Linguista

Peter Lax访谈（2005阿贝尔奖）

彼得·D·拉克斯是2005年挪威科学与文学院阿贝尔奖的获得者。 2005年5月24日，在奥斯陆举行的阿贝尔奖庆祝活动之前，拉克斯接受了奥尔堡大学的马丁·劳森和挪威科技大学的克里斯蒂安·斯考的采访。 本次采访最初发表于2005年9月的《欧洲数学会通讯》第24-31页。

劳森与斯考： 我们谨代表挪威和丹麦数学会，祝贺您荣获2005年阿贝尔奖。 您于1941年，年仅15岁时从匈牙利来到美国。仅仅三年后，即1944年，您应征入伍。您没有被派往海外前线，而是在1945年被派往洛斯阿拉莫斯参与曼哈顿计划，建造第一颗原子弹。作为一个年轻人，来到洛斯阿拉莫斯参与如此重大的事业，并遇到那么多传奇的著名科学家：费米、贝特、西拉德、维格纳、泰勒、费曼等物理学家，以及冯·诺伊曼和乌拉姆等数学家，这一定是一段令人敬畏的经历。这段经历如何塑造了您对数学的看法，并影响了您在数学领域研究方向的选择？

拉克斯： 事实上，我在1949年获得博士学位后，回到洛斯阿拉莫斯待了一年，之后又作为顾问在那里度过了许多夏天。我第一次在洛斯阿拉莫斯度过的时光，尤其是后来的接触，塑造了我的数学思维。首先，这是作为科学团队一员的经历——不仅仅是数学家，还有不同观点的人——目标不是一个定理，而是一个产品。这是书本上学不到的，必须亲身参与，因此我敦促我的学生至少在洛斯阿拉莫斯做一暑期访问学者。洛斯阿拉莫斯有一个非常活跃的访问学者项目。其次，正是在那里——那是在20世纪50年代——我开始深刻认识到计算对于科学和数学的极端重要性。在冯·诺伊曼的影响下，洛斯阿拉莫斯在20世纪50年代和60年代初一度是计算科学领域无可争议的领导者。

研究贡献

劳森与斯考： 我们可以稍后再谈计算机吗？首先是一些关于您在数学领域主要研究贡献的问题：您在非线性偏微分方程理论方面做出了杰出贡献。对于双曲守恒律系统的理论和数值解，您的贡献是决定性的，更不用说您对不连续性（即所谓的激波）传播的理解所做的贡献了。您能用几句话描述一下您是如何克服数学这一领域巨大障碍和困难的吗？

拉克斯： 嗯，当我开始研究它时，我深受两篇论文的影响。一篇是埃伯哈德·霍普夫关于伯格斯方程粘性极限的论文，另一篇是冯·诺伊曼-里奇特迈耶关于人工粘性的论文。通过研究这些例子，我能够看出一般理论可能是什么样子。

劳森与斯考： 20世纪60年代，克鲁斯卡尔和扎布斯基关于孤子在科特韦赫-德弗里斯（KdV）方程解中作用的惊人发现，以及随后几位人士同样惊人的解释——即KdV方程是完全可积的——代表了非线性偏微分方程理论的一次革命性发展。您以一种巧妙的原创视角进入了这个领域，引入了所谓的拉克斯对，它解释了反散射变换如何应用于像KdV这样的方程，以及应用于数学物理中其他核心的非线性方程，例如西涅-戈尔登方程和非线性薛定谔方程。您能谈谈您认为这个理论对于数学物理和应用有多重要，以及您如何看待这个领域的未来吗？

拉克斯： 或许我应该首先指出，孤子相互作用这一惊人现象是通过数值计算发现的，正如冯·诺伊曼多年前所预测的那样，即计算将揭示非常有趣的现象。由于我是克鲁斯卡尔的好朋友，我很早就了解了他的发现，这引发了我的思考。很明显，存在无限多个守恒量，所以我问自己：你如何能一次性生成无限多个守恒量？我想，如果你有一个变换能够保持算子的谱，那么那就会是这样一种变换，结果证明这是一个非常有成果的想法，适用范围相当广泛。

现在你问它有多重要？我认为它非常重要。毕竟，从信号传输技术的角度来看，通过孤子进行信号传输非常重要，并且是跨洋传输领域一项有前途的未来技术。这是由贝尔实验室的杰出工程师林恩·莫勒诺尔（Linn Mollenauer）开发的。它尚未投入实践，但总有一天会。有趣的是，经典信号理论完全是线性的，而孤子信号传输的主要特点是方程是非线性的。这是其实际重要性的一个方面。

至于理论上的重要性：KdV方程是完全可积的，随后又发现了数量惊人的其他完全可积系统。完全可积系统可以真正意义上被“解决”，就像普通大众使用“解决”这个词一样。当一位数学家说他解决了问题时，他的意思是他知道解的存在性、唯一性，但通常仅此而已。

现在的问题是：完全可积系统是非可积系统解行为的例外，还是其他系统也有类似行为，只是我们无法分析？在这里，我们的指南很可能是柯尔莫戈洛夫-阿诺德-莫泽定理，该定理指出，接近完全可积系统的系统其行为就像完全可积系统一样。当然，“接近”在证明定理时是一回事，在做实验时又是另一回事。这是数值实验揭示事物的另一个方面。所以我确实认为，研究完全可积系统也将为更一般系统的行为提供线索。

谁能在1965年猜到完全可积系统会变得如此重要呢？

劳森与斯考： 下一个问题是关于您1957年发表的开创性论文《振荡初值问题的渐近解》。许多人认为这篇论文是傅里叶积分算子的起源。论文中哪个新观点被证明如此富有成果？

拉克斯： 这是一种微局部分析的描述。它结合了宏观和微观上看待问题的方法。它结合了这两个方面，这赋予了它力量。微局部分析观点的数值实现是通过小波和类似的方法，这些方法在数值上非常强大。

劳森与斯考： 我们可以谈谈您与拉尔夫·菲利普斯长达三十多年的断断续续的合作吗？——关于散射理论，并将其应用于多个领域。您能评论一下这次合作，以及您认为最重要的成果是什么吗？

拉克斯： 那是我一生中最快乐的事情之一！拉尔夫·菲利普斯是我们这个时代最伟大的分析学家之一，我们建立了非常亲密的友谊。我们对散射过程有了一种新的看法，即传入和传出子空间。可以说，我们从酉群中“雕刻”出一个半群，其无穷小生成元几乎包含了关于散射过程的所有信息。因此，我们将其应用于声波和电磁波被势和障碍物经典散射的问题。继法捷耶夫和帕夫洛夫的一个非常有趣的发现之后，我们研究了自守函数的谱理论。我们进一步阐述了它，并且有了一种全新的处理爱森斯坦级数的方法，例如，通过平移表示获得谱表示。我们甚至能够——效仿法捷耶夫和帕夫洛夫——思考黎曼猜想初露端倪的情形。

劳森与斯考： 那一定很令人兴奋！

拉克斯： 是的！这种方法是否会导致黎曼猜想的证明，即像人们所说的那样，纯粹通过剔除所有驻波来用衰减信号来表述它，这是不太可能的。黎曼猜想是一个非常难以捉摸的东西。你可能还记得在《皮尔·金特》中有一个神秘的角色，博伊格（Boyg），他无论皮尔·金特走到哪里都会挡住他的去路。黎曼猜想就像博伊格！

