文德林·维尔纳Wendelin Werner

文德林·维尔纳出生时父母住在德国，但在他一岁时搬到了法国。他们在法国定居，维尔纳在那里长大。他有一个哥哥Benjamin Werner，1966年6月出生于慕尼黑。Benjamin Werner，一位杰出的计算机科学研究员，当然也是在法国长大的。维尔纳在九岁时取得了法国国籍。他在巴黎西南、靠近凡尔赛的布克法德中学学习。这所以学术卓越闻名的学校，直到1981年迁入自己的校舍之前，一直在凡尔赛的奥什中学办学。离开布克法德中学后，维尔纳前往靠近凡尔赛宫的奥什中学为大学学习做准备。他于1987年考入巴黎高等师范学院，并于1991年毕业，获得数学学位，期间对概率论产生了特别兴趣。他留在巴黎，在皮埃尔和玛丽·居里大学，即巴黎第六大学，师从Jean-Francois Le Gall进行博士研究，后者是他的学位论文导师。

1993年，维尔纳被任命为法国国家科学研究中心（CNRS）的永久研究员，就在他答辩关于平面布朗运动的学位论文并获得博士学位之前。他在CNRS工作到1997年，然后被任命为巴黎南大学11分校的数学教授。他继续在奥赛理学院的数学实验室进行研究，该实验室由巴黎南大学和CNRS联合运营。此外，2005年，他被任命为巴黎高等师范学院的数学教授。

维尔纳卓越的数学贡献使他获得了许多极具声望的奖项。1998年，他在剑桥大学被授予Rollo Davidson奖，该奖由Rollo Davidson信托人每年颁发给概率论领域的早期职业研究者。1999年，维尔纳被授予巴黎科学院的Doisteau-Émile Blutet奖，并荣幸受邀在法兰西学院开设Cours Peccot讲座。2000年7月，他在巴塞罗那举行的欧洲数学大会上获得了欧洲数学会奖。该奖表彰不超过32岁的年轻研究者的杰出数学贡献。2001年，他被图卢兹数学研究所授予皮埃尔·德·费马奖：-

……因其关于布朗运动相交指数及其在理论物理学中影响的工作。

他的工作继续得到更多奖项的认可，2003年，他又获得了巴黎科学院颁发的另一奖项，即他们的雅克·埃尔布朗奖。2005年，他在加利福尼亚大学乔治·伯克利分校获得了Line and Michel Loeve国际概率奖。该奖每两年颁发一次，表彰45岁以下的概率论研究者的杰出贡献。2006年的工业与应用数学学会的乔治·波利亚奖被联合授予维尔纳及其合作者康奈尔大学的Greg Lawler和微软公司的奥代·施拉姆：-

Lawler、施拉姆和维尔纳因在随机查尔斯·娄威纳演化（SLE）的发展与应用方面的开创性工作而获奖。特别值得注意的是，他们严格建立了统计物理中出现的若干二维格点模型临界标度极限的存在性与共形不变性。

同样在2006年，维尔纳获得了数学家所能获得的最负盛名的奖项，即约翰·查尔斯·菲尔兹奖章。授奖词称该奖章授予：-

……以表彰他在随机查尔斯·娄威纳演化的发展、二维布朗运动的几何以及共形场论方面的贡献。

在获得菲尔兹奖后的一次采访中，维尔纳被问及能否用简单的语言解释他所研究的一个问题。以下是他的回答：-

拿一把剪刀，在一张纸上完全随机地剪出一个形状。关于这个形状你能说些什么？问题的一部分是要弄清“完全随机”这一概念的含义，因为存在无穷多种可能性。研究这类问题的一个动机来自物理学：如果你考虑一个物理系统并升高其温度，那么在特定的温度值处，它的宏观行为会发生突然变化：液体变成蒸汽，铁失去其自发磁化等等。经验观察表明，当一个系统恰好处于这样的“临界”温度时，它可以表现出随机的宏观特征。例如，如果该系统是平面的，那么两个相可能共存，而分隔对应于每个相的区域的线便是随机环，正如用剪刀剪出的那些一样。

