路易斯·卡法雷利Luis Caffarelli

路易斯·卡法雷利的父母是卡法雷利（1915年3月18日出生于阿根廷布宜诺斯艾利斯）和Hilda Delia Cespi（1922年9月9日出生于阿根廷布宜诺斯艾利斯）。卡法雷利 Sr是一名机械工程师，在拉普拉塔湾的航运业工作。卡法雷利和Hilda有三个孩子：Maria Luisa 卡法雷利、卡法雷利 Ángel 卡法雷利（本传记的主人公）和Alicia Caffarelli。卡法雷利在采访[19]中谈到了他的早年经历：-

我们住在布宜诺斯艾利斯一个不错的中产阶级区。在我还是个孩子的时候，我经常和朋友们踢足球。我们很喜欢的另一个游戏是扔一个球或其他东西，看谁能让它最接近墙壁或一条线。

他就读于布宜诺斯艾利斯国立学院，该校最初是17世纪的一所耶稣会学校。作为一所国立学院，它的历史可追溯到1863年，并于1911年并入布宜诺斯艾利斯大学。到卡法雷利在那里学习时，它已在人文和科学方面赢得了极好的声誉。优秀的数学和科学教师使这些科目成为他最喜欢的科目。他于1966年从国立学院毕业，并于1967年3月进入布宜诺斯艾利斯大学。他写道[12]：-

当时军政府领导着国家，许多顶尖科学家都在国外。尽管如此，大学里剩下的教职员工和学生都非常敬业。

他在[19]中谈到了他在大学的讲师：-

作为一名学生，我深受Luis Santaló（1911-2001）、Manuel Balanzat（1912-1994）和Carlos Segovia（1937-2007）的影响和启发。Santaló和Balanzat都是因西班牙内战而移居阿根廷的西班牙数学家。Santaló在积分几何和几何概率方面做出了重要贡献，而Balanzat则从事泛函分析研究。他们与Rey Pastor（1888-1962）和Pi Calleja（1907-1986）共同在布宜诺斯艾利斯大学建立了一个极好的本科和研究生数学课程，形成了一个非常强大的分析、几何和代数几何团队。调和分析学家Segovia是布宜诺斯艾利斯大学的杰出毕业生，他于1967年在芝加哥大学在阿尔伯托·考尔德伦（1920-1988）的指导下获得博士学位。虽然Segovia在年龄上比Santaló和Balanzat更接近我，但他一直是一个强有力的支持者。

他大约有一年半的时间同时学习数学和物理，但最终他选择完全专注于数学。他于1969年获得理学硕士学位，并继续攻读博士学位的研究。他写道[15]：-

我接受的是Calderón学派实分析的训练，我的博士学位论文和一些其他文章是关于特殊多项式级数的可和性与共轭的。

我们应该稍微谈谈“Calderón学派”。有两位同父异母的兄弟都姓Calderón，即Calixto Pedro Calderón和他年长得多的同父异母哥哥阿尔伯托·考尔德伦。两人有同一个父亲Pedro Calderón，但在Alberto的母亲去世后，Pedro再婚娶了Matilde Garcia，她正是Calixto的母亲。Calixto生于1939年12月，比阿尔伯托·考尔德伦小20岁。他于1969年从布宜诺斯艾利斯大学获得博士学位，仅比卡法雷利早两年，而卡法雷利是他的第一位博士学生。1974年，Calixto Calderón前往伊利诺伊大学芝加哥分校，在那里工作直到2000年退休。

1971年，卡法雷利向布宜诺斯艾利斯大学提交了他的学位论文Sobre conjugación y sumabilidad de series de Jacobi Ⓣ（论卡尔·古斯塔夫·雅各布·雅可比级数的共轭与可和性），并于1972年被授予博士学位。他与他的学位论文导师Calixto Pedro Calderón进行了联合研究，他们合写了两篇论文：Weak type estimates for the Hardy-Littlewood maximal functions（1974-74）；以及On Abel summability of multiple Jacobi series （1974）。

在布宜诺斯艾利斯期间，他与Irene Martínez Gamba（生于1957年）结婚，她也是一位杰出的数学家；他们有三个孩子Alejandro 卡法雷利、Nicolas Caffarelli和Mauro Caffarelli。

1973年，在获得博士后奖学金后，他前往明尼苏达大学，与Eugene Fabes和Calixto Calderón合作；他在那里一直待到1983年。他说[56]：-