劳森与斯考： 您今天对哪些特定领域或问题最感兴趣？

拉克斯： 我对零色散极限有一些想法。

纯粹数学与应用数学

劳森与斯考： 我们可以向您提出一个或许有争议的问题吗：纯粹数学与应用数学。在数学界，偶尔会听到这样的说法：非线性偏微分方程理论虽然深刻且在应用中通常非常重要，但却充斥着丑陋的定理和蹩脚的论证。另一方面，在纯粹数学中，美和审美占据主导地位。英国数学家G.H.哈代是这种态度的一个极端例子，但今天仍然可以遇到这种情况。您对此有何回应？这会让您生气吗？

拉克斯： 我不太容易生气。我曾经对我们的一位院长生过气，他是个可怕的混蛋，破坏性的骗子。我还对占领柯朗研究所并试图烧毁我们计算机的暴民非常生气。科学上的分歧不会激怒我。但我认为这种观点绝对是错误的。我认为保罗·哈尔莫斯曾经声称应用数学即使不是糟糕的数学，至少也是丑陋的数学，但我认为我可以指出阿贝尔委员会的那些引文，它们都强调了我工作的优雅性！

现在谈谈哈代：哈代写《一个数学家的辩白》时，他已是垂暮之年，年纪老迈，我想他当时心脏病发作，身体虚弱，情绪非常低落。所以这一点应该考虑到。关于这本书本身：化学家弗雷德里克·索迪曾对此进行了非常严厉的批评，他是同位素的共同发现者之一——他与卢瑟福分享了诺贝尔奖。他看到哈代对其数学的无用性感到自豪，于是写道：“世界因这种与世隔绝的小丑行径而作呕。”这非常严厉，因为哈代是个非常好的人。

我的朋友乔·凯勒（Joe Keller），一位最杰出的应用数学家，曾被要求定义应用数学，他想出了这个说法：“纯粹数学是应用数学的一个分支。”如果你仔细想想，这是对的。数学最初，比如说牛顿之后，是为了解决物理学中出现的非常具体的问题而设计的。后来，这些学科独立发展，成为纯粹数学的分支，但它们都来自应用背景。正如冯·诺伊曼指出的那样，过了一段时间，这些独立发展的纯粹分支需要新的经验材料来注入活力，比如一些科学问题、实验事实，尤其是某些数值证据。

劳森与斯考： 在数学史上，阿贝尔和伽罗瓦可能是第一批可以被称为“纯粹数学家”的伟大数学家，他们对任何“应用”数学本身都不感兴趣。然而，阿贝尔确实解决了一个积分方程，后来被称为“阿贝尔积分方程”，并且阿贝尔给出了一个显式解，这顺便说一句，可能是数学史上第一次一个积分方程被表述并解决。有趣的是，通过简单的重新表述，可以证明阿贝尔积分方程及其解等价于拉东变换，而拉东变换是现代医学断层扫描技术的数学基础。

在数学史上，纯粹数学成果和定理的这种完全出乎意料的实际应用比比皆是——从伽罗瓦的工作发展而来的群论是另一个显著的例子。您对这种现象有何看法？数学中深刻而重要的理论和定理最终会找到实际应用，例如在物理科学中，这是真的吗？

拉克斯： 嗯，正如你所指出的，这种情况经常发生：以尤金·维格纳在量子力学中应用群论为例。而且这种情况发生得太频繁了，不可能是巧合。尽管，也许有人会说，其他没有找到应用的理论和定理被遗忘了。对于数学史家来说，研究这种现象可能会很有趣。但我确实相信数学具有一种神秘的统一性，它确实连接了看似不同的部分，这是数学的荣耀之一。

劳森与斯考： 您曾说过洛斯阿拉莫斯是计算动力学的发源地，我想可以肯定地说，20世纪40年代美国的战争努力推动并加速了这一发展。高速计算机的出现以何种方式改变了数学的完成方式？未来高速计算机将在数学中扮演什么角色？

拉克斯： 它扮演了几个角色。一个是我们从克鲁斯卡尔和扎布斯基发现孤子中看到的，如果没有计算证据，这是不可能发现的。同样，费米-帕斯塔-乌拉姆的回归现象也是一个非常引人注目的事情，无论有没有计算机，都可能（或不可能）被发现。这是一个方面。

但另一个方面是：在过去，要获得数值结果，如果你的计算是手工完成的，或者通过简单的计算机器完成的，你就必须进行极其剧烈的简化。而进行何种剧烈简化的才能，是一种特殊的才能，并不吸引大多数数学家。今天，你处于一个完全不同的境地。你不必在数值上攻击问题之前，把它放在一个普罗克拉斯提斯之床上（Procrustean bed，意指强求一致，削足适履）并将其肢解。而且我认为这吸引了更多的人来研究应用中的数值问题——你真的可以使用完整的理论。它振兴了线性代数这个学科，这个学科作为研究课题在20世纪20年代已经消亡了。突然之间，执行这些运算的实际算法变得重要起来。它充满了惊喜，比如快速矩阵乘法。在我新版的线性代数书中，我将增加一章关于对称矩阵特征值的数值计算。 你知道这是一个老生常谈的说法，由于计算机速度的提高，四十年前需要一个月才能完成的问题，现在几分钟甚至几秒钟就能完成。大部分的速度提升，至少在公众看来，都归功于计算机速度的提高。但如果你仔细观察，实际上只有一半的速度提升是由于这种速度的提高。另一半是由于巧妙的算法，而发明巧妙的算法需要数学家。所以让数学家参与进来非常重要，他们现在也参与进来了。

劳森与斯考： 您能否举一些个人例子，说明应用视角的的问题和方法如何引发了“纯粹”的数学研究和成果？反过来，您的非线性偏微分方程理论，特别是您对不连续性如何传播的解释，是否曾有过商业上的应用？特别是在对挪威如此重要的石油勘探方面！

拉克斯： 是的，石油勘探使用由爆炸产生的信号，这些信号通过地球和油藏传播，并在远处的测站记录下来。这被称为反演问题。如果你知道物质密度的分布和相关的波速，那么你就可以计算信号如何传播。反演问题是，如果你知道信号如何传播，那么你想从中推断出物质的分布。由于信号是不连续的，你需要不连续性传播的理论。否则，它有点类似于医学成像问题，也是一个反演问题。这里的信号不是穿过地球而是穿过人体，但问题有相似之处。但毫无疑问，在解决反演问题之前，你必须非常了解正演问题。

匈牙利数学

劳森与斯考： 现在来谈谈一些与您个人经历相关的问题。第一个是关于您对解决您自己称之为“轻松数学”（Mathematics Light）类型问题的兴趣和天赋。仅举几例，早在您十七岁的时候，您就为一个由埃尔德什提出的、与某个多项式不等式（早期由伯恩斯坦证明）相关的问题给出了一个优雅的解。在您职业生涯的后期，您研究了所谓的波利亚函数，它将单位区间连续映射到一个直角三角形上，并且您发现了它惊人的可微性。在您早年在匈牙利接受的数学教育中，问题解决是否受到特别鼓励，这对您后来的职业生涯有何影响？

拉克斯： 是的，解决问题被认为是激发有才华年轻人的捷径，我很高兴得知挪威这里有一个成功的高中竞赛，获胜者今天早上受到了表彰。但过了一段时间，人们不应该只停留在解决问题上，应该拓宽视野。不过，我偶尔还是会回到解决问题上来。