我们提到的第一个重要奖项是与剑桥大学相关的Rollo Davidson奖。2001年，维尔纳应邀在剑桥大学丘吉尔学院作第二届Rollo Davidson讲座。他作了题为Random planar curves and conformal invariance的讲座，而这次演讲的摘要极好地体现了他当时参与的一个研究项目的特点：-

理解平面上某些自然的极长随机曲线的行为，看似是一个简单的问题，却引出了深刻的问题，其中一些问题至今仍未解决。例如，理论物理学家已经预测（这仍然是一个未解决的问题），在方格$\mathbb{Z} \times \mathbb{Z}$上长度为$N$的自回避曲线的数目$a(N)$渐近地像$C^{N} N^{11/32}$那样增长，其中C是某个常数。更一般地，理论物理学家……利用与数学的若干分支（概率论、复变量、无穷维索菲斯·李代数的表示论）相关的考虑，对统计物理中各种二维系统（如自回避游走、临界渗流、简单随机游走的交点）的临界指数的存在性和取值做出了预测。

2005年4月，维尔纳在加州理工学院做了第四届Thomas Wolff纪念数学讲座。他系列演讲的标题是Two-dimensional continuous random systems，这些演讲的摘要再次很好地反映了他与Greg Lawler和施拉姆合作中产生显著影响的那些主题：-

概率论和统计力学通常关注大型随机系统的某个泛函的行为，其微观随机输入有时会相互作用。在许多情况下，当系统非常大时，输出接近确定性。然而，在“临界情形”下，结果在任何尺度上都可能是随机的。微观随机性于是产生宏观连续随机性，自然地人们试图从数学上理解它。这在一般情况下相当困难，但在某些情况下，由于额外的数学结构而成为可能，并将这些随机结构与数学的其他部分联系起来。一个例子是临界二维粒子系统的标度极限，其中粒子局部相互作用。对这些现象的理解与复分析和表示论有关，正如理论物理学家所预测的那样。在过去几年中，在更好地严格理解这些模型方面已经取得了数学进展。

除了上面提到的那些之外，维尔纳还受邀做了其他几个冠名讲座，例如乌得勒支的马克·卡茨讨论班、麻省理工学院的James and Marylin Simons讲座、巴塞罗那的Levy讲座和斯德哥尔摩的Goran Gustafsson讲座。他在许多国际会议上做了全会报告，包括印度班加罗尔和巴西里约热内卢的会议。

最后，我们引用Charles M 马克斯·纽曼在马德里国际数学家大会上所做的讲座的引言，其中描述了维尔纳的贡献，这些贡献使他获得了该大会颁发的约翰·查尔斯·菲尔兹奖章[3]：-

维尔纳的工作有几个方面增加了我对这次活动的喜悦。其中之一是他接受的是概率论学者的训练，1993年在巴黎在Jean-François Le Gall的指导下获得博士学位，学位论文涉及平面布朗运动……直到现在，概率论在约翰·查尔斯·菲尔兹奖章中一直没有代表，因此我非常高兴能在这里见证这一历史的改变。……维尔纳的工作，连同他的合作者如Greg Lawler、施拉姆和Stas 弗拉迪米尔·斯米尔诺夫，涉及将概率论和共形映射理论应用于统计物理中的基本问题……第二个喜悦之源是我相信，这连同近年来的其他工作，代表了数学与物理学之间一般性互动的一个分水岭。也就是说，像维尔纳这样的数学家不仅为物理学文献中已有的论断提供严格证明，而且更进一步，对基本现象提供了相当新的概念性理解——在这种情况下，是物理系统在其临界点处内在随机结构的直接几何图像（至少在二维情形）。一个简单但重要的例子是渗流……第三个喜悦之源涉及维尔纳大部分工作的合作性质。美丽而富有成果的数学可以源于许多不同的个人工作风格。但那种高度互动的风格，维尔纳连同Lawler、施拉姆和其他合作者是其主要典范，对我们许多人来说既有益于心灵，又能产生比各部分之和更强的工作。看到约翰·查尔斯·菲尔兹奖章授予这种风格的工作，是一个有希望的迹象。

