查尔斯·费夫曼Charles Fefferman

查尔斯·费夫曼的父母是Arthur Stanley Fefferman（1918-2006）和Liselott Ruth Stern（1921-2008）。Arthur Fefferman出生于纽约布鲁克林，1941年在哥伦比亚大学获得经济学硕士学位。他于1945年被任命到华盛顿特区的财政部，正是在华盛顿特区，他的儿子费夫曼于1949年出生。Arthur Fefferman攻读经济学博士学位，并于1950年从社会研究新学院（现新学院大学）获得该学位。Liselott 莫里茨·亚伯拉罕·斯特恩在德国长大，她所就读的学校热衷于严格的纪律，但数学教得很差。因此她讨厌数学。Arthur Fefferman在高中时喜欢数学并且成绩优异，但被告知这对经济学家来说没有必要，甚至在大学里连一门微积分课程都没有修过。然而，当他的两个儿子热情地学习这门学科时，他给予了极大的支持。费夫曼有一个弟弟Robert Fefferman，他于1972年在马里兰大学获得数学学士学位，并于1975年在普林斯顿大学获得博士学位，其学位论文为A Theory of Entropy in Fourier Analysis。

关于费夫曼的父母及其早年生活的更多信息，请参见THIS LINK。

费夫曼，他的朋友和同事都叫他查理，是个神童。他说[41]：-

当我还是个小男孩时，我对儿童科学感兴趣：火箭如何工作之类的事情，但我不满足于简单的解释，所以我从公共图书馆借了一本物理教科书，但我什么都看不懂。我当时九岁，我父亲告诉我：“你当然看不懂这本书，里面有数学！”于是我问他我能不能学数学。我当时上四年级，他给我买了一本四年级数学教科书。那就是开始。

事实上，他在十二岁之前就已经掌握了微积分。他解释了他是如何如此迅速地达到那个水平的[41]：-

我一两天就读完了四年级数学教科书。我父亲不相信我，所以他问了我几个问题，意识到我理解了。然后他给我买了一本五年级教科书，我一两天就读完了，如此继续，直到我学习微积分，那花的时间超过了一两天。但我只是一个小男孩在学习微积分，所以很明显我有天赋。

费夫曼一家住在马里兰州蒙哥马利县银泉，这是华盛顿特区北部的住宅郊区。费夫曼的父母意识到他们十二岁的儿子在数学方面有非凡的天赋，就带他到离家不远的马里兰大学接受辅导。这个故事在[5]中有叙述：-

吉姆·胡梅尔是复变函数方面的权威，1961年7月的一个早晨，他坐在自己狭小的办公室里。他接到了费弗曼家的电话。他们有一个热切的六年级学生，像吃糖果一样消耗数学教科书。胡梅尔教授愿意和他谈谈吗？胡梅尔教授愿意。于是费夫曼太太开着他们可靠的道奇车把查理从银泉送到大学公园。“我不害怕，”查理回忆道，“但我真的很兴奋。”在那个夏天剩下的时间里，两人每周见面一次进行数学讨论。查理确实很聪明，所以胡梅尔邀请他下一个夏天再来。在第二个夏天的某个时候，胡梅尔意识到查理是一个真正的数学家，他已经“像数学家一样思考”。这一点出现在查理对整数的朱塞佩·皮亚诺公理提出了一个深入的问题时。

其他大学教授开始对这位才华横溢的年轻人产生兴趣。他于1962年进入塔科马帕克的东部初中，但从未上过高中，因为他于1963年十四岁时进入了马里兰大学。如此年轻的人入学需要打破大学规定，但教授们艰难地说服当局打破了他们的规定。同年，他的父亲离开政府税收分析办公室，成为后来成为美国人寿保险理事会经济分析主任。费夫曼很快在马里兰大学给人留下了非凡的印象。在弗瑞兹·约翰 Horvath的微积分课上两周后，很明显这些内容对费夫曼来说太基础了，他被安排到约翰 Brace的荣誉微积分课程中。15岁时，也就是他在马里兰大学学习的第二年，他写了他的第一篇数学论文[41]：-

那时我有一位出色的数理逻辑教授：Carol Karp，她对如果你能用无限长的句子说话会怎样感兴趣。有一个问题关于根据无限长句子中哪种无限性可以区分多少不同的事物。我本应在课堂上展示一个非常复杂的解决方案，但我无法理解那个非常复杂的证明，所以我想出了自己的证明，结果发现它可以被推广。我的教授非常支持，说：“你为什么不把这个做出来并进一步深入呢？”然后又说：“你为什么不把它写下来？”之后她说：“我们把它寄给期刊看看会怎样，”结果被接受了。