回到波利亚函数的可微性问题：我与波利亚相当熟悉，1946年曾上过他的暑期课程。可微性问题是这样产生的：我当时在教一门实变函数课程，我介绍了波利亚的面积填充曲线的例子，并给学生们布置了证明它处处不可微的作业。没有人做这个作业，于是我坐下来研究，发现情况更复杂。

匈牙利有一个传统，就是寻找最简单的证明。你可能熟悉埃尔德什的“那本书”（The Book）的概念。那是上帝保存所有定理及其最佳证明的书。埃尔德什对一个证明的最高评价是它出自“那本书”。人们可能会做得过火，但在我获得博士学位后不久，我了解了哈恩-巴拿赫定理，我认为可以用它来证明格林函数的存在性。这是一个非常简单的论证——我相信这是最简单的——所以它出自“那本书”。而且我想我有一个布劳威尔不动点定理的证明，用的是微积分和变量代换。这可能是最简单的证明，同样出自“那本书”。我认为所有这些都是匈牙利传统的一部分。但我们不能做得过火。

劳森与斯考： 在法西斯主义、纳粹主义和反犹太主义在欧洲兴起后，有一大批杰出的匈牙利物理学家和数学家（他们都是犹太背景）不得不逃往美国。您如何解释匈牙利这种非凡的卓越文化，它培养出了像德海韦西、西拉德、维格纳、泰勒、冯·诺伊曼、冯·卡门、埃尔德什、塞格、波利亚以及您自己这样一些最杰出的人物？

拉克斯： 约翰·卢卡奇写过一本非常有趣的书，书名是《布达佩斯1900：一座城市及其文化的历史肖像》，书中记载了中产阶级的崛起、商业的崛起、工业的崛起、科学的崛起、文学的崛起。它由许多因素推动：长期的和平，大量渴望提升社会地位的犹太人口从东方涌入，以及知识传统。你知道，在数学领域，波利亚伊是匈牙利人的文化英雄，这就是为什么数学被特别视为一种光荣的职业。

劳森与斯考： 但是是谁培养了这种人才的蓬勃发展，如此引人注目？

拉克斯： 或许应该把很多功劳归于尤利乌斯·柯尼希（Julius Kőnig），他的名字你可能不知道。我相信他是克罗内克的学生，但他也学习了康托尔的集合论并做出了一些基本贡献。我认为他在培养数学人才方面很有影响力。他的儿子是一位非常杰出的数学家，丹尼斯·柯尼希（Denes Kőnig），实际上是现代图论之父。然后出现了一些非凡的人物。例如，利奥波德·费耶（Leopold Fejér）具有巨大的影响力。像匈牙利这样的小国，职位太少，无法容纳那么多人，所以他们不得不出国。部分原因也是反犹太主义。

关于利奥波德·费耶的任命有一个有趣的故事，他是第一个被提名为布达佩斯大学教授的犹太人。当时有人反对。那时神学院有一位非常杰出的神学家，名叫伊格纳修斯·费耶（Ignatius Fejér）。费耶的原名是魏斯（Weiss）。因此，一位非常清楚费耶原名是魏斯的反对者尖锐地问道：你提议的这位利奥波德·费耶教授，他与我们尊敬的同事伊格纳修斯·费耶神父有亲戚关系吗？而推动这项任命的伟大物理学家厄特沃什（Eőtvős）则面不改色地回答：“私生子。”这就结束了这件事。

劳森与斯考： 然后他得到了那份工作？

拉克斯： 他得到了那份工作。

改变人类事务进程的涂鸦

劳森与斯考： 数学家斯坦尼斯瓦夫·乌拉姆参与了曼哈顿计划，并被认为是氢弹之父之一。他在其自传《一个数学家的奇遇》中写道：“看到黑板上或一张纸上的几笔涂鸦，就能改变人类事务的进程，这至今仍让我感到惊奇不已。”您有同感吗？您对广岛和长崎原子弹爆炸的受害者，以及结束第二次世界大战的原子弹爆炸事件有何感想？

拉克斯： 嗯，让我先回答最后一个问题。我当时在军队里，我们所有在军队里的人都预计会被派往太平洋参与入侵日本的行动。你还记得诺曼底登陆造成的巨大伤亡。与入侵日本本土相比，那简直是小巫见大巫。你还记得冲绳岛和硫磺岛上的惨烈屠杀。日本人会抵抗到最后一个人。原子弹结束了这一切，使得入侵变得不必要。我不相信那些修正主义历史学家的说法：“哦，日本已经被打败了，他们无论如何都会投降的。”我没有看到任何证据。

还有一点我曾经和一位参与原子弹计划的人提起过。如果世界没有看到一颗炸弹能造成多大的破坏，它还会对核战争感到恐惧吗？世界因其使用而被“接种”了疫苗，从而不再使用核武器。我并不是说单凭这一点就能证明其合理性，当然也不是其使用的理由。但我认为这是一个历史事实。

现在谈谈改变历史的涂鸦：当然，没有涂鸦，狭义相对论或量子力学在今天将是不可想象的。顺便说一句，乌拉姆是一位非常有趣的数学家。他是个点子王。大多数数学家喜欢将自己的想法付诸实践。他更喜欢抛出想法。他的好朋友罗塔甚至暗示他没有技术能力或耐心去实现它们。但如果是这样的话，那就是乌拉姆将残疾转化为巨大优势的一个例子。我从他那里学到了很多。

劳森与斯考： 我们很惊讶地得知，一位十八岁的移民竟然被允许在二战期间参与一项绝密的、决定性的武器研发项目。

拉克斯： 战争造成了紧急情况。曼哈顿计划的许多领导人都是外国人，所以外国人身份并不是障碍。

合作与工作风格

劳森与斯考： 您的主要工作地点一直是纽约的库朗数学科学研究所，它是纽约大学的一部分。您在20世纪70年代担任了八年的所长。您能描述一下是什么让这个由德国难民理查德·库朗在20世纪30年代创建的研究所从早期就成为一个非常特殊的地方，拥有独特的精神和氛围吗？今天的库朗研究所是否仍然是一个与众不同的特殊地方？

拉克斯： 回答你的第一个问题，库朗的个性无疑是决定性的。库朗对数学的看法非常广阔，他对专业化持怀疑态度。他希望数学尽可能广泛地发展，这就是为什么应用课题和纯粹数学会并驾齐驱地被研究，而且常常是由同一些人研究。这使得库朗研究所在其成立之初以及20世纪40年代、50年代和60年代都是独一无二的。从那时起，其他一些中心也开始尊重和研究应用数学。我很高兴地说，这种最初的精神仍然存在于库朗研究所。我们仍然有许多感兴趣的应用领域，如安迪·马伊达（Andy Majda）领导下的气象学和气候学，罗伯特·科恩（Robert Kohn）等人领导下的固态和材料科学，以及流体动力学。但我们也有微分几何以及偏微分方程的一些纯粹方面，甚至还有一些代数。

我对库朗研究所目前的运作方式非常满意。现在是第三代人在管理它，库朗所灌输的那种精神——一种家庭般的感觉——依然存在。我很高兴地注意到，许多挪威数学家在库朗研究所接受了培训，后来成为他们领域的领导者。