当然，在获得约翰·查尔斯·菲尔兹奖章之后，维尔纳继续获得各种荣誉。例如，2008年他当选为法国科学院院士，同年1月，他担任美国数学会学术报告会主讲人，在圣迭戈会议中心就Random conformally invariant pictures做了一系列讲座。

我们在结束这篇传记时不能不提到，维尔纳 在数学之外有一个也许令人惊讶的生活，作为演员在 1982 年的电影 La Passante du Sans-Souci Ⓣ（《路人》）中扮演了一个角色。他还参与政治问题，于 2009 年 2 月发表了一封致法国总统 Nicolas Sarkozy 的公开信。这封信批评了 Sarkozy 政府的政策，在 维尔纳 看来，这些政策导致了政府与研究人员之间信任的破裂。

Wendelin Werner's parents were living in Germany when he was born but moved to France when he was one year old. They settled in France and Wendelin was brought up there. He has an older brother, Benjamin Werner, who was born in Munich in June 1966. Benjamin Werner, an outstanding researcher in computer science, was, of course, also brought up in France. Wendelin took French nationality at the age of nine. He studied at the Lycée Franco-Allemand of Buc, lying south-west of Paris and close to Versailles. This school, with a reputation for academic excellence, had operated in the Lycée Hoche in Versailles until 1981 when it moved into its own premises. After leaving the Lycée Franco-Allemand, it was to the Lycée Hoche, close to the Palace of Versailles, that Werner went to prepare for his university studies. He matriculated at the École Normale Supérieure in 1987 and graduated with a degree in mathematics in 1991 having become especially interested in probability theory. Remaining in Paris, he undertook research for his doctorate at the Université Pierre-et-Marie-Curie, Université Paris VI, with Jean-Francois Le Gall as his thesis advisor.

In 1993 Werner was appointed as a permanent researcher at the Centre National de la Recherche Scientifique (CNRS) just before he defended his thesis on planar Brownian Motion for which he was awarded his doctorate. He worked at the CNRS until 1997 when he was named as professor of mathematics at the Université Paris-Sud 11. He continued to undertake research at the Laboratoire de mathématiques of the Orsay Faculty of Science, a laboratory jointly run by the Université Paris-Sud and CNRS. In addition, in 2005 he was appointed as Professor of Mathematics at the École Normale Supérieure.

Werner's remarkable mathematical contributions have led to him receiving many highly prestigious prizes. In 1998 he was awarded the Rollo Davidson Prize at the University of Cambridge which is awarded annually to early-career researchers in probability by the Rollo Davidson trustees. In 1999 Werner was awarded the Doisteau-Émile Blutet prize from the Academy of Sciences in Paris and was honoured with an invitation to give the Cours Peccot at the Collège de France. In July 2000 he received the European Mathematical Society prize at the European Congress of Mathematics in Barcelona. This prize recognizes excellent mathematical contributions by young researchers not older than 32 years. He was awarded the Fermat prize in 2001 by the Institute of Mathematics of Toulouse:-

... for his works on the intersection exponents of Brownian motion and their impact in theoretical physics.

His work continued to be recognised by further awards and, in 2003, he received another award from the Academy of Sciences in Paris, namely their Jacques Herbrand prize. He received the Line and Michel Loeve International Prize in Probability at the University of California, Berkeley, in 2005. This prize, awarded every second year, recognises outstanding contributions by researchers in probability who are under 45 years old. The Society for Industrial and Applied Mathematics' George Polya Prize for 2006 was awarded jointly to Werner and his collaborators Greg Lawler of Cornell University and Oded Schramm of the Microsoft Corporation:-

Lawler, Schramm and Werner received the prize for their groundbreaking work on the development and application of stochastic Loewner evolution (SLE). Of particular note is the rigorous establishment of the existence and conformal invariance of critical scaling limits of a number of 2D lattice models arising in statistical physics.

Also in 2006 Werner received the most prestigious award that a mathematician can receive, namely a Fields Medal. The citation states that the Medal was awarded:-

... for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.