这就是论文Cardinally maximal sets of non-equivalent order types（1967年）。

在马里兰大学，费夫曼修了许多数学和物理课程。然而，他的教授们想拓展这个年轻男孩，而标准课程肯定做不到这一点，所以他们给了他许多超出班级正常作业的问题和项目。费夫曼意识到他受到了非常特殊的待遇，许多教授给他个人辅导。他说[5]：-

我认为我有幸去那里，我将永远感激数学系做得非常出色。微妙之处在于，系里的成员本可以推我太快，或者另一方面，拖我后腿。他们两者都没有做。我相信他们为我的发展找到了恰到好处的节奏。

1966年，17岁时，他以最高荣誉毕业，获得数学和物理学学士学位。他获得了为期3年的国家科学基金会奖学金进行研究。如果他不是从初中直接上大学，他本会就读的高中是蒙哥马利布莱尔，在这个阶段，学校授予他荣誉文凭。

从费夫曼大学毕业后，费夫曼在普林斯顿大学师从埃利亚斯·施泰因进行研究生工作。费夫曼对他的学位论文导师[41]赞不绝口：-

我的学位论文导师，埃利亚斯·施泰因，对我产生了巨大的影响；我认为他可能是世界上最好的高等数学教师。他影响了我所学的内容，塑造了我的品味，并教会了我面对困难问题时的乐观精神。

他在采访中举了一个例子[37]：-

我记得有一次他和我要用到戈弗雷·哈罗德·哈代和利特尔伍德的某个不等式。它有点晦涩，我们俩都不知道证明。我们去了图书馆（这当然远在互联网之前），戈弗雷·哈罗德·哈代的全集已被借出。我想——将死了，我们得等它还回来。但埃利亚斯·施泰因说：“没关系，如果戈弗雷·哈罗德·哈代和利特尔伍德能证明它，那么我们也能证明它。”我们回到他的办公室，几个小时内我们就得到了证明。那种你能做到的精神——埃利亚斯·施泰因能给你这种精神。他还传达了许多许多其他东西。他有一种绝妙的解释数学的风格，其中几乎所有的解释努力都用于找到正确的观点并解释问题的本质。

费夫曼于1969年因其题为Inequalities for Strongly Regular Convolution Operators的学位论文而获得博士学位。他基于学位论文的材料发表了两篇论文，On some singular convolution operators（1969）和Inequalities for strongly singular convolution operators （1970）。在这两篇论文中，费夫曼都包含了以下致谢：-

我深深感谢我的导师和老师，埃利亚斯·施泰因，感谢他让我注意到这些问题，以及他的许多有益建议和批评。

这些论文并非他的首批出版物，因为除了上面提到的那篇论文外，他已经发表了A Radon-Nikodym theorem for finitely additive set functions （1967）和$L_{p}$ spaces over finitely additive measures（1968），两者都发表在Pacific Journal of Mathematics上。

在普林斯顿读研究生期间，费夫曼 确实有数学之外的兴趣。特别是他对政治感兴趣，当参议员尤金·麦卡锡于1967年11月宣布他将在民主党初选中挑战 罗杰·林登 B 约翰逊时，费夫曼 涉足了他的竞选活动。我们注意到麦卡锡反对美国卷入越南战争。值得指出的是，费夫曼 也支持乔治·麦戈文作为1972年总统选举民主党候选人的竞选活动。

费夫曼 在1969-70年间在普林斯顿讲课。他于1970年转到芝加哥大学担任助理教授，一年后的1971年，他在那里被提升为正教授，这使他成为美国有史以来被任命的最年轻的正教授。他在1970年获得了阿尔弗雷德·P·斯隆基金会奖学金，并在1971年获得了北约博士后奖学金。他在1971年发表了五篇论文，即：On the divergence of multiple Fourier series；Characterizations of bounded mean oscillation；Some maximal inequalities；The multiplier problem for the ball；和 On the convergence of multiple Fourier series。同样在1971年，他被授予 拉斐尔·塞勒姆 奖，奖金为5000法国法郎。该奖项由法国银行家和著名数学家 拉斐尔·塞勒姆 的继承人设立，授予在 约瑟夫·傅里叶 级数理论及相关问题方面有杰出工作的年轻数学家。

1973年，费夫曼 回到普林斯顿，并于1974年10月被介绍给朱莉·Anne Albert。她像费弗曼一样，曾是一个神童，但朱莉是音乐神童，不是数学神童。她年仅9岁时就在茱莉亚学院学习小提琴，当费弗曼遇到她时，她正在纽约一所高中教音乐。他们于1975年结婚，当时朱莉放弃了她的教学生涯。他们有两个女儿，妮娜·海蒂·费夫曼 和莱妮·费夫曼。妮娜成为一名计算生物学家，应用数学来建模复杂的生物系统。莱妮学习音乐，获得普林斯顿大学的博士学位，是一位全国知名的作曲家。

