柯蒂斯·麦克马伦Curtis T. McMullen

柯蒂斯·麦克马伦（被称为Curt）在成长过程中在美国各地搬过几次家，但他基本上是在佛蒙特州夏洛特长大的。他最初在俄亥俄州哥伦布附近上阿灵顿的温德米尔小学就读，然后在佛蒙特州夏洛特的夏洛特中央学校就读，之后在佛蒙特州海恩斯堡的尚普兰谷联合高中学习，距离夏洛特仅约12公里。他于1976年进入马萨诸塞州西部威廉斯敦的威廉姆斯学院。他说，在威廉姆斯学院读本科期间[6]：-

……我去斯坦福呆了一年，从来自耶路撒冷的访问教授Benjamin Weiss那里上了一门很棒的分析课程。这真的让我对分析感到兴奋。然后我回到威廉姆斯，与Bill Oliver密切合作。他在我的数学教育中非常有影响力；正是从他那里，我第一次学到了在数学中使用字典的想法，将其作为不同领域或不同理论发展之间的一种类比，以试图指导我的工作。

他于1980年从威廉姆斯学院毕业，获得学士学位，作为告别演说者，以最优等成绩毕业，获得数学最高荣誉，并额外专注于物理学。他获得了Herchel Smith奖学金，前往英国剑桥大学学习，并在剑桥伊曼纽尔学院度过了1980-81学年。他在数学荣誉学位考试第二部分中名列第一。然后他回到美国，于1981年在哈佛大学开始研究。他在[6]中解释了他是如何设法从哈佛获得博士学位的，丹尼斯·苏利文是他的论文导师，尽管丹尼斯·苏利文并不在哈佛工作：-

毕业前，我曾用计算机与戴维·芒福德一起研究克莱因群，并对这个课题产生了兴趣。但实际上，我最终是与丹尼斯·苏利文一起完成了我的学位论文，他当时是纽约城市大学和法国高等科学研究所的教授。所以我很幸运，在研究生生涯的最后一年，由戴维·芒福德把我介绍给了他，那时我既没有导师也没有论文题目。我去了法国，在高等科学研究所与丹尼斯·苏利文一起工作了一个学期，并在那里遇到了斯蒂芬·斯梅尔，他给了我一个很好的论文问题：用迭代法解多项式方程。

在1980年至1985年的每个夏天，麦克马伦都在纽约约克敦高地的IBM沃森研究中心进行组合研究和计算机编程。他在那里的工作涉及VLSI设计问题：图论、逻辑综合、布尔最小化和稀疏矩阵处理。在约克敦高地[6]：-

……曼德尔布罗和戴维·芒福德几乎是在合作；曼德尔布罗为戴维·芒福德提供了约克敦高地的计算机使用权限，后者正在绘制克莱因群极限集的这些美丽图片。作为一个熟悉约克敦计算机世界的人，我开始为他工作，担任他的计算机程序员，帮助他绘制这些图片等等。

正如他在上述引文中所提到的，麦克马伦在1984年秋天作为访问学者在巴黎附近的比雷-伊维特的高等科学研究所度过。他的第一篇论文The Hausdorff dimension of general Sierpinski carpets于1984年发表，早于他提交学位论文。在1984-85学年，他获得了阿尔弗雷德·P·斯隆博士论文奖学金。他向哈佛大学提交了学位论文Families of Rational Maps and Iterative Root-Finding Algorithms，并于1985年6月被授予博士学位。

麦克马伦在其学位论文中考虑的问题长期以来一直很重要。一旦人们正确理解了哪些多项式方程可以用根式求解，对于那些不存在公式的多项式方程，就仍然存在通过迭代程序求根的问题。艾萨克·牛顿曾给出这样一种方法，他的迭代程序对所有二次多项式和初始点一般都收敛，但对三次多项式方程则并非如此。斯蒂芬·斯梅尔曾问，对每个$n$，是否存在一种对所有初始点一般都收敛的迭代程序。麦克马伦在其学位论文中为三次多项式给出了这样一种程序，但证明了对于次数大于三的情形，不存在这样的迭代程序。

完成博士学位后，麦克马伦于1985-86学年在麻省理工学院度过，此前被任命为C L E Moore数学讲师。他提交了更多论文供发表，Area and Hausdorff dimension of Julia sets of entire functions于1987年发表。一篇与其学位论文同名的论文也在1987年发表。他于1986-87年作为普林斯顿高等爱德华·斯图迪的成员度过，随后于1987年被任命为普林斯顿大学助理教授，并获得NSF博士后奖学金。该奖学金持续到1990年，但他还在1988年获得Alfred P Sloan奖学金，并于1988年至1993年担任总统青年研究员。1990年，他被提升为普林斯顿大学正教授，但在同一年，接受了加州大学乔治·伯克利分校的正教授职位。