从布宜诺斯艾利斯的炎热夏天到明尼苏达的冬天是一个巨大的变化，但在Fabes和Walter Littman的慷慨帮助下，我爱上了明尼阿波利斯。

2009年，卡法雷利谈到了他1973年到达明尼苏达大学的情况[1]：-

我于1973年1月来到美国明尼苏达大学。那时没有电子邮件，没有传真，甚至电话也非常昂贵。但我在明尼苏达和中西部发现了一群非凡的人。我的同事们极其慷慨、专注和友好，他们教会了我所知道的许多东西。当我开始我的研究计划时，他们分享想法并给予我指导。

在1963-74年于明尼苏达大学担任博士后研究员之后，他于1975年被任命为助理教授，1977年升为副教授，1979年成为正教授。他在[15]中写到了他在明尼苏达大学期间所进行的研究，试图向普通读者解释他的高深成果：-

我刚到不久，就参加了汉斯·卢伊的一系列精彩讲座，并对非线性偏微分方程、变分不等式和自由边界问题产生了兴趣。这类问题的一个简单例子是盒子里的气球（或空腔中的液滴）。如果气球自由悬浮在空气中，其形状的首次近似将由一个规定的平均曲率方程（一个轻度非线性方程）给出，我们可以从气球试图最小化构型能量这一事实推导出来（一个无约束变分问题）。如果被限制在盒子内，气球的表面在自由时和压在壁上时的行为会有所不同（一个强非线性微分方程），在两者之间形成一条分离曲线（自由边界）。在这一领域，我广泛研究了与固-液界面、射流和空化流以及多孔介质中的气体和液体流动相关的数学问题。

卡法雷利的论文Surfaces of minimum capacity for a knot发表于1975年，但第二年对卡法雷利来说在发表方面是显著的一年，因为那一年他发表了六篇论文：Certain multiple valued harmonic function；On the Hölder continuity of multiple valued harmonic functions；The regularity of elliptic and parabolic free boundaries；（与Néstor M Rivière合作）On the rectifiability of domains with finite perimeter；（与Néstor M Rivière合作）Smoothness and analyticity of free boundaries in variational inequalities；以及The smoothness of the free surface in a filtration problem。他继续保持着惊人的发表记录，1977年有三篇论文，其中两篇是与Néstor M Rivière合写的，1978年有五篇论文，其中四篇是与Avner Friedman合写的。

卡法雷利继续在明尼苏达大学担任教授直到1983年，他在1980-82两年间担任理查·科朗特研究所的教授。这导致了他的研究兴趣发生了变化，正如他在[15]中所解释的：-

1980年，我受邀加入理查·科朗特研究所的教职，在那里我在路易·尼伦伯格的建议和合作下发展了新的兴趣：流体动力学和完全非线性方程。我们一直研究的一个领域是三维克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯流（粘性不可压缩流体流动演化的模型），我们证明了流动的速度至多在空间和时间中的一维测度为零的集合上（即小于一条曲线）变为无穷大。（根据Shefer最近的例子，这是一个近乎最优的结果）。我们还广泛研究了n维欧几里得空间中超曲面的性质，其中规定了它们主曲率之间的椭圆关系（即平均曲率方程、实或复空间中的加斯帕尔·蒙日-安德烈-马里·安培方程等）。

1982年，卡法雷利被比萨高等师范学校授予圭多·史坦穆奇亚奖（详情见[62]）。该奖项联合授予了六位数学家，卡法雷利及其合著者，以表彰这六篇系列论文：Brownian Motion and Harnack inequality for Schrödinger operators；Existence and regularity for a minimum problem with free boundary；Axially symmetric jet flows；Asymmetric jet flows；Jet flows with gravity；以及Asymptotic behaviour of minimum problems with bilateral obstacles。这是授予卡法雷利的众多奖项和奖金中的第一个。

1983年，他被任命为芝加哥大学教授，担任该职位三年。在这些年里，他继续因其杰出贡献而获得重大奖项。1984年，他被美国数学会授予马克希莫·博谢奖[44]：-

... 因他在非线性偏微分方程方面的深刻和基础性工作，特别是他在自由边界问题、涡旋理论和正则性理论方面的工作。

1986年，卡法雷利 离开芝加哥，被任命为普林斯顿高等 爱德华·斯图迪 研究所教授。1988年10月31日，教皇 弗瑞兹·约翰 保罗二世向 卡法雷利 颁发了 宗座科学院 庇护十一世金质奖章。我们在上文已多次引用 卡法雷利 在那一场合的演讲。这里让我们引用他关于自己当时（1988年）研究兴趣所说的话 [15]：-