在[5]中，写于Nina出生后但Laine出生前，Carl Bode写道：-

他们住在一栋三卧室的牧场风格房子里；一间小卧室用作Charlie的书房。……为了保持身材，Charlie做加拿大空军体操。

1976年，他被授予Alan T Waterman奖，是第一个获得此奖的人。这是美国国家科学基金会为35岁或以下科学家颁发的最高荣誉奖。该奖项包括一枚奖章和每年50,000美元的资助，为期三年，用于在获奖者选择的美国机构进行高级研究或研究。Feffermen留在普林斯顿，在那里，他能够专注于研究，而不必承担教学职责。

费夫曼通过找到经典低维结果的正确推广，贡献了几项革新，修订了多维复分析的研究。他在偏微分方程、约瑟夫·傅里叶分析方面的工作，特别是收敛性、乘子、发散、奇异积分和戈弗雷·哈罗德·哈代空间，使他在1978年于赫尔辛基举行的国际数学家大会上获得了约翰·查尔斯·菲尔兹奖章。他曾在前一次于1974年8月在温哥华举行的国际数学家大会上作全会报告，当时他作了Recent Progress in Classical Fourier Analysis的演讲。

1979年，费夫曼描述了他喜欢的工作方式（见[5]）：-

我喜欢连续几个小时躺在沙发上，专心思考形状、关系和变化——很少思考数字本身。我在脑海中探索一个又一个想法，丢弃大部分。当一个概念最终看起来有希望时，我就准备在纸上尝试它。但首先我起身给婴儿换尿布。……新想法不容易找到。如果你幸运地正在研究一个实际上正确的想法，可能需要很长时间才能知道它是正确的。相反，如果你走进了一条死胡同，也可能需要很长时间才能发现。你最终可能会说：“哎呀，我多年来一直在研究错误的东西。”一个好的数学家必须有勇气投入大量工作然后将其抛弃。

费夫曼在[3]中以非数学家可能理解的方式描述了他研究的两个主题：-

我想描述我的两项贡献。第一个是挂谷集与约瑟夫·傅里叶分析之间的联系。挂谷集是平面中的奇怪形状。人们可以将一根1英寸长的针旋转整整360度，同时保持针完全在挂谷集内；然而挂谷集的面积可以小到任意程度。约瑟夫·傅里叶分析是研究复杂振动如何分解为简单振动。例如，小提琴弦的复杂运动由基音、第一泛音、第二泛音等组成。如果移除频率，小提琴弦的声音会退化。部分原因是因为小提琴弦是一维的。照片是二维图像，也由类似于弦的基音和泛音的简单部分构建而成。因为照片是二维的，它可能失焦，但当移除其高频时，它会变得清晰聚焦。这是因为挂谷集的存在。我在1970年代发现了这一点。维度高于二的挂谷集继续提出挑战性问题。本书中的照片完美聚焦。其次，我花了多年时间研究关于原子的数学问题。任何量子力学教科书都解释了一个电子和一个质子如何结合成一个氢原子。教科书不会告诉你为什么十亿、十亿、十亿个质子结合成许多氢原子。这是一个更难的问题，涉及大量数学；完整的解决方案尚不清楚。我通过将问题简化为系统能量的估计做出了贡献。

1984年，他被任命为普林斯顿大学的Herbert Jones讲席。费夫曼于1992年被授予斯特凡·伯格曼奖。颁奖委员会的引文如下（见[4]）：-

费夫曼对斯特凡·伯格曼核的研究做出了极其重要的贡献，并引发了该主题的许多活动。……[费夫曼的三篇论文]中的每一篇都是“杰作”，每一篇不仅包含关于斯特凡·伯格曼核及其应用的结果，而且还发展了极具原创性的思想和技术，这些都非常重要……

在他的回复中，费夫曼说[4]：-

我感谢评选委员会授予我斯特凡·伯格曼奖。斯特凡·伯格曼的思想对我的工作产生了重大影响。它们继续为分析提供深刻而重要的问题。

除了上述奖项外，费夫曼还获得了许多荣誉。这些包括1972年当选美国艺术与科学院，1979年当选国家科学院，以及1989年当选美国哲学学会。他获得了马里兰大学（1979年）、诺克斯学院（1981年）、巴伊兰大学（1985年）、马德里大学（1990年）和华威大学（2018年）的荣誉学位。2008年，他被授予美国数学会[1]的马克希莫·博谢纪念奖：-

... 由于他对分析不同领域的许多基础性贡献，包括他最近在哈斯勒·惠特尼扩展问题上的工作。他在这一领域的重要工作包含在他的论文《惠特尼扩展定理的一个尖锐形式》，数学年刊161（2005），509-577，以及《$C^{m}$的惠特尼扩展问题》，数学年刊164（2006），313-359中。