劳森与斯考： 您已经告诉我们您与拉尔夫·菲利普斯的合作。总的来说，浏览您的出版物清单以及您和您的合作者命名的定理和方法，很明显您与许多数学家进行了广泛的合作。这种思想共享对您来说是否是一种特别成功，或许也是一种愉快的前进方式？

拉克斯： 当然，当然。数学毕竟是一种社会现象。合作是一种心理上和有趣的现象。我的一个朋友，维拉·约翰-施泰纳（Vera John-Steiner），写了一本关于它的书（《创造性合作》）。一个解决方案的两半由两个不同的人提供，然后一些非常美妙的东西就产生了。

劳森与斯考： 许多数学家在攻克某些难题时都有非常独特的工作风格。您如何描述自己独特的思考、工作和写作方式？是偏向于趣味性还是勤奋型？或者两者兼而有之？

拉克斯： 菲利普斯认为我懒。他是大萧条时期的产物，那段时期对人们施加了某种严格的纪律。他认为我工作不够努力，但我认为我确实努力了！

劳森与斯考： 有时数学上的洞察力似乎依赖于突如其来的灵感。您自己的职业生涯中有过这样的例子吗？在您看来，这种突如其来的灵感的背景是什么？

拉克斯： 这个问题让我想起了一个关于德国数学家肖特基（Schottky）的故事，当他七八十岁的时候。当时有一个庆祝活动，在像我们现在这样的采访中，他被问到：“您将您的创造力和生产力归功于什么？”这个问题让他非常困惑。最后他说：“但是先生们，如果一个人思考数学五十年，他总得想出点什么吧！”希尔伯特的情况则不同。这是我从库朗那里听来的一个故事。那是一个类似的场合。在他七十岁生日时，有人问他将自己伟大的创造力和独创性归功于什么。他立刻回答说：“我把它归功于我非常糟糕的记性。”他真的必须重新构建一切，然后它就变成了别的东西，更好的东西。所以也许我应该说的就这些了。我介于这两个极端之间。顺便说一句，我记性很好。

教学

劳森与斯考： 您也一直从事微积分教学。例如，您与您的妻子安娜莉（Anneli）合著了一本微积分教科书。在这方面，您曾就微积分应如何向初学者讲授表达过强烈的意见。您能详细说明一下吗？

拉克斯： 我们的微积分教材非常不成功，尽管其中包含许多优秀的想法。部分原因是某些材料的呈现方式学生难以吸收。一本微积分教材必须经过精心打磨，而我没有那个耐心。安娜莉本来会有，但我恐怕对她太霸道了。有时我梦想重做它，因为里面的想法，以及我后来的想法，仍然有效。

当然，曾经有过一场微积分改革运动，也出版了一些好书，但我认为它们并非最终答案。首先，这些书太厚了，常常超过1000页。把这样的书交到一个几乎拿不动它的毫无戒心的学生手中是不公平的。他们对此的反应会是：“哦，我的天，我必须把这里面的所有东西都学会吗？”嗯，并非所有内容都在里面！其次，如果你把它与旧的标准，比如说托马斯（Thomas）的教材相比，它并没有那么不同——也许是主题和概念的顺序不同。

例如，在我的微积分书中，我主张用一致连续性代替点态连续性。这比在一点定义连续性，然后再说函数在每一点都连续要容易解释得多。那样你会失去学生；里面的量词太多了。但是数学界非常保守：“连续性是逐点定义的，所以就应该是这样！”

我还想强调其他一些事情：毫无疑问，这些新书中都有应用。但是应用应该突出。在我的书中，有专门讨论应用的章节，就应该这样——它们应该被重点介绍。我还有很多其他的想法。我仍然梦想重做我的微积分教材，并且正在寻找一个好的合作者。我最近遇到一个人，他对原版书表示赞赏，所以也许如果我有精力的话，这个梦想可以实现。我还有其他事情要做，比如我的线性代数教材的第二版，以及修订一些关于双曲方程的旧讲义。但即使我能找到一个微积分教材的合作者，它会被接受吗？不清楚。1873年，戴德金提出了一个重要的问题：“什么是实数，实数应该是什么？”不幸的是，就微积分学生而言，他给出了错误的答案。正确的答案是：无穷小。我不知道这样的笑话效果如何。

领导大型机构

劳森与斯考： 您曾多次担任大型组织的负责人：1972-1980年担任库朗研究所所长，1977-1980年担任美国数学会主席，1980-1986年担任国家科学委员会（National Science Board）下设的所谓拉克斯小组（Lax Panel）的负责人。您能谈谈在这些时期需要做出的一些最重要的决定吗？

拉克斯： 美国数学会主席只是个象征性人物。他的影响力在于任命委员会成员。拥有广泛的人脉和合理的判断力会有所帮助。美国数学会秘书埃弗雷特·皮彻（Everett Pitcher）给了我很大帮助。

至于担任库朗研究所所长，我是在纽约大学最糟糕的时候开始我的所长任期的。他们刚刚关闭了工程学院，这意味着工程学院的数学家被转移到了库朗研究所。那正是杰克·施瓦茨（Jack Schwartz）在库朗创办计算机科学系的时候。有一群工程师想在信息学（informatics）领域开展活动，这是工程师们对同一事物的称呼。作为所长，我非常努力地阻止了这件事。我认为如果大学有两个计算部门，那对大学来说会非常糟糕——对我们的计算机科学系来说肯定会非常糟糕。其他事情：嗯，我在亚历山大·乔林（Alexander Chorin）的推荐下聘用了查理·佩斯金（Charlie Peskin），对此我非常高兴。同样，在鲍勃·科恩（Bob Kohn）的推荐下聘用了西尔万·卡佩尔（Sylvain Cappell）。这两次都是巨大的成功。

我的失败是什么？嗯，也许在计算机科学系成立时，我应该坚持非常高的招聘标准。我们需要人来教课，但事后看来，我认为我们应该在招聘方面更加克制。我们本可以成为第一的计算机科学系。现在质量已经大大提高——我们有一位出色的系主任，玛格丽特·赖特（Margaret Wright）。

在国家科学委员会任职是我最愉快的行政经历。它是国家科学基金会（NSF）的决策机构，所以我了解了决策意味着什么。大多数时候，它只是意味着点头说“是”，偶尔说“不”。但有时会出现机会之窗，而拉克斯小组就是对这种情况的回应。你看，我通过自己的经验以及对大规模计算感兴趣的朋友（特别是抱怨此事的保罗·加拉贝迪安）的经验注意到，大学的计算科学家无法使用超级计算机。在某个时候，政府（只有它有足够的钱购买这些超级计算机）停止将它们安置在大学。相反，它们被安置在国家实验室和工业实验室。除非你碰巧在那里有合作的朋友，否则你无法使用。从计算科学进步的角度来看，这非常糟糕，因为最有才华的人都在大学里。那时，由于ARPANET（后来成为互联网的典范）的出现，远程访问和计算成为可能。所以我成立的小组强烈建议NSF建立计算中心，并且这个建议得到了采纳。我对我们成就的引述是爱默生的一句话的改写：“没有什么能抵抗一个迟到了十年的想法的力量。”