In an interview after being awarded the Fields Medal, Werner was asked if he could explain, in simple terms, one of the problems he worked on. Here is his reply:-

Take scissors and cut completely at random a shape in a piece of paper. What can you say about this shape? Part of the question is to make sense of notion "completely at random" because there are infinitely many possibilities. One motivation to study this type of question comes from physics: If you consider a physical system and raise its temperature, then at certain values of the temperature, there occurs sudden change of its macroscopic behaviour: Liquid becomes vapour, iron looses its spontaneous magnetisation etc. It has been observed empirically that when a system is exactly at such a "critical" temperature, it can exhibit random macroscopic features. For instance, if the system is planar, then the two phases may coexist and the lines separating the regions corresponding to each of the phases are then random loops, just as those cut out by scissors.

The first major prize we mentioned was the Rollo Davidson prize which is associated with the University of Cambridge. In 2001 Werner was invited to deliver the Second Rollo Davidson Lecture in Churchill College, Cambridge. He gave the lecture Random planar curves and conformal invariance and his abstract for this talk gives an excellent feel for one of the research projects he was involved in at this time:-

Understanding the behaviour of certain natural very long random curves in the plane is a seemingly simple question that has turned out to raise deep questions, some of which remain unsolved. For instance, theoretical physicists have predicted (and this is still an open problem) that the number $a(N)$ of self-avoiding curves of length $N$ on the square lattice $\mathbb{Z} \times \mathbb{Z}$ grows asymptotically like $C^{N} N^{11/32}$ for some constant C. More generally, theoretical physicists ... have made predictions concerning the existence and values of critical exponents for various two-dimensional systems in statistical physics (such as self-avoiding walks, critical percolation, intersections of simple random walk) using considerations related to several branches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras).

In April 2005 Werner delivered the 4th Thomas Wolff Memorial Lectures in Mathematics at the California Institute of Technology. The title for his series of talks was Two-dimensional continuous random systems and again the abstract for these talks gives good insight into topics on which he was making a remarkable impact in collaboration with Greg Lawler and Oded Schramm:-

Probability theory and statistical mechanics often focus on the behaviour of a functional of a large random system, with microscopic random inputs that sometimes interact with each other. In many cases, the output is close to being deterministic when the system is very large. In "critical cases," the outcome can however be random at any scale. The microscopic randomness then gives rise to a macroscopic continuous randomness and it is natural to try to understand it mathematically. This turns out to be rather difficult in general, but in some cases, it is possible thanks to an additional mathematical structure, and relates these random structures to other parts of mathematics. An example is given by the scaling limit of critical two-dimensional particle systems, where particles interact locally. The understanding of these phenomena is related to complex analysis and to representation theory, as predicted by theoretical physicists. Mathematical progress has been made towards a better rigorous understanding of these models during the last years.

Werner has been invited to give several other named lectures, in addition to those mentioned above, such as the Mark Kac seminar in Utrecht, the James and Marylin Simons lectures at the Massachusetts Institute of Technology, the Levy lecture in Barcelona, and the Goran Gustafsson lectures in Stockholm. He has given plenary lectures at many international conferences including ones in Bangalore, India, and Rio de Janeiro, Brazil.

Finally we quote from the introduction to the lecture Charles M Newman gave at the International Congress of Mathematicians in Madrid describing Werner's contributions which had led to him receiving the Fields Medal at this Congress [3]:-

There are a number of aspects of Werner's work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion ... Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history. ... Werner's work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Mapping Theory to fundamental issues in Statistical Physics ... A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are providing quite new conceptual understanding of basic phenomena - in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation ... Yet a third source of pleasure concerns the collaborative nature of much of Werner's work. Beautiful and productive mathematics can be the result of many different personal work styles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work.

Of course Werner has continued to receive honours following the award of the Fields Medal. For example, in 2008 he was elected to the French Academy of Sciences and, in January of the same year, he was the American Mathematical Society Colloquium Lecturer, giving his series of lectures on Random conformally invariant pictures at San Diego Convention Center.

We must not end this biography without mentioning that Werner has had a, perhaps surprising, life outside mathematics as an actor playing a role in the 1982 film La Passante du Sans-Souci Ⓣ. He has also involved himself with political issues, publishing an open letter to President Nicolas Sarkozy of France in February 2009. The letter criticises policies of Sarkozy's government which have, in Werner's opinion, led to a breakdown in trust between government and researchers.