在他的回复中，费夫曼说[1]：-

我感谢被选为马克希莫·博谢奖，并感谢对我关于哈斯勒·惠特尼问题工作的认可。那个问题及其密切相关的问题多年来一直吸引着我。在解决它们时，我得到了来自几个人的美丽、高度原创思想的至关重要的帮助。

2009年，费夫曼成为伦敦数学会的荣誉成员。引文写道[44]：-

费夫曼的贡献和思想对现代分析、微分方程、数学物理和几何学的发展产生了影响，他最近的工作包括对哈斯勒·惠特尼扩张问题的敏锐（可计算）解法。

费夫曼继续在普林斯顿工作，1999年至2002年期间，他担任普林斯顿大学数学系主任。2012年，他与C Robin Graham合作出版了The Ambient Metric一书。出版商的描述如下：-

本书发展并应用了共形几何中的环境度量理论。这是一个n + 2维的亨德里克·洛伦兹度量，它编码了n维中的一个共形度量类。环境度量还有另一种体现形式，即儒勒·昂利·庞加莱度量，这是一个n + 1维的度量，其共形无穷远为共形流形。在这种体现中，该构造在物理学中的AdS/CFT对应中发挥了核心作用。

Michael G Eastwood在评论[21]中写道：-

这本专著已被热切期待了超过25年！[C L 费夫曼和C R Graham，'Astérisque' 1985，Numéro Hors Série，95-116]的发表开创了共形流形上局部不变量理论的革命，正如[C L 费夫曼，'Adv. in Math.' 31 (2) (1979)，131-262]在CR流形上所做的那样。……总的来说，这本细致的阐述值得等待！

截至2023年10月，费夫曼在MathSciNet上列出了约220篇已发表成果。他还通过编辑工作做出了重大贡献，担任以下期刊的编委：《偏微分方程通讯》；《数学进展》；《伊比利亚美洲数学杂志》；《约瑟夫·傅里叶分析与应用杂志》；《美国国家科学院院刊》；以及《分析杂志》。他曾在许多机构担任访问职位，包括：马里兰大学威尔逊·埃尔金斯访问教授；加州理工学院；理查·科朗特数学科学研究所，纽约大学；法国巴黎大学；约斯塔·米塔格-莱弗勒研究所，瑞典尤斯霍尔姆；以色列雷霍沃特魏茨曼研究所；以色列拉马特甘巴伊兰大学；以及西班牙马德里大学。他与西班牙数学界的联系始于他在芝加哥担任年轻教授时，成为他第一个博士生的导师，这位学生是Antonio Córdoba。两人成为亲密朋友，合作发表了多篇论文，与西班牙的紧密联系由此形成。

2017年，费夫曼被授予沃尔夫奖。颁奖词结尾如下[12]：-

费夫曼因在多个领域做出重大贡献而被授予沃尔夫奖，这些领域包括多复变、偏微分方程和次椭圆问题。他将新的基本技巧引入调和分析，并探索了它们在流体动力学、谱几何和数学物理等广泛领域中的应用。这对经典方程，如克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程和莱昂哈德·欧拉方程的正则性问题产生了重大影响。他解决了与偏微分方程解的精细结构相关的重大问题。

关于沃尔夫奖颁奖词的更多细节，见THIS LINK。

2022年，费夫曼被授予BBVA知识前沿奖。费夫曼由马德里自治大学数学分析荣休教授Antonio Córdoba等人提名该奖项。Córdoba在提名中写道，费夫曼[40]：-

……作为一位全才而脱颖而出。数学家通常是在一两个领域做出基础性贡献；费夫曼则在调和分析、偏微分方程、量子力学问题以及流体力学领域做出了基础性贡献，在流体力学中他提出了一个结果，为理解湍流指明了全新的方向。正是这种广阔的视野使费夫曼成为如此杰出的一位数学家。

该奖项的颁奖词指出，费夫曼[40]：-

……引入了开创性的技巧，用于研究函数的精细结构以及偏微分方程解的行为，包括流体动力学中出现的那些方程。

得知获奖消息后，费夫曼在接受采访时解释说，对他而言，在不同领域之间跳跃是第二天性：-

我感觉不是我挑选问题，而是问题挑选我。我听说一个问题，它如此引人入胜，以至于我无法停止思考它。如果它恰好属于我以前没有涉足过的领域，但我认为自己有机会参与其中，或许能做出点什么，那么我就会去尝试。