劳森与斯考： 美国的许多数学研究都由国防部（DOD）、能源部（DOE）、原子能委员会、国家安全局（NSA）的合同资助。这种相互依赖是否互惠互利？是否存在陷阱？

拉克斯： 恐怕我们的领导人不再意识到科学活力与技术先进性之间微妙而密切的联系。

个人兴趣

劳森与斯考： 您能谈谈您与数学没有直接关系的兴趣和爱好吗？

拉克斯： 我喜欢诗歌。匈牙利诗歌特别优美，但英国诗歌可能更美。我喜欢打网球。现在我的膝盖有点不稳，跑不动了，但也许可以换掉——我还到不了那一步。我的儿子和三个孙子都是网球爱好者，所以我可以和他们打双打。我喜欢阅读。我擅长写作。唉，这些天我写的都是讣告——写讣告总比被人写讣告要好。

劳森与斯考： 您也写过日本俳句吗？

拉克斯： 你说对了。我从马歇尔·斯通（Marshall Stone）的一篇好文章中得到了这个想法——我忘了具体是哪篇了——他在文章中写道，数学语言极其浓缩，就像俳句一样。我想我应该更进一步，用俳句来表达一个数学思想。（见下面彼得·拉克斯的俳句。）

劳森与斯考： 拉克斯教授，非常感谢您代表挪威、丹麦和欧洲数学会接受这次采访！

拉克斯： 我感谢你们。

速度依尺寸， 色散来平衡， 哦，孤独的辉煌。

访谈原文

Interview with Peter D. Lax

Martin Raussen and Christian Skau

Peter D. Lax is the recipient of the 2005 Abel Prize of the Norwegian Academy of Science and Letters. On May 24, 2005, prior to the Abel Prize celebrations in Oslo, Lax was interviewed by Martin Raussen of Aalborg University and Christian Skau of the Norwegian University of Science and Technology. This interview originally appeared in the European Mathematical Society Newsletter, September 2005, pages 24-31.

Martin Raussen is associate professor of mathematics at Aalborg University, Denmark. His email address is raussen@math.aau.dk. Christian Skau is professor of mathematics at Norwegian University of Science and Technology, Trondheim, Norway. His email address is csk@math.ntnu.no.

Raussen & Skau: On behalf of the Norwegian and Danish Mathematical Societies we would like to congratulate you on winning the Abel Prize for 2005. You came to the U.S. in 1941 as a fifteen-year-old kid from Hungary. Only three years later, in 1944, you were drafted into the U.S Army. Instead of being shipped overseas to the war front, you were sent to Los Alamos in 1945 to participate in the Manhattan Project, building the first atomic bomb. It must have been awesome as a young man to come to Los Alamos to take part in such a momentous endeavor and to meet so many legendary famous scientists: Fermi, Bethe, Szilard, Wigner, Teller, Feynman, to name some of the physicists, and von Neumann and Ulam, to name some of the mathematicians. How did this experience shape your view of mathematics and influence your choice of a research field within mathematics?

Lax: In fact, I returned for a year's stay at Los Alamos after I got my Ph.D. in 1949 and then spent many summers as a consultant. The first time I spent in Los Alamos, and especially the later exposure, shaped my mathematical thinking. First of all, it was the experience of being part of a scientific team—not just of mathematicians, but people with different outlooks—with the aim being not a theorem, but a product. One cannot learn that from books, one must be a participant, and for that reason I urge my students to spend at least a summer as a visitor at Los Alamos. Los Alamos has a very active visitor's program. Secondly, it was there—that was in the 1950s—that I became imbued with the utter importance of computing for science and mathematics. Los Alamos, under the influence of von Neumann, was for a while in the 1950s and the early 1960s the undisputed leader in computational science.

Research Contributions

R & S: May we come back to computers later? First some questions about some of your main research contributions to mathematics: You have made outstanding contributions to the theory of nonlinear partial differential equations. For the theory and numerical solutions of hyperbolic systems of conservation laws your contribution has been decisive, not to mention your contribution to the understanding of the propagation of discontinuities, so-called shocks. Could you describe in a few words how you were able to overcome the formidable obstacles and difficulties this area of mathematics presented?

Lax: Well, when I started to work on it I was very much influenced by two papers. One was Eberhard Hopf's on the viscous limit of Burgers' equation, and the other was the von Neumann-Richtmyer paper on artificial viscosity. And looking at these examples I was able to see what the general theory might look like.

R & S: The astonishing discovery by Kruskal and Zabusky in the 1960s of the role of solitons for solutions of the Korteweg-deVries (KdV) equation, and the no less astonishing subsequent explanation given by several people that the KdV equation is completely integrable, represented a revolutionary development within the theory of nonlinear partial differential equations. You entered this field with an ingenious original point of view, introducing the so-called Lax-pair, which gave an understanding of how the inverse scattering transform applies to equations like the KdV, and also to other nonlinear equations which are central in mathematical physics, like the sine-Gordon and the nonlinear Schrödinger equation. Could you give us some thoughts on how important you think this theory is for mathematical physics and for applications, and how do you view the future of this field?

(Photo: Knut Falch/Scanpix) Copyright: Det Norske Videnskaps-Akademi/Abelprisen. Peter D. Lax was interviewed by Martin Raussen and Christian Skau at the Hotel Continental in Oslo.

Lax: Perhaps I should start by pointing out that the astonishing phenomenon of the interaction of solitons was discovered by numerical calculations, as was predicted by von Neumann some years before, namely that calculations will reveal extremely interesting phenomena. Since I was a good friend of Kruskal, I learned early about his discoveries, and that started me thinking. It was quite clear that there are infinitely many conserved quantities, and so I asked myself: How can you generate all at once an infinity of conserved quantities? I thought if you had a transformation that preserved the spectrum of an operator then that would be such a transformation, and that turned out to be a very fruitful idea, applicable quite widely. Now you ask how important is it? I think it is pretty important. After all, from the point of view of technology for the transmission of signals, signalling by solitons is very important and a promising future technology in trans-oceanic transmission. This was developed by Linn Mollenauer, a brilliant engineer at Bell Labs. It has not yet been put into practice, but it will be some day. The interesting thing about it is that classical signal theory is entirely linear, and the main point of soliton signal transmission is that the equations are nonlinear. That's one aspect of the practical importance of it. As for the theoretic importance: the KdV equation is completely integrable, and then an astonishing number of other completely integrable systems were discovered. Completely integrable systems can really be solved in the sense that the general population uses the word solved. When a mathematician says he has solved the problem he means he knows the solution exists, that it's unique, but very often not much more. Now the question is: Are completely integrable systems exceptions to the behavior of solutions of non-integrable systems, or is it that other systems have similar behavior, only we are unable to analyze it? And here our guide might well be the Kolmogorov-Arnold-Moser theorem which says that a system near a completely integrable system behaves as if it were completely integrable. Now, what near means is one thing when you prove theorems, another when you do experiments. It's another aspect of numerical experimentation revealing things. So I do think that studying completely integrable systems will give a clue to the behavior of more general systems as well. Who could have guessed in 1965 that completely integrable systems would become so important?

R & S: The next question is about your seminal paper "Asymptotic solutions of oscillating initial value problems” from 1957. This paper is considered by many people to be the genesis of Fourier Integral Operators. What was the new viewpoint in the paper that proved to be so fruitful?

Lax: It is a micro-local description of what is going on. It combines looking at the problem in the large and in the small. It combines both aspects, and that gives it its strengths. The numerical implementation of the micro-local point of view is by wavelets and similar approaches, which are very powerful numerically.