由拉斐尔·塞勒姆的遗孀创立的拉斐尔·塞勒姆奖，每年颁发给一位年轻数学家，该数学家被认为在拉斐尔·塞勒姆感兴趣的领域即分析中做出了杰出工作。麦克马伦解释说[6]：-

……1991年，我获得了拉斐尔·塞勒姆奖，这是一个分析领域的奖项；我很高兴能以这种方式得到认可，因为我真的很喜欢这个领域——作为数学家，这是我的第一个领域。事实上，我在研究生时写的副论文是关于拉斐尔·塞勒姆数的，而这个奖项是为了纪念拉斐尔·塞勒姆，所以它对我有个人意义。我从未期望过能得到那种认可……

1994年，他被任命为伯克利的米勒教授，并在1996至1998年间担任伯克利的校长讲席教授。1998年，他离开伯克利，被任命为哈佛大学教授。2001年，他被任命为哈佛大学的玛丽亚·莫尔斯·卡伯特教授，至今仍担任此职位。从2001年到2007年的每个夏天，他都在波恩的马克斯·普朗克数学研究所担任访问职位。

让我们提及麦克马伦撰写的两部专著。Complex dynamics and renormalization（1994年）由Gregery T Buzzard评论，他的评论开头如下：-

这本书非常清晰易读地呈现了单变量复动力系统研究中使用的许多主要思想和技术。

在详细介绍了每一章的内容之后，他这样结束他的评论：-

这本书非常清晰地呈现了大量的思想，必将成为复动力系统文献中的宝贵补充。

1996年，麦克马伦出版了Renormalization and 3-manifolds which fiber over the circle ，该书由Athanase Papadopoulos评论：-

在这部专著中，作者对一种理论进行了全面的研究，该理论将两个近期且非常深刻的定理并行起来，涉及几何与动力学。这两个定理分别是威廉·瑟斯顿关于存在双曲度量的定理，该度量在具有伪德米特里·阿诺索夫单值性的圆上纤维化的三维流形上，以及丹尼斯·苏利文关于实二次映射的重整化映射收敛性的定理。这里给出的证明是新的，并且使用了与威廉·瑟斯顿和丹尼斯·苏利文各自原始证明不同的论证。这两个定理之间的关系很难用简洁的形式陈述，但至少可以说的一点是，这两个定理都可以被视为不动点定理。

1998年，麦克马伦在柏林国际数学家大会上获得了约翰·查尔斯·菲尔兹奖。除了我们上面描述的他学位论文中的工作外，引文还提到了他在曼德尔布罗集上的工作[2]：-

这个集描述了可用于模拟复杂自然现象（如天气或流体流动）的动态系统。关注点在于系统在哪里发散以及哪些点向平衡中心移动。这两个极端之间的边界就是所谓的茹利亚集，以法国数学家茹利亚命名，他在二十世纪初为动态系统理论奠定了基础。曼德尔布罗集显示了使茹利亚集连通的参数，即在数学上具有吸引力的参数。这种描述非常粗略，但当时还没有更好的边界集特征描述。然而，麦克马伦 T 麦克马伦取得了重大进展，他表明可以部分基于曼德尔布罗集来决定哪个相关的动态系统是“双曲的”，因此可以更详细地描述。对于这些系统，有一套完善的理论可用。麦克马伦的结果在六十年代就已被怀疑，但此前没有人能够证明对茹利亚集的这种精确刻画。

在[6]中，他讲述了一个关于菲尔兹奖的精彩故事：-

我有一个关于我从柏林回来的故事。机场安检员操作金属探测器时，我的背包通过机器时她拦住了我。她说：“打扰一下，你背包里有什么？”我说：“是一枚金牌。”她有点怀疑地说：“嗯哼。”于是我从包里拿出来。她有点懊恼地说：“哦，非常好；这是你的吗？”我说：“嗯哼！”