让我们以费夫曼的女儿Lainie 费夫曼的话作为结束，她于2015年从普林斯顿大学获得音乐博士学位[31]：-

费夫曼正在获得音乐作曲博士学位，她选择让她的父亲费夫曼——Herbert E Jones Jr. '43数学大学讲席教授——为她披上学位服，因为正是他对音乐的热爱启发了她的道路。费夫曼学位论文的最后几章考察了奥地利作曲家Franz Schubert的声乐套曲《冬之旅》的录音，这是她父亲在她童年时期钟爱的一套作品。她回忆父亲跟着这套声乐套曲的录音一起哼唱，是“一段珍贵的记忆，支撑我度过了许多漫长的工作日”。费夫曼说：“从小到大，我父亲向我展示了对优美专辑的热情，甚至在我自己意识到之前。”她的导师是Paul Lansky，William Shubael Conant荣休教授。她说：“我父亲极其欣赏美——无论是优美数学的优雅，还是他周围世界的美学。正是这种对世界的审美化以及这种对美的热爱，我始终随身携带，它是我作为作曲家声音中最大的成分之一。强烈而简单的想法——多亏了我可爱的父亲，它们现在是我的热情所在。”

Charles Fefferman's parents were Arthur Stanley Fefferman (1918-2006) and Liselott Ruth Stern (1921-2008). Arthur Fefferman was born in Brooklyn, New York, and was awarded a Master's Degree in economics from Columbia University in 1941. He was appointed to the Treasury Department in Washington D.C. in 1945 and it was in Washington D.C. that his son Charles was born in 1949. Arthur Fefferman studied for a doctorate in economics which he was awarded in 1950 from the New School for Social Research (now New School University). Liselott Stern was brought up in Germany where the schools she attended were keen on strict discipline but taught mathematics very badly. As a consequence she hated mathematics. Arthur Fefferman had enjoyed and excelled in mathematics at high school but had been told that it was not necessary for an economist and had not taken even a calculus course at university. However, he was extremely supportive of both his sons as they enthusiastically studied the subject. Charles has a younger brother Robert Fefferman who received his bachelor's degree in mathematics from the University of Maryland in 1972 and his Ph.D. from Princeton University in 1975 for his thesis A Theory of Entropy in Fourier Analysis.

For more information about Fefferman's parents and his early years, see THIS LINK.

Charles Fefferman, known to his friends and colleagues as Charlie, was a child prodigy. He said [41]:-

When I was a little boy I was interested in children's science: how rockets work and things like that, but I wasn't satisfied with simple explanations, so I checked a Physics textbook out of the public library and I couldn't understand anything. I was nine years old, and my father told me: "Of course you can't understand this book, there's Mathematics in it!" So I asked him if I could study Mathematics. I was in the fourth grade, and he bought me a 4th grade Mathematics textbook. That was the beginning.

In fact he had mastered the calculus before the age of twelve. He explained how he got so rapidly to that level [41]:-

I read the 4th grade Mathematics textbook in a day or two. My father didn't believe me, so he asked me a few questions and realised that I understood. Then he bought me a 5th grade textbook, and I read it in a day or two, and so on until I studied Calculus and that took longer than a day or two. But I was just a little boy studying Calculus so it was obvious that I had talent.

The Fefferman family were living in Silver Spring, Montgomery County, Maryland, a northern residential suburb of Washington, D.C. Fefferman's parents, realising that their twelve year old son had a remarkable talent for mathematics, took him to the University of Maryland, which was near their home, for tutoring. The story is related in [5]:-

Jim Hummel, an authority on complex variables, was sitting in his tiny office one morning in July 1961. He got a phone call from the Feffermans. They had this avid 6th grader who was consuming mathematics textbooks like candy. Would Professor Hummel talk with him? Professor Hummel would. So Mrs Fefferman drove Charlie from Silver Spring to College Park in their trusty Dodge. "I wasn't scared," Charlie recalls, "but I was really excited." Once a week for the rest of the summer, the two met for mathematical talk. Charlie was bright, indeed, so Hummel invited him back for the next summer. At a certain point during that second summer, Hummel realised that Charlie was a genuine mathematician, that he was already "thinking like a mathematician." The point came when Charlie asked a searching question about the Peano axioms for integers.

The other university professors began to take an interest in the talented youngster. He entered Eastern Junior High in Takoma Park in 1962 but never went to High School for he entered the University of Maryland in 1963 when he was fourteen years old. It required the university rules to be broken for someone so young to enter but, with difficulty, the professors managed to persuade the authorities to break their rules. In the same year his father left the government Office of Tax Analysis to become director of economic analysis for what became the American Council of Life Insurance. Charles was soon making a remarkable impression at the University of Maryland. After two weeks in John Horvath's calculus class it was clear that the material was far too elementary for Fefferman and he was put into John Brace's honours calculus course. At the age of 15, when in his second year of study at the University of Maryland, he wrote his first mathematics paper [41]:-

At that time I had a wonderful professor of mathematical logic: Carol Karp, who was interested in what you might you say if you could speak in infinitely long sentences. There was a question about how many different things could be distinguished depending on what kind of infinity was in the infinitely long sentence. I was supposed to present a very complicated solution in class and I couldn't understand the very complicated proof, so I thought about my own proof and it turned out that it could be generalised. My professor was very supportive and said: "Why don't you work this out and go further?," and then: "Why don't you write it down?" After that she said: "Let's send it to a journal and see what happens," and it was accepted.

