program are found in one of the deepest results in number theory, the Law of Quadratic Reciprocity, which was first proved by Carl Friedrich Gauss in 1801. This law allows one to describe, for any positive integer d, the primes p for which the congruence $x^{2} = d mod p$ has a solution. Despite many proofs of this law (Gauss himself produced six different proofs), it remains one of the most mysterious facts in number theory. The search for generalizations of the Law of Quadratic Reciprocity stimulated a great deal of research in number theory in the nineteenth century. Landmark work by Emil Artin in the 1920s produced the most general reciprocity law known up to that time. One of the original motivations behind the Langlands Program was to provide a complete understanding of reciprocity laws.

The global Langlands correspondence for $GL_{n}$ proved by Lafforgue provides a complete understanding of reciprocity laws for function fields. Lafforgue established, for any given function field, a precise link between the representations of its Galois groups and the automorphic forms associated with the field. He built on work of 1990 Fields Medalist Vladimir Drinfeld, who in the 1970s proved the global Langlands correspondence for $GL_{2}$.

In 2003 Lafforgue became Chevalier de la Légion d'Honneur and was elected to the Paris Académie des Sciences. Also in 2003 his major text Chirurgie des grassmanniennes (Surgery on Grassmannians) was published by the American Mathematical Society. However, a new chapter in his life was about begin, namely his passionate involvement in the problem of education [4]:-

My specific interest in the topic of education began a few years ago when I signed a petition defending Greek and Latin as academic subjects, as they were in grave danger. Struck by this dramatic situation, denounced by just a handful of teachers, I began inquiring more into it, reading books by people of different ideological orientations, joined by their seriousness about the work, and by their passion for school and the future of young people. This reading shook me profoundly - Latin and Greek are just the tip of the iceberg! In France, even the teaching of the French language itself was at risk. The new French school no longer had anything to do with the one I had known only twenty-five years ago.

On 15 May 2004, Lafforgue gave the address A mathematician and the classics to a conference organised to support the teaching of Latin and Greek in secondary schools. From that time on he became highly involved in educational issues. On Tuesday, 8 November 2005, the Haut Conseil de l'Education (High Committee for Education) was set up, replacing several other educational bodies. Its remit included defining the contents of the knowledge and skills that all children in France will need to have acquired by the age of 16. It also has a remit to define the specifications for the University Institutes for Teachers. Jacques Chirac, President of the French Republic, nominated Laurent Lafforgue to the High Committee for Education which held its first meeting on Thursday, 17 November 2005. The day after Lafforgue was asked to resign from the Committee because he had questioned the need to take advice from the so-called "experts of the Ministry of National Education". In a letter to the President of the High Committee for Education, Lafforgue had asked [1]:-

Does [the Committee] wish to "entrust the same experts whose policies have led to the present disaster of our schools with the task of elaborating the future policies", or will it have the salutary will to "radically break the ties with all the present educrats" and "work [...] on developing policy advice that the government may use to save our educational system from a complete and definite destruction?"

We should end this biography by giving some details of Lafforgue's views on education. We encourage the reader to look at the article [11] in detail for it gives an excellent account of these views. We quote from the article a paragraph in which he identifies the problems with schools in France:-

Students, all the students, are the primary victims of the destruction of the school. This destruction has resulted from educational policies of all the governments of the last few decades. It is not the teachers who are responsible for it, for they are victims themselves: firstly in that they have been prevented from teaching correctly, by the publication of national curricula which are increasingly disorganized, incoherent and emptied of content; then because the knowledge gaps accumulated by their students over the course of years have made the conditions of teaching ever more difficult, and have exposed them to incidents of increasing incivility and violence on the part of adolescents who have never been taught either the elementary understandings, the habits of work, or the self-control which are indispensable to the progress of their studies; and finally because the younger generation of teachers has suffered from an already degraded educational program, so that their own understanding is less certain than that of their elders, and, with the exception of some well tempered characters, has been disoriented by the absurd training so prodigally distributed by the teachers colleges.

Finally, quoting from the same article, let us see how Lafforgue believes mathematics should be taught in schools:-

Let us take the example of the four operations: addition, subtraction, multiplication and division. Their teaching has been considerably retarded and neglected in recent decades, to where the majority of middle school children do not know the multiplication tables and many high school students are unable to add two fractions. These operations have been neglected because of the emergence of calculators, and the belief that an operation carried out by a machine can be the same thing as an operation carried out by a human spirit. It is the same thing as to result - supposing that one has calculated correctly and not made an error of fingering, and with the reservation that there are, just the same, many occasions where the calculator doesn't replace a mental calculation: I recently received a letter from a grandfather whose granddaughter had been fired after several hours as a salesgirl in a market because she was unable to make change. But above all, a calculator which one has programmed to perform certain operations knows only those operations for which it has been programmed. Whereas those same operations acquired and mastered by a student becomes nourishment for his spirit, empowers him, is digested by him, is made his own, enlarges and awakens his mathematical faculties and power. A familiarity with numbers, and similarly as to geometric objects, that permits the life that has been given him to enter, little by little, into the world of mathematics.