论文[1]列出了麦克马伦贡献的领域：-

他对动力系统理论的各个分支做出了重要贡献，例如多项式方程的算法研究、索菲斯·李群的格点分布研究、双曲几何、全纯动力学以及区间映射的重整化。

除了1991年的拉斐尔·塞勒姆奖和1998年获得的约翰·查尔斯·菲尔兹奖章外，麦克马伦还因其杰出贡献获得了大量其他荣誉。其中包括：1998年当选为美国艺术与科学院会士，1999年被威廉姆斯学院授予荣誉理学博士学位，2007年当选为国家科学院成员。他还受邀做过许多特别讲座，例如（自2000年以来）：罗尔夫·内万林纳学术报告会，赫尔辛基（2000年）；American Mathematical Society学术报告系列讲座，华盛顿特区（2000年）；杰出讲座系列，布朗大学（2001年）；American Mathematical Society罗斯讲座，波士顿（2002年）；Namboodiri讲座，芝加哥大学（2003年）；Alaoglu讲座，加州理工学院，帕萨迪纳（2003年）；数学工作会议，马克斯·普朗克研究所，波恩（2003年）；Bowen讲座，加州大学，伯克利（2004年）；艾力斯·科尔钦讲座，哥伦比亚大学，纽约（2005年）；海因茨·霍普夫讲座，苏黎世联邦理工学院（2005年）；数学工作会议，马克斯·普朗克研究所，波恩（2007年）；Ziwet讲座，密歇根大学，安娜堡（2008年）以及Nielsen讲座，CTQM，丹麦奥胡斯（2008年）。

尽管麦克马伦对最新数学研究做出了惊人的贡献，但他还是为学童们做了精彩的讲座，例如他在2002年关于From Triangles to Infinity的弗拉基米尔·阿诺尔德罗斯讲座。以下是讲座的报道：-

麦克马伦通过向听众提问来引入话题：如果一只狮子和一个人都在一个封闭的环形区域内，狮子应该走什么路径才能抓住人。稍后在讲座中，他要求听众中的学生用相互咬合的三角形组装多面体，条件是每个顶点必须有固定数量的三角形相交。正如讲座标题所示，麦克马伦涉及了许多不同的数学领域，包括：皮埃尔·德·费马最后定理、芝诺悖论、双曲几何和球面几何、调和级数以及密铺。在讲座接近尾声时，麦克马伦展示了一条人可以躲避狮子的路径，并利用无穷级数的结果来证明该路径的有效性。挤满波士顿科学博物馆礼堂的师生们充分享受了讲座的主题和讲授方式。许多学生在讲座结束后找到麦克马伦提问，有些甚至索要签名。

最近，他于2008年2月在波士顿举行的美国科学促进会年会上就The Geometry of 3-Manifolds做了一场讲座：-

在这场专题讲座中，麦克马伦对儒勒·昂利·庞加莱猜想做了精彩的描述……麦克马伦解释了该猜想中的许多拓扑学思想，它如何与宇宙的形状相关联，并概述了格里戈里·佩雷尔曼对该猜想的证明。

让我们以斯蒂芬·斯梅尔在[10]中的话结束这篇传记：-

我……想强调，[麦克马伦的]工作涵盖了我们丰富文化中许多路径交叉处的那类数学的广阔领域。麦克马伦不是动力系统学家，不是分析学家，也不是几何学家。他是一位数学家。

Curtis McMullen (known as Curt) moved around the United States a bit as he was growing up, but he basically grew up in Charlotte, Vermont. He first attended Windermere Elementary School in Upper Arlington, near Columbus, Ohio, then Charlotte Central School in Charlotte, Vermont, before studying at Champlain Valley Union High School in Hinesburg, Vermont, only about 12 km from Charlotte. He entered Williams College, Williamstown, Western Massachusetts, in 1976. He said that during his undergraduate years at Williams College [6]:-

... I went to Stanford for a year and took a great real analysis course from Benjamin Weiss who was a visiting professor from Jerusalem. And that really got me excited about analysis. Then I went back to Williams and I worked closely with Bill Oliver. He was very influential in my mathematical education; it was from him that I first learned this idea of using dictionaries in mathematics to use as a sort of analogy between different fields or different theoretical developments to try to guide my work.

He graduated from Williams College with a B.A. in 1980 as a Valedictorian, Summa Cum Laude, with Highest Honours in Mathematics and an additional concentration in Physics. He was awarded a Herchel Smith fellowship to study at the University of Cambridge in England and he spent the academic year 1980-81 at Emmanuel College, Cambridge. He was placed first in Part II of the Mathematical Tripos. He then returned to the United States to begin research in 1981 at Harvard University. He explains in [6] how he managed to obtain a Ph.D. from Harvard with Dennis Sullivan as his thesis advisor, although Sullivan did not work at Harvard:-

I had been doing some computer work with David Mumford on Kleinian groups before I graduated, and I got interested in that subject. But I actually ended up writing my thesis with Dennis Sullivan, who at that time was a professor at City University in New York and the Institut des Hautes Études Scientiques at France. So I was very lucky that David Mumford introduced me to him in the last year of my graduate career, at which point I had no advisor and no thesis topic. And I went to France and worked with Sullivan at Institut des Hautes Études Scientiques for a semester, and I met Steve Smale there who gave me this nice thesis problem on solving polynomial equations by iteration.