This is the paper Cardinally maximal sets of non-equivalent order types (1967).

At the University of Maryland, Fefferman took lots of mathematics and physics courses. However, his professors wanted to stretch the young boy and certainly the standard courses were not doing that so they gave him lots of problems and projects above the normal work of the class. Fefferman realised that he was receiving very special treatment with so many of the professors giving him personal tuition. He said [5]:-

I think I was privileged to go there and I'll always be grateful to the mathematics department for doing a delicate job very well. The delicate part came because the members of the department could have pushed me too fast or, on the other hand, held me back. They didn't do either one. I believe they found exactly the right pace for my development.

In 1966, at the age of 17, he graduated with a B.S., with the highest distinction, majoring in Mathematics and Physics. He was awarded a 3-year National Science Foundation Fellowship to undertake research. The High School he would have attended if he had not gone straight to University from Junior High was Montgomery Blair and, at this stage, the school presented him with an honorary diploma.

After graduating from the University of Maryland, Fefferman undertook postgraduate work at Princeton University supervised by Elias Stein. Fefferman was full of praise for his thesis advisor [41]:-

My thesis advisor, Elias Stein, has had a tremendous influence on me; I think he was probably the best teacher of advanced mathematics in the world. He influenced me about what I learnt and formed my taste, and he taught me his spirit of optimism when facing hard problems.

He gives an example in the interview [37]:-

I remember once he and I had to use some inequality of Hardy and Littlewood. It was a little obscure and neither of us knew the proof. We went to the library (this of course was long before the internet) and the complete works of Hardy had been checked out. I thought - checkmate, we have to wait till it comes back. But Eli said, "Never mind, if Hardy and Littlewood can prove it, then we can prove it too." We went back to his office and in a couple of hours we got the proof. That spirit that you can do it - Eli can give you that. There were many many other things he conveyed. He has a wonderful style of explaining maths in which almost all the effort of explaining goes into finding the right point of view and explaining the essence of the problem.

Fefferman was awarded his PhD in 1969 for his thesis entitled Inequalities for Strongly Regular Convolution Operators. He published two papers based on the material of his thesis, On some singular convolution operators (1969) and Inequalities for strongly singular convolution operators (1970). In both of these papers Fefferman included the following acknowledgement:-

I am deeply grateful to my adviser and teacher, E M Stein, for bringing these problems to my attention, and for his many helpful suggestions and criticisms.

These papers were not his first publications for, in addition to the paper mentioned above, he had already published A Radon-Nikodym theorem for finitely additive set functions (1967) and $L_{p}$ spaces over finitely additive measures (1968), both appearing in the Pacific Journal of Mathematics.

While a graduate student at Princeton, Fefferman did have interests outside mathematics. In particular he was interested in politics and, when Senator Eugene McCarthy announced in November 1967 that he would challenge Lyndon B Johnson in the Democratic primaries, Fefferman dabbled in his campaign. We note that McCarthy opposed American involvement in the Vietnam War. It is worth pointing out that Fefferman also supported George McGovern's campaign as the Democratic candidate for the 1972 presidential election.

Fefferman lectured at Princeton for the years 1969-70. He moved to the University of Chicago in 1970 as an assistant professor and, one year later in 1971, he was promoted to full professor there, earning him the distinction of becoming the youngest full professor ever appointed in the United States. He had a Alfred P Sloan Foundation Fellowship in 1970 and a NATO Postdoctoral Fellowship in 1971. He had published five papers in 1971, namely: On the divergence of multiple Fourier series; Characterizations of bounded mean oscillation; Some maximal inequalities; The multiplier problem for the ball; and On the convergence of multiple Fourier series. Also in 1971 he was awarded the Salem Prize of 5,000 French Francs. This prize was established by the heirs of Raphael Salem, the French banker and famous mathematician, to be awarded to a young mathematician for outstanding work connected with the theory of Fourier series and related problems.

In 1973 Fefferman returned to Princeton and, in October 1974, he was introduced to Julie Anne Albert. She, like Feffermen, had been a child prodigy but Julie was a musical prodigy not a mathematical one. She had studied violin at the Juilliard School when she was only 9 years of age and, when Feffermen met her, she was teaching music in a New York high school. They were married in 1975 and at that time Julie gave up her teaching career. They had two daughters, Nina Heidi Fefferman and Lainie Fefferman. Nina became a computational biologist who applies mathematics to model complex biological systems. Lainie studied music, was awarded a Ph.D. by Princeton, and is a nationally known composer.