R & S: May we touch upon your collaboration with Ralph Phillips—on and off over a span of more than thirty years—on scattering theory, applying it in a number of settings. Could you comment on this collaboration, and what do you consider to be the most important results you obtained?

Lax: That was one of the great pleasures of my life! Ralph Phillips is one of the great analysts of our time and we formed a very close friendship. We had a new way of viewing the scattering process with incoming and outgoing subspaces. We were, so to say, carving a semi-group out of the unitary group, whose infinitesimal generator contained almost all the information about the scattering process. So we applied that to classical scattering of sound waves and electromagnetic waves by potentials and obstacles. Following a very interesting discovery of Faddeev and Pavlov, we studied the spectral theory of automorphic functions. We elaborated it further, and we had a brand new approach to Eisenstein series for instance, getting at spectral representation via translation representation. And we were even able to contemplate—following Faddeev and Pavlov—the Riemann hypothesis peeking around the corner.

R & S: That must have been exciting!

Lax: Yes! Whether this approach will lead to the proof of the Riemann hypothesis, stating it, as one can, purely in terms of decaying signals by cutting out all standing waves, is unlikely. The Riemann hypothesis is a very elusive thing. You may remember in Peer Gynt there is a mystical character, the Boyg, which bars Peer Gynt's way wherever he goes. The Riemann hypothesis resembles the Boyg!

R & S: Which particular areas or questions are you most interested in today?

Lax: I have some ideas about the zero dispersion limit.

Pure and Applied Mathematics

R & S: May we raise a perhaps contentious issue with you: pure mathematics versus applied mathematics. Occasionally one can hear within the mathematical community statements that the theory of nonlinear partial differential equations, though profound and often very important for applications, is fraught with ugly theorems and awkward arguments. In pure mathematics, on the other hand, beauty and aesthetics rule. The English mathematician G.H. Hardy is an extreme example of such an attitude, but it can be encountered also today. How do you respond to this? Does it make you angry?

Lax: I don't get angry very easily. I got angry once at a dean we had, terrible son of a bitch, destructive liar, and I got very angry at the mob that occupied the Courant Institute and tried to burn down our computer. Scientific disagreements do not arouse my anger. But I think this opinion is definitely wrong. I think Paul Halmos once claimed that applied mathematics was, if not bad mathematics, at least ugly mathematics, but I think I can point to those citations of the Abel Committee dwelling on the elegance of my works! Now about Hardy: When Hardy wrote A Mathematician's Apology he was at the end of his life, he was old, I think he had suffered a debilitating heart attack, he was very depressed. So that should be taken into account. About the book itself: There was a very harsh criticism by the chemist Frederick Soddy, who was one of the co-discoverers of the isotopes—he shared the Nobel Prize with Rutherford. He looked at the pride that Hardy took in the uselessness of his mathematics and wrote: "From such cloistral clowning the world sickens.” It was very harsh because Hardy was a very nice person. My friend Joe Keller, a most distinguished applied mathematician, was once asked to define applied mathematics and he came up with this: "Pure mathematics is a branch of applied mathematics." Which is true if you think a bit about it. Mathematics originally, say after Newton, was designed to solve very concrete problems that arose in physics. Later on, these subjects developed on their own and became branches of pure mathematics, but they all came from applied background. As von Neumann pointed out, after a while these pure branches that develop on their own need invigoration by new empirical material, like some scientific questions, experimental facts, and, in particular, some numerical evidence.

R & S: In the history of mathematics, Abel and Galois may have been the first great mathematicians that one may describe as “pure mathematicians”, not being interested in any “applied” mathematics as such. However, Abel did solve an integral equation, later called “Abel's integral equation”, and Abel gave an explicit solution, which incidentally may have been the first time in the history of mathematics that an integral equation had been formulated and solved. Interestingly, by a simple reformulation one can show that the Abel integral equation and its solution are equivalent to the Radon Transform, the mathematical foundation on which modern medical tomography is based. Examples of such totally unexpected practical applications of pure mathematical results and theorems abound in the history of mathematics—group theory that evolved from Galois' work is another striking example. What are your thoughts on this phenomenon? Is it true that deep and important theories and theorems in mathematics will eventually find practical applications, for example in the physical sciences?

Lax: Well, as you pointed out, this has very often happened: Take for example Eugene Wigner's use of group theory in quantum mechanics. And this has happened too often to be just a coincidence. Although, one might perhaps say that other theories and theorems which did not find applications were forgotten. It might be interesting for a historian of mathematics to look into that phenomenon. But I do believe that mathematics has a mysterious unity which really connects seemingly distinct parts, which is one of the glories of mathematics.

R & S: You have said that Los Alamos was the birthplace of computational dynamics, and I guess it is safe to say that the U.S. war effort in the 1940s advanced and accelerated this development. In what way has the emergence of the high-speed computer altered the way mathematics is done? Which role will high-speed computers play within mathematics in the future?

Lax: It has played several roles. One is what we saw in Kruskal's and Zabusky's discovery of solitons, which would not have been discovered without computational evidence. Likewise the Fermi-Pasta-Ulam phenomenon of recurrence was also a very striking thing which may or may not have been discovered without the computer. That is one aspect. But another is this: in the old days, to get numerical results you had to make enormously drastic simplifications if your computations were done by hand, or by simple computing machines. And the talent of what drastic simplifications to make was a special talent that did not appeal to most mathematicians. Today you are in an entirely different situation. You don't have to put the problem on a Procrustean bed and mutilate it before you attack it numerically. And I think that has attracted a much larger group of people to numerical problems of applications—you could really use the full theory. It invigorated the subject of linear algebra, which as a research subject died in the 1920s. Suddenly the actual algorithms for carrying out these operations became important. It was full of surprises, like fast matrix multiplication. In the new edition of my linear algebra book I will add a chapter on the numerical calculation of the eigenvalues of symmetric matrices. You know it's a truism that due to increased speed of computers, a problem that took a month forty years ago can be done in minutes, if not seconds today. Most of the speed-up is attributed, at least by the general public, to increased speed of computers. But if you look at it, actually only half of the speed-up is due to this increased speed. The other half is due to clever algorithms, and it takes mathematicians to invent clever algorithms. So it is very important to get mathematicians involved, and they are involved now.

R & S: Could you give us personal examples of how questions and methods from applied points of view have triggered “pure” mathematical research and results? And conversely, are there examples where your theory of nonlinear partial differential equations, especially your explanation of how discontinuities propagate, have had commercial interests? In particular, concerning oil exploration, so important for Norway!

Lax: Yes, oil exploration uses signals generated by detonations that are propagated through the earth and through the oil reservoir and are recorded at distant stations. It's a so-called inverse problem. If you know the distribution of the densities of materials and the associated waves' speeds, then you can calculate how signals propagate. The inverse problem is that if you know how signals propagate, then you want to deduce from it the distribution of the materials. Since the signals are discontinuities, you need the theory of propagation of discontinuities. Otherwise it's somewhat similar to the medical imaging problem, also an inverse problem. Here the signals do not go through the earth but through the human body, but there is a similarity in the problems. But there is no doubt that you have to understand the direct problem very well before you can tackle the inverse problem.

Hungarian Mathematics

R & S: Now to some questions related to your personal history. The first one is about your interest in, and great aptitude for, solving problems of a type that you call "Mathematics Light” yourself. To mention just a few, already as a seventeen-year-old boy you gave an elegant solution to a problem that was posed by Erdős and is related to a certain inequality for polynomials, which was earlier proved by Bernstein. Much later in your career you studied the so-called Pólya function which maps the unit interval continuously onto a right-angled triangle, and you discovered its amazing differentiability properties. Was problem solving specifically encouraged in your early mathematical education in your native Hungary, and what effect has this had on your career later on?