During each of the summers from 1980 to 1985 McMullen undertook combinatorial research and computer programming at the IBM Watson Research Center, Yorktown Heights, New York. His work there involved VLSI design problems: graph theory, logic synthesis, boolean minimization, and sparse matrix processing. At Yorktown Heights [6]:-

... Mandelbrot and Mumford were almost collaborating; Mandelbrot was furnishing access to computers at Yorktown Heights to Mumford, who was drawing these beautiful pictures of limit sets of Kleinian groups. As somebody who was conversant with the computer world at Yorktown, I started working for him as his computer programmer, helping him draw these pictures and so forth.

As he mentions in the above quote, McMullen spent the autumn of 1984 as a visitor at the Institut des Hautes Études Scientiques, Bures-sur-Yvette near Paris. His first paper The Hausdorff dimension of general Sierpinski carpets was published in 1984 before he submitted his thesis. In the academic year 1984-85 he held an Alfred P Sloan Doctoral Dissertation Fellowship. He submitted his thesis Families of Rational Maps and Iterative Root-Finding Algorithms to Harvard University and was awarded his doctorate in June 1985.

The problems which McMullen considered in his thesis had long been important. Once a proper understanding was achieved of which polynomial equations could be solved by radicals, there remained the problem of finding the roots of a polynomial equation by an iterative procedure for those for which no formula existed. Newton had produced such a method and his iterative procedure generally converged for all quadratic polynomials and initial points, but this was not the case for polynomial equations of degree three. Stephen Smale had asked whether, for each $n$, there existed an iterative procedure which generally converged for all initial points. In his thesis McMullen produced such a procedure for polynomials of degree three, but showed that for degree greater than three no such iterative procedure existed.

After completing his doctorate, McMullen spent the academic year 1985-86 at the Massachusetts Institute of Technology having been appointed a C L E Moore Instructor in Mathematics. He submitted further papers for publication and Area and Hausdorff dimension of Julia sets of entire functions was published in 1987. A paper with the same title as his thesis also appeared in 1987. He spent the year 1986-87 as a member of the Institute for Advanced Study at Princeton, then in 1987 he was appointed as an Assistant Professor at Princeton University and awarded a NSF Postdoctoral Fellowship. This fellowship ran until 1990 but he also held an Alfred P Sloan Fellowship in 1988 and was a Presidental Young Investigator from 1988 to 1993. In 1990 he was promoted to full professor at Princeton but, in the same year, accepted a full professorship at the University of California, Berkeley.

The Salem Prize, founded by the widow of Raphael Salem, is awarded every year to a young mathematician judged to have done outstanding work in Salem's field of interest, namely analysis. McMullen explains that [6]:-

... in 1991, I won the Salem Prize, which is a prize in Analysis; I was pleased to be recognized that way because I really love the field - it was my first, as a mathematician. In fact, I had written my minor thesis as a graduate student on Salem numbers, and this prize is in honour of Raphael Salem, so it has personal meaning for me. I had never expected to get any recognition of that kind ...

In 1994 he was named Miller Professor at Berkeley and held a Chancellor's Professorship at Berkeley during 1996-1998. He left Berkeley in 1998 when appointed to a professorship at Harvard University. In 2001 he was named Maria Moors Cabot Professor at Harvard, a position he continues to hold. In each of the summers from 2001 to 2007 he held a visiting position at the Max-Planck-Institute für Mathematik in Bonn.

Let us mention two monographs written by McMullen. Complex dynamics and renormalization (1994) was reviewed by Gregery T Buzzard who begins his review as follows:-

This book provides a very clear and readable presentation of many of the main ideas and techniques used in the study of complex dynamics in one variable.

After giving details of the contents of each chapter, he ends his review:-

This book presents a great many ideas very clearly and should prove to be a valuable addition to the complex dynamics literature.

In 1996 McMullen published Renormalization and 3-manifolds which fiber over the circle which was reviewed by Athanase Papadopoulos:-

In this monograph, the author presents a comprehensive study of a theory which brings into parallel two recent and very deep theorems, involving geometry and dynamics. These are Thurston's theorem on the existence of hyperbolic metrics on three-manifolds which fiber over the circle with pseudo-Anosov monodromy, and Sullivan's theorem on the convergence of the renormalization map for real quadratic mappings. The proofs that are given here are new and use different arguments than Thurston's and respectively Sullivan's original proofs. The relation between the two theorems is difficult to state in a concise form, but one thing that can be said at least is that both theorems can be thought of as fixed-point theorems.