In [5], written after the birth of Nina but before the birth of Laine, Carl Bode writes:-

They live in a three-bedroom ranch style house; one small bedroom serves as Charlie's study. ... To stay in shape Charlie does Canadian air force exercises.

In 1976 he was awarded the Alan T Waterman Award, being the first to receive this award. It is the United States' highest honorary award for scientists 35 years of age or younger selected by the National Science Foundation. The award consisted of a medal and a grant of $50,000 a year for three years for advanced study or research at a U.S. institution of the recipient's choice. Feffermen remained at Princeton where, for three years, he was able to concentrate on research and not have teaching duties.

Fefferman contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalisations of classical low-dimensional results. His work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978. He had been a plenary speaker at the previous International Congress of Mathematicians held in Vancouver in August 1974 when he gave the lecture Recent Progress in Classical Fourier Analysis.

In 1979 Fefferman described the way that he liked to work (see [5]):-

I like to lie down on the sofa for hours at a stretch thinking intently about shapes, relationships and change - rarely about numbers as such. I explore idea after idea in my mind, discarding most. When a concept finally seems promising, I'm ready to try it out on paper. But first I get up and change the baby's diaper. ... New ideas are not easy to find. If you are lucky enough to be working on an idea which is actually right, it can take a long time before you know that it's right. Conversely, if you are going up a blind alley, it can also take a long time before you find out. You can end up saying "Oops, I've been working for years on something wrong." A good mathematician must have the courage to take a lot of work and throw it away.

Fefferman described two topics that he worked on in a way that might be understood by non-mathematicians in [3]:-

I would like to describe two of my contributions. The first is a connection between Kakeya sets and Fourier analysis. Kakeya sets are strange shapes in the plane. One can turn a 1-inch-long needle through a full 360 degrees, keeping the needle entirely inside a Kakeya set; yet the area of a Kakeya set is as small as you please. Fourier analysis is the study of how complicated vibrations break up into simple ones. For example, the complicated motion of a violin string is made up of a fundamental note, a first overtone, a second overtone, and so on. The sound of the violin string is degraded if the frequencies are removed. In part, that's because the violin string is one-dimensional. A photograph is a two-dimensional image, also built up from simple pieces analogous to the fundamental note and overtones of a string. Because a photo is two-dimensional, it may be out of focus yet come into sharp focus when its high frequencies are removed. That's because of the existence of Kakeya sets. I discovered this in the 1970s. Kakeya sets in dimension higher than two continue to present challenging problems. The photos in this book are in perfect focus. Secondly, I've spent many years on mathematical problems about atoms. Any quantum mechanics textbook explains why one electron and one proton combine to make one hydrogen atom. The textbook won't tell you why a billion, billion, billion protons combine to make lots of hydrogen atoms. That's a much harder problem, which entails a lot of mathematics; the full solution isn't yet known. I made a contribution by reducing the problem to an estimate for the energy of the system.

In 1984 he was appointed Herbert Jones Professor at Princeton. Fefferman was awarded the Bergman Prize in 1992. The citation from the awarding committee read (see [4]):-

Charles Fefferman has made enormously important contributions to the study of the Bergman kernel and has initiated much of the activity in the topic. ... Each of [three of Fefferman's papers] is a 'tour-de-force', each contains not only results about the Bergman kernel and its applications but each also develops highly original ideas and techniques which are of great importance...

In his reply Fefferman said [4]:-

I am grateful to the selection committee for awarding me the Bergman Prize. Bergman's ideas have been a major influence in my work. They continue to provide deep, important problems for analysis.

In addition to the awards mentioned above, Fefferman had received many honours. These include election to the American Academy of Arts and Sciences (1972), election to the National Academy of Science in 1979 and election to the American Philosophical Society in 1989. He has received honorary degrees from the University of Maryland (1979), Knox College (1981), Bar-Ilan University (1985), the University of Madrid (1990), and the University of Warwick (2018). In 2008 he was awarded the Bôcher Memorial Prize from the American Mathematical Society [1]:-

... for his many fundamental contributions to different areas of analysis, including his recent work on the Whitney extension problem. His important work in this area is contained in his papers 'A sharp form of Whitney's extension theorem', Annals of Math. 161 (2005), 509-577, and 'Whitney's extension problem for $C^{m}$', Annals of Math. 164 (2006), 313-359.

In his reply Fefferman said [1]:-

I am grateful for my selection for the Bôcher Prize and for the recognition of my work on Whitney's problem. That question and its close relatives have fascinated me for years. In solving them, I've had crucial help in the form of beautiful, highly original ideas due to several people.

In 2009 Fefferman was made an honorary member of the London Mathematical Society. The citation reads [44]:-

Professor Charles Fefferman's contributions and ideas have had an impact on the development of modern analysis, differential equations, mathematical physics and geometry, with his most recent work including his sharp (computable) solution of the Whitney extension problem.