我当前的研究兴趣包括在接近奇异极限时对奇异扰动的均匀估计，例如对于接近不连续面的水平面及其对不连续面稳定性和用于模拟极限问题的数值方法的稳定性的影响。

卡法雷利在高等斯图迪研究院工作了十年，然后于1994年，因为他想念与研究生一起工作，他回到了理查·科朗特研究所，在那里度过了三年。1997年，他搬到德克萨斯大学奥斯汀分校，被任命为Sid W Richardson基金会董事讲席。奖项继续表彰他研究贡献的极高质量。2003年，他被授予阿根廷Premio Konex, Platino y Brillantes[43]：-

... 作为过去十年中他国家最重要的科学家。

Swedish Academy of Sciences于2005年授予他罗尔夫·肖克奖[52]：-

... 因其对非线性偏微分方程理论的重要贡献。

American Mathematical Society 于2009年授予他 Leroy P 凯瑟琳·斯蒂尔 数学终身成就奖。颁奖词开头写道 [1]：-

卡法雷利是研究非线性偏微分方程（PDE）的世界最伟大的数学家之一。这是一个困难的领域：非线性偏微分方程的解很少有精确公式，精确的代数计算也很少能得出有用的表达式。相反，研究者通常必须借助泛函分析来为许多重要方程构建“广义”解。剩下的便是证明这些弱解正则性的深刻且极其技术性的问题，而公认的正则性理论最高权威是卡法雷利。他的突破如此之多，又如此技术性，以至于无法在此简单复述。但正是卡法雷利关于“自由边界”问题的工作，首先展现了他的深刻洞见。自由边界问题不仅要求找到某些偏微分方程的解，还要求找到方程成立的那个区域本身。卡法雷利的大量工作完全主导了这一领域，从他早期关于障碍问题的论文开始。...

2012年，他当选为美国数学会会士。同年，卡法雷利获得了著名的沃尔夫数学奖[23]：-

... 因其在偏微分方程方面的工作。

导致他获得沃尔夫奖的工作在23中有描述：-

卡法雷利屡次取得非常深刻的突破。他早期关于自由边界问题的工作，是首次展现其非凡才能与直觉的领域。自由边界问题关乎寻找方程的解以及方程成立的区域。在一系列开创性论文中，卡法雷利提出了一种新颖的方法论，经过若干真正令人惊叹的技术估计——这些估计逐步改进了解与边界的正则性——最终在非常温和的假设下导向完全正则性。尽管理论复杂，论证却是初等的，充满了优美的几何直观和对分析技巧的掌握。卡法雷利的第二个基础性贡献是对完全非线性椭圆偏微分方程（包括著名的加斯帕尔·蒙日-安德烈-马里·安培方程）的研究，他对此进行了革命性的改造。结果是，尽管方程是非线性的，但就正则性而言，它们的行为如同线性方程一样。……卡法雷利的另一个基础性贡献是他与Kohn及路易·尼伦伯格关于三维空间中不可压缩克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程解的部分正则性的合作工作。尽管解的完全正则性仍然未知且可能极为困难，卡法雷利-Kohn-路易·尼伦伯格证明了奇异集必须具有严格小于一的抛物费利克斯·豪斯多夫维数。特别地，奇异纤维不可能出现。……卡法雷利还在均匀化以及具有非局部耗散的方程方面做出了深刻的工作。这样的例子还可以继续列举。卡法雷利是偏微分方程解的正则性方面的世界顶尖专家。

2018年，卡法雷利成为工业与应用数学学会（SIAM）的会士。引文指出[47]：-

卡法雷利，德克萨斯大学奥斯汀分校，因在非线性偏微分方程正则性理论、自由边界问题、完全非线性方程及非局部扩散方面的开创性贡献而受到表彰。

同年2018年，他荣获由总部位于香港的邵逸夫基金会颁发的邵逸夫数学科学奖[8]：-

……因其在偏微分方程方面的开创性工作，包括为诸如加斯帕尔·蒙日-安德烈-马里·安培方程等非线性方程以及诸如障碍问题等自由边界问题创建正则性理论，这些工作影响了该领域整整一代研究者。