Lax: Yes, problem solving was regarded as a royal road to stimulate talented youngsters, and I was very pleased to learn that here in Norway they have a successful high-school contest, where the winners were honored this morning. But after a while one shouldn't stick to problem solving, one should broaden out. I return to it every once in a while, though. Back to the differentiability of the Pólya function: I knew Pólya quite well having taken a summer course with him in 1946. The differentiability question came about this way: I was teaching a course on real variables, and I presented Pólya's example of an area-filling curve, and I gave as homework to the students the problem of proving that it's nowhere differentiable. Nobody did the homework, so then I sat down and I found out that the situation was more complicated. There was a tradition in Hungary to look for the simplest proof. You may be familiar with Erdős' concept of The Book. That's The Book kept by the Lord of all theorems and the best proofs. The highest praise that Erdős had for a proof was that it was out of The Book. One can overdo that, but shortly after I had gotten my Ph.D., I learned about the Hahn-Banach theorem, and I thought that it could be used to prove the existence of Green's function. It's a very simple argument—I believe it's the simplest—so it's out of The Book. And I think I have a proof of Brouwer's Fixed Point Theorem, using calculus and just change of variables. It is probably the simplest proof and is again out of The Book. I think all this is part of the Hungarian tradition. But one must not overdo it.

R & S: There is an impressive list of great Hungarian physicists and mathematicians of Jewish background that had to flee to the U.S. after the rise of fascism, Nazism and anti-Semitism in Europe. How do you explain this extraordinary culture of excellence in Hungary that produced people like de Hevesy, Szilard, Wigner, Teller, von Neumann, von Karman, Erdős, Szegő, Pólya, yourself, to name some of the most prominent ones?

Lax: There is a very interesting book written by John Lukacs with the title "Budapest 1900: A Historical Portrait of a City and its Culture", and it chronicles the rise of the middle class, rise of commerce, rise of industry, rise of science, rise of literature. It was fueled by many things: a long period of peace, the influx of mostly Jewish population from the East eager to rise, and intellectual tradition. You know in mathematics, Bolyai was a cultural hero to Hungarians, and that's why mathematics was particularly looked upon as a glorious profession.

R & S: But who nurtured this fantastic flourishing of talent, which is so remarkable?

Lax: Perhaps much credit should be given to Julius Kőnig, whose name is probably not known to you. He was a student of Kronecker, I believe, but he also learned Cantor's set theory and made some basic contribution to it. I think he was influential in nurturing mathematics. His son was a very distinguished mathematician, Denes Kőnig, really the father of modern graph theory. And then there arose extraordinary people. Leopold Fejér, for instance, had enormous influence. There were too many to fill positions in a small country like Hungary, so that's why they had to go abroad. Part of it was also anti-Semitism. There is a charming story about the appointment of Leopold Fejér, who was the first Jew proposed for a professorship at Budapest University. There was opposition to it. At that time there was a very distinguished theologian, Ignatius Fejér, in the Faculty of Theology. Fejér's original name was Weiss. So one of the opponents, who knew full well that Fejér's original name had been Weiss, said pointedly: This professor Leopold Fejér that you are proposing, is he related to our distinguished colleague Father Ignatius Fejér? And Eőtvős, the great physicist who was pushing the appointment, replied without batting an eyelash: “Illegitimate son." That put an end to it.

R & S: And he got the job?

Lax: He got the job.

Scribbles That Changed the Course of Human Affairs

R & S: The mathematician Stanislaw Ulam was involved with the Manhattan Project and is considered to be one of the fathers of the hydrogen bomb. He wrote in his autobiography Adventures of a Mathematician: “It is still an unending source of surprise for me to see how a few scribbles on a blackboard, or on a sheet of paper, could change the course of human affairs.” Do you share this feeling? And what are your feelings about what happened to Hiroshima and Nagasaki, to the victims of the explosions of the atomic bombs that brought an end to World War II?

Lax: Well, let me answer the last question first. I was in the army, and all of us in the army expected to be sent to the Pacific to participate in the invasion of Japan. You remember the tremendous slaughter that the invasion of Normandy brought about. That would have been nothing compared to the invasion of the Japanese mainland. You remember the tremendous slaughter on Okinawa and Iwo Jima. The Japanese would have resisted to the last man. The atomic bomb put an end to all this and made an invasion unnecessary. I don't believe reversionary historians who say: "Oh, Japan was already beaten, they would have surrendered anyway." I don't see any evidence for that. There is another point which I raised once with someone who had been involved with the atomic bomb project. Would the world have had the horror of nuclear war if it had not seen what one bomb could do? The world was inoculated against using nuclear weaponry by its use. I am not saying that alone justifies it, and it certainly was not the justification for its use. But I think that is a historical fact. Now about scribbles changing history: Sure, the special theory of relativity, or quantum mechanics, would be unimaginable today without scribbles. Incidentally, Ulam was a very interesting mathematician. He was an idea man. Most mathematicians like to push their ideas through. He preferred throwing out ideas. His good friend Rota even suggested that he did not have the technical ability or patience to work them out. But if so, then it's an instance of Ulam turning a disability to tremendous advantage. I learned a lot from him.

R & S: It is amazing for us to learn that an eighteen-year-old immigrant was allowed to participate in a top-secret and decisive weapon development during WWII.

Lax: The war created an emergency. Many of the leaders of the Manhattan Project were foreigners, so being a foreigner was no bar.

Collaboration. Work Style

R & S: Your main workplace has been the Courant Institute of Mathematical Sciences in New York, which is part of New York University. You served as its director for an eight-year period in the 1970s. Can you describe what made this institute, which was created by the German refugee Richard Courant in the 1930s, a very special place from the early days on, with a particular spirit and atmosphere? And is the Courant Institute today still a special place that differs from others?

Lax: To answer your first question, certainly the personality of Courant was decisive. Courant saw mathematics very broadly, he was suspicious of specialization. He wanted it drawn as broadly as possible, and that's how it came about that applied topics and pure mathematics were pursued side by side, often by the same people. This made the Courant Institute unique at the time of its founding, as well as in the 1940s, 1950s, and 1960s. Since then there are other centers where applied mathematics is respected and pursued. I am happy to say that this original spirit is still present at the Courant Institute. We still have large areas of applied interest, meteorology and climatology under Andy Majda, solid state and material science under Robert Kohn and others, and fluid dynamics. But we also have differential geometry as well as some pure aspects of partial differential equations, even some algebra. I am very pleased how the Courant Institute is presently run. It's now the third generation that's running it, and the spirit that Courant instilled in it—kind of a family feeling—still prevails. I am happy to note that many Norwegian mathematicians received their training at the Courant Institute and later rose to become leaders in their field.

R & S: You told us already about your collaboration with Ralph Phillips. Generally speaking, looking through your publication list and the theorems and methods you and your collaborators have given name to, it is apparent that you have had a vast collaboration with a lot of mathematicians. Is this sharing of ideas a particularly successful, and maybe also joyful, way of advancing for you?