Fefferman continues to work at Princeton and from 1999 to 2002 he was Chair of the Department of Mathematics at Princeton University. In 2012, in collaboration with C Robin Graham, he published the book The Ambient Metric. The publisher's description is as follows:-

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n + 2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n + 1 dimensions having the conformal manifold as its conformal infinity. In this realisation, the construction has played a central role in the AdS/CFT correspondence in physics.

Michael G Eastwood writes in a review [21]:-

This monograph has been keenly anticipated for more than 25 years! The publication of [C L Fefferman and C R Graham, 'Astérisque' 1985, Numéro Hors Série, 95-116] initiated a revolution in the theory of local invariants on a conformal manifold just as [C L Fefferman, 'Adv. in Math.' 31 (2) (1979), 131-262] had done on CR manifolds. ... Overall, this careful exposition has been well worth the wait!

As of October 2023, Fefferman has around 220 published items listed in MathSciNet. He has also made major contributions through his editorial work being on the editorial boards of: Communications in Partial Differential Equations; Advances in Mathematics; Revista Mat. Iberoamericana; Journal of Fourier Analysis and Applications; Proceedings of the National Academy of Sciences; and Journal d'Analyse. He has held Visiting Positions at many institutions including: Wilson Elkins Visiting Professorship, University of Maryland; California Institute of Technology; Courant Institute of Mathematical Sciences, New York University; University of Paris, France; Mittag-Leffler Institute, Djursholm, Sweden; Weitzmann Institute, Rehovot, Israel; Bar-Ilan University, Ramat-Gan, Israel; and University of Madrid, Spain. His relation with the mathematical community of Spain began when as a young professor at Chicago he became advisor to his first Ph.D. student who was Antonio Córdoba. The two became close friends, collaborated on various papers and the strong link with Spain was formed.

In 2017 Fefferman was awarded the Wolf Prize. The citation ends as follows [12]:-

Charles Fefferman is awarded the Wolf Prize for making major contributions to several fields, including several complex variables, partial differential equations and subelliptic problems. He introduced new fundamental techniques into harmonic analysis and explored their application to a wide range of fields including fluid dynamics, spectral geometry and mathematical physics. This had a major impact on regularity questions for classical equations such as the Navier-Stokes equation and the Euler equation. He solved major problems related to the fine structure of solutions to partial differential equations.

For more details of the citation for the Wolf Prize, see THIS LINK.

In 2022 Fefferman was awarded the BBVA Frontiers of Knowledge award. Fefferman was nominated for the award by, among others, Antonio Córdoba, Professor Emeritus of Mathematical Analysis at the Universidad Autónoma de Madrid. In his nomination Córdoba wrote that Fefferman [40]:-

... stands out as an all-rounder. The normal thing is for a mathematician to make fundamental contributions in one or two areas; Fefferman has made them in harmonic analysis, partial differential equations, problems in quantum mechanics, and in the field of fluid mechanics, where he came up with a result that suggested a whole new tack for understanding turbulence.It is this broad sweep that makes Fefferman such an exceptional mathematician.

The citation for the award states that Fefferman [40]:-

... has introduced ground-breaking techniques to study the detailed structure of functions and the behaviour of solutions to partial differential equations, including those arising in fluid dynamics.

In an interview after hearing of the award, Fefferman explained that, for him, jumping between fields is second nature:-

I have the feeling that I don't pick the problems, they pick me. I hear about a problem and it is so fascinating that I cannot stop thinking about it. If it happens to be in a field I have not worked in before but I think I have a chance to get involved and maybe do something, then I try.

Let us end with the words of Fefferman's daughter Lainie Fefferman when graduating with a Ph.D. in music from Princeton University in 2015 [31]:-

Lainie Fefferman, who is receiving her Ph.D. in music composition, elected to have her father, Charles Fefferman, the Herbert E Jones Jr. '43 University Professor of Mathematics, present her hood as it was his love of music that inspired her path. The final chapters of Lainie Fefferman's dissertation examined recordings of the song cycle "Winterreise" by Austrian composer Franz Schubert, which was a beloved collection of her father's during her childhood. Recollections of him singing along to a recording of the song cycle are "a treasured memory that kept me going through many long workdays," she said. "Growing up, my dad showed me a passion for beautiful albums even before I knew I had any myself," said Lainie Fefferman, whose adviser was Paul Lansky, the William Shubael Conant Professor, Emeritus. "My dad is a huge appreciator of beauty - in the elegance of good math and in the aesthetics of the world around him," she said. "It is that aestheticizing of the world and that love of beauty I carry with me always, and it is one of the biggest ingredients in my voice as a composer. Strong, simple ideas - they're my passion now thanks to my lovely dad."