该奖项是世界上金额最大的数学奖项之一，奖金为120万美元。

除上述荣誉外，卡法雷利还获得了许多荣誉。他先后当选为美国艺术与科学院（1986年）、National Academy of Sciences（1991年）、Pontifical Academy of Sciences、Argentina Mathematical Union、Accademia Nazionale dei Lincei等机构的成员。他应邀在比萨高等师范学校作恩里科·费米讲座（1998年），并于1993年担任美国数学会学术报告会主讲人。Pontifical Academy of Sciences对其研究的总结如下[32]：-

卡法雷利从事非线性分析研究，主要研究源于几何和力学的非线性偏微分方程。他对自由边界问题和奇异摄动问题进行了广泛研究。他研究过当本构关系或守恒量（温度、压力、密度）在所考虑变量的某个值处发生不连续变化时自然出现的自由边界问题。典型例子包括固液界面、火焰传播中的已燃与未燃混合物以及多孔介质中的流动。理解解的几何结构及其界面的稳定性对于选择和评估模拟方法以及理解模型本身都很重要。另一个研究领域是完全非线性方程和最优传输。完全非线性方程出现在优化和最优控制中。最近人们还结合最优传输和最优天线设计对它们进行了研究。其他感兴趣的领域包括不可压缩流、调和映射和极小曲面理论，以及最近关于非线性随机均匀化的研究。

尼尔斯·阿贝尔奖被公认为授予数学家的最高奖项。该奖于2023年颁发给卡法雷利[16]：-

……因其对非线性偏微分方程正则性理论的奠基性贡献，包括自由边界问题和加斯帕尔·蒙日-安德烈-马里·安培方程。

关于新闻稿、引文以及卡法雷利获得该奖项的其他信息，请参见THIS LINK。

在访谈57中，路易·尼伦伯格被问到：“卡法雷利作为数学家是什么样的人？”他回答说：-

非凡的直觉，非常了不起。我们现在已经好几年没有合作了，但当我们合作时，我很难跟上他。他不知怎么地立刻看到别人看不到的东西，但他解释起来有困难。他说得少，写得也少，所以当我们在黑板上工作时，我总是说：“卡法雷利，请多写点，多写下来。”有一次我对他说：“卡法雷利，用《圣经》的话说，‘经上记在哪里呢？’”有人说他曾听过一个报告，其中卡法雷利证明了偏微分方程中的某个东西——什么都没用！不知怎么地，他就能凭空想出想法。他真的很了不起——而且是个非常好的人。

卡法雷利工作的数量和质量都令人震惊。我们提供截至2023年9月底的数据。MathSciNet列出了卡法雷利的322篇出版物，其中三篇出现在2022年，一篇在2023年。这些论文在7,236位不同作者的11,499篇出版物中被引用20,200次。他指导了30多名博士生。在提供这些数据时，我们注意到他的妻子Irene Martínez Gamba在MathSciNet中列出了99篇出版物，最早的一篇发表于1986年。这99篇论文在975位不同作者的954篇出版物中被引用1,715次。她于1989年在芝加哥大学获得博士学位，学位论文为Asymptotic behavior at the boundary of a semiconductor device in two space dimensions。她拥有德克萨斯大学W A Tex Moncrief, Jr.计算工程与科学III讲席。

卡法雷利令人难以置信的数学产出，很难看出他有多少时间用于爱好。在访谈19中被问及这些时，他说：-

我喜欢做饭，但我妻子是比我更好的厨师。如果有时间，我也喜欢踢足球。我弹钢琴。我主要喜欢古典音乐。

让我们以卡法雷利在[1]中的一句话结束：-

多年来，我有机会加入一些出色的机构，并与世界各地非凡的科学家结交和合作。这带来了更多机会，让我指导非常有才华的年轻人，他们用新思想激发了我的研究。

Luis Caffarelli's parents were Luis Caffarelli (born 18 March 1915 in Buenos Aires, Argentina) and Hilda Delia Cespi (born 9 September 1922 in Buenos Aires, Argentina). Luis Caffarelli Sr was a mechanical engineer who worked in the shipping industry in the Río de la Plata bay. Luis and Hilda had three children, Maria Luisa Caffarelli, Luis Ángel Caffarelli (the subject of this biography) and Alicia Caffarelli. Luis spoke about his early years in the interview [19]:-

We lived in a nice middle class area of Buenos Aires. At the time when I was a kid I would play soccer a lot with my friends. Another game we enjoyed a lot was to throw a ball, or something else, and see who got it closest to a wall or to a line.