Lax: Sure, sure. Mathematics is a social phenomenon after all. Collaboration is a psychological and interesting phenomenon. A friend of mine, Vera John-Steiner, has written a book (Creative Collaboration) about it. Two halves of a solution are supplied by two different people, and something quite wonderful comes out of it.

R & S: Many mathematicians have a very particular work style when they work hard on certain problems. How would you characterize your own particular way of thinking, working, and writing? Is it rather playful or rather industrious? Or both?

Lax: Phillips thought I was lazy. He was a product of the Depression, which imposed a certain strict discipline on people. He thought I did not work hard enough, but I think I did!

R & S: Sometimes mathematical insights seem to rely on a sudden unexpected inspiration. Do you have examples of this sort from your own career? And what is the background for such sudden inspiration in your opinion?

Lax: The question reminds me of a story about a German mathematician, Schottky, when he reached the age of seventy or eighty. There was a celebration of the event, and in an interview like we are having, he was asked: "To what do you attribute your creativity and productivity?" The question threw him into great confusion. Finally he said: "But gentlemen, if one thinks of mathematics for fifty years, one must think of something!" It was different with Hilbert. This is a story I heard from Courant. It was a similar occasion. At his seventieth birthday he was asked what he attributed his great creativity and originality to. He had the answer immediately: "I attribute it to my very bad memory." He really had to reconstruct everything, and then it became something else, something better. So maybe that is all I should say. I am between these two extremes. Incidentally, I have a very good memory.

Teaching

R & S: You have also been engaged in the teaching of calculus. For instance, you have written a calculus textbook with your wife Anneli as one of the co-authors. In this connection you have expressed strong opinions about how calculus should be exposed to beginning students. Could you elaborate on this?

Lax: Our calculus book was enormously unsuccessful, in spite of containing many excellent ideas. Part of the reason was that certain materials were not presented in a fashion that students could absorb. A calculus book has to be fine-tuned, and I didn't have the patience for it. Anneli would have had it, but I bullied her too much, I am afraid. Sometimes I dream of redoing it because the ideas that were in there, and that I have had since, are still valid. Of course, there has been a calculus reform movement and some good books have come out of it, but I don't think they are the answer. First of all, the books are too thick, often more than 1,000 pages. It's unfair to put such a book into the hands of an unsuspecting student who can barely carry it. And the reaction to it would be: “Oh, my God, I have to learn all that is in it?" Well, all that is not in it! Secondly, if you compare it to the old standards, Thomas, say, it's not so different—the order of the topics and concepts, perhaps. In my calculus book, for instance, instead of continuity at a point, I advocated uniform continuity. This you can explain much more easily than defining continuity at a point and then say the function is continuous at every point. You lose the students; there are too many quantifiers in that. But the mathematical communities are enormously conservative: “Continuity has been defined pointwise, and so it should be!" Other things that I would emphasize: To be sure there are applications in these new books. But the applications should all stand out. In my book there were chapters devoted to the applications, that's how it should be—they should be featured prominently. I have many other ideas as well. I still dream of redoing my calculus book, and I am looking for a good collaborator. I recently met someone who expressed admiration for the original book, so perhaps it could be realized, if I have the energy. I have other things to do as well, like the second edition of my linear algebra book, and revising some old lecture notes on hyperbolic equations. But even if I could find a collaborator on a calculus book, would it be accepted? Not clear. In 1873, Dedekind posed the important question: "What are, and what should be, the real numbers?" Unfortunately, he gave the wrong answer as far as calculus students are concerned. The right answer is: infinidecimals. I don't know how such a joke will go down.

Heading Large Institutions

R & S: You were several times the head of large organizations: director of the Courant Institute in 1972–1980, president of the American Mathematical Society in 1977–1980, leader of what was called the Lax Panel on the National Science Board in 1980–1986. Can you tell us about some of the most important decisions that had to be taken in these periods?

Lax: The president of the American Mathematical Society is a figurehead. His influence lies in appointing members of committees. Having a wide friendship and reasonable judgement are helpful. I was very much helped by the secretary of the American Mathematical Society, Everett Pitcher. As for being the director of the Courant Institute, I started my directorship at the worst possible time for New York University. They had just closed down their School of Engineering, and that meant that mathematicians from the engineering school were transferred to the Courant Institute. This was the time when the Computer Science Department was founded at Courant by Jack Schwartz. There was a group of engineers that wanted to start activity in informatics, which is the engineers' word for the same thing. As a director I fought very hard to stop that. I think it would have been very bad for the university to have had two computing departments—it certainly would have been very bad for our Computer Science Department. Other things: Well, I was instrumental in hiring Charlie Peskin at the recommendation of Alexander Chorin. I was very pleased with that. Likewise, hiring Sylvain Cappell at the recommendation of Bob Kohn. Both were enormous successes. What were my failures? Well, maybe when the Computer Science Department was founded I should have insisted on having a very high standard of hiring. We needed people to teach courses, but in hindsight I think we should have exercised more restraint in our hiring. We might have become the number one computer science department. Right now the quality has improved very much—we have a wonderful chairwoman, Margaret Wright. Being on the National Science Board was my most pleasant administrative experience. It's a policy-making body for the National Science Foundation (NSF), so I found out what making policy means. Most of the time it just means nodding "yes", and a few times saying “no”. But then there are sometimes windows of opportunity, and the Lax Panel was a response to such a thing. You see, I noticed through my own experience and that of my friends who are interested in large scale computing (in particular, Paul Garabedian, who complained about it), that university computational scientists had no access to the supercomputers. At a certain point the government, which alone had enough money to purchase these supercomputers, stopped placing them at universities. Instead they went to national labs and industrial labs. Unless you happened to have a friend there with whom you collaborated, you had no access. That was very bad from the point of view of the advancement of computational science, because the most talented people were at the universities. At that time accessing and computing at remote sites became possible thanks to ARPANET, which then became a model for the Internet. So the panel that I established made strong recommendation that the NSF establish computing centers, and that was followed up. My quote on our achievement was a paraphrase of Emerson: "Nothing can resist the force of an idea that is ten years overdue."

R & S: A lot of mathematical research in the U.S. has been funded by contracts from DOD (Department of Defense), DOE (Department of Energy), the Atomic Energy Commission, the NSA (National Security Agency). Is this dependence of mutual benefit? Are there pitfalls?

Lax: I am afraid that our leaders are no longer aware of the subtle but close connection between scientific vigor and technological sophistication.

Personal Interests

R & S: Would you tell us a bit about your interests and hobbies that are not directly related to mathematics?

Lax: I love poetry. Hungarian poetry is particularly beautiful, but English poetry is perhaps even more beautiful. I love to play tennis. Now my knees are a bit wobbly, and I can't run anymore, but perhaps these can be replaced—I'm not there yet. My son and three grandsons are tennis enthusiasts so I can play doubles with them. I like to read. I have a knack for writing. Alas, these days I write obituaries—it's better to write them than being written about.

R & S: You have also written Japanese haikus?

Lax: You're right. I got this idea from a nice article by Marshall Stone—I forget exactly where it was—where he wrote that the mathematical language is enormously concentrated, it is like haikus. And I thought I would take it one step further and actually express a mathematical idea by a haiku. (See Peter Lax's haiku below.)

R & S: Professor Lax, thank you very much for this interview on behalf of the Norwegian, the Danish, and the European Mathematical Societies!

Lax: I thank you.

Speed depends on size Balanced by dispersion Oh, solitary splendor.