In 1998 McMullen received a Fields Medal at the International Congress of Mathematicians in Berlin. In addition to the work in his thesis which we have described above, the citation mentions his work on the Mandelbrot set [2]:-

This set describes dynamic systems which can be used to model complicated natural phenomena such as weather or fluid flow. The point of interest is where a system drifts apart and which points move towards centres of equilibrium. The border between these two extremes is the so-called Julia set, named after the French mathematician Gaston Julia, who laid the foundations for the theory of dynamic systems early in the twentieth century. The Mandelbrot set shows the parameters for which the Julia set is connected, i.e. is mathematically attractive. This description is very crude, but a better characteristic of the boundary set was not available. Curtis T McMullen made a major advance, however, when he showed that it is possible to decide in part on the basis of the Mandelbrot set which associated dynamic system is "hyperbolic" and can therefore be described in more detail. For these systems a well-developed theory is available. McMullen's results were suspected already in the sixties, but nobody had previously been able to prove this exact characterization of the Julia set.

In [6] he tells a nice story about the Fields Medal:-

I have a story about when I was coming back from Berlin. The security guard in the airport running the metal detector stopped me when my backpack went through the machine. She said, "Excuse me, what do you have in your backpack here?" I said, "It's a gold medal." She said, a little dubiously, "Mmm hmm." So I took it out of my pack. A little chagrined, she said "Oh, very nice; is it yours?" I said "Mmm hmm!"

The paper [1] list the areas of McMullen's contributions:-

He has made important contributions to various branches of the theory of dynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of a Lie group, hyperbolic geometry, holomorphic dynamics and the renormalization of maps of the interval.

In addition to the Salem Prize in 1991 and the Fields Medal he was awarded in 1998, McMullen had received a large number of other distinctions for his outstanding contributions. Among these we note that he was elected a Fellow of the American Academy of Arts and Sciences in 1998, was given an Honorary Doctor of Science Degree by Williams College in 1999, and was elected a Member of the National Academy of Sciences in 2007. He has also been invited to give special lectures such as (since 2000): Nevanlinna Colloquium, Helsinki (2000); American Mathematical Society Colloquium Lectures, Washington DC (2000); Distinguished Lecture Series, Brown University (2001); American Mathematical Society Ross Lecture, Boston (2002); Namboodiri Lectures, University of Chicago (2003); Alaoglu Lecture, California Institute of Technology, Pasadena (2003); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2003); Bowen Lectures, University of California, Berkeley (2004); Kolchin Lecture, Columbia University, New York (2005); Hopf Lectures, Eidgenossische Technische Hochschule, Zürich (2005); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2007); Ziwet Lectures, University of Michigan, Ann Arbor (2008) and Nielsen Lecture, CTQM, Aarhus, Denmark (2008).

Despite his stunning contributions to the latest mathematical research, McMullen has given excellent lectures to school children as for example the Arnold Ross Lecture he gave in 2002 on From Triangles to Infinity. Here is a report of the lecture:-

McMullen motivated the talk by asking the audience what path a lion should take to capture a human, if both are in an enclosed ring. A little later in the talk, he asked students in the audience to assemble polyhedra using interlocking triangles, given the constraint that a fixed number of triangles have to meet at each vertex. As the title of the talk suggests, there were many different areas of mathematics touched on by McMullen, including: Fermat's Last Theorem, Zeno's Paradoxes, hyperbolic and spherical geometry, the harmonic series, and tiling. Near the end of his talk, McMullen showed a path that a human could take to elude the lion and used results about infinite series to demonstrate the path's effectiveness. The teachers and students who filled the Boston Museum of Science auditorium thoroughly enjoyed the subject of the talk and the manner in which it was delivered. Many students sought out McMullen after his talk to ask questions, and some even asked for his autograph.

More recently he gave a lecture The Geometry of 3-Manifolds to the annual meeting of the American Association for the Advancement of Science in Boston in February 2008:-

In this topical lecture, McMullen gave a wonderful description of the Poincaré conjecture ... McMullen explained many topological ideas in the conjecture, how it relates to the shape of the universe, and gave an overview of Grigori Perelman's proof of the conjecture.

Let us end this biography by quoting Stephen Smale's words from [10]:-

I ... would like to emphasise that [McMullen's] work has encompassed a large realm of the kind of mathematics that lies at the cross-section of many paths of our rich culture. McMullen is not a dynamicist, not an analyst nor a geometer. He is a mathematician.