He attended the Colegio Nacional de Buenos Aires which began its existence as a Jesuit school in the 17th century. As a National College it dates from 1863, and was incorporated into the University of Buenos Aires in 1911. By the time Caffarelli studied there it had gained an excellent reputation for both Humanities and Sciences. Excellent teachers of mathematics and science led to these topics becoming the subjects he most enjoyed. He graduated from the Colegio Nacional in 1966 and entered the University of Buenos Aires in March 1967. He wrote [12]:-

The military led the country at that time and many of the top scientists were abroad. Still, the remaining faculty and students at the university were very dedicated.

He spoke in [19] about his lecturers at the University:-

As a student I was heavily influenced and inspired by Luis Santaló (1911-2001), Manuel Balanzat (1912-1994) and Carlos Segovia (1937-2007). Santaló and Balanzat were both Spanish mathematicians who moved to Argentina as a consequence of the Spanish Civil War. Santaló made important contributions to integral geometry and geometric probability, while Balanzat worked in functional analysis. They built, jointly with Rey Pastor (1888-1962) and Pi Calleja (1907-1986), a superb undergraduate and graduate mathematics programme at the University of Buenos Aires, generating a very strong group in analysis, geometry and algebraic geometry. The harmonic analyst Segovia was a prominent graduate from Universidad de Buenos Aires who did his PhD at the University of Chicago in 1967 with Alberto Calderón (1920-1988). While closer to me in age than Santaló and Balanzat, Segovia was always a strong support.

For about a year and a half he studied both mathematics and physics but finally he chose to concentrate totally on mathematics. He was awarded his Master of Science degree in 1969 and continued undertaking research for a Ph.D. degree. He wrote [15]:-

I was trained in the Calderón school of real analysis and wrote my Ph.D. dissertation and some other articles on summability and conjugation of series of special polynomials.

We should say a little about the 'Calderón school'. There are two half-brothers named Calderón, namely Calixto Pedro Calderón and his much older half-brother Alberto Pedro Calderón. Both had the same father, Pedro Calderón, but after Alberto's mother died, Pedro remarried Matilde Garcia who was Calixto's mother. Calixto, born in December 1939, was 20 years younger than Alberto Calderón. He was awarded his doctorate in 1969 from the University of Buenos Aires, only two years before Luis Caffarelli who was his first Ph.D. student. In 1974 Calixto Calderón went to the University of Illinois at Chicago where he worked until he retired in 2000.

In 1971 Caffarelli submitted his thesis Sobre conjugación y sumabilidad de series de Jacobi Ⓣ to the University of Buenos Aires and he was awarded a Ph.D. in 1972. He undertook joint research with his thesis advisor Calixto Pedro Calderón and they wrote two joint papers: Weak type estimates for the Hardy-Littlewood maximal functions (1974-74); and On Abel summability of multiple Jacobi series (1974).

While in Buenos Aires he married Irene Martínez Gamba (born 1957) who is also an exceptional mathematician; they have three children Alejandro Caffarelli, Nicolas Caffarelli and Mauro Caffarelli.

In 1973, having been awarded a postdoctoral fellowship, he went to the University of Minnesota to work with Eugene Fabes and Calixto Calderón; he remained there until 1983. He said [56]:-

From the hot Buenos Aires summer to the Minnesota winter was a dramatic change, but with the generous help of Fabes and Walter Littman I got to love Minneapolis.

In 2009 Caffarelli spoke about his arrival at the University of Minnesota in 1973 [1]:-

I came to the United States to the University of Minnesota in January of 1973. There was no email, no fax, and even the telephone was very expensive. But I found at Minnesota and in the midwest an extraordinary group of people. My colleagues were extremely generous, dedicated, and friendly, and they taught me much of what I know. They shared their ideas and gave me guidance as I began my research programme.

After the Postdoctoral Fellowship which he held at the University of Minnesota in 1963-74, he was appointed as an Assistant Professor in 1975, an Associate Professor in 1977, and a full professor in 1979. He wrote in [15] about the research he undertook while at the University of Minnesota attempting to explain his high-powered results to a general audience:-

Shortly after my arrival, I attended a fascinating series of lectures by Hans Lewy and became interested in nonlinear partial differential equations, variations inequalities and free-boundary problems. An elementary example of this type of a problem would be a balloon inside a box (or a drop inside a cavity). If the balloon were suspended freely in the air, a first approximation to its shape would be given by a prescribed mean curvature equation (a mildly non-linear equation) that we could deduce from the fact that the balloon tries to minimise the energy of the configuration (an unconstrained variational problem). If constrained to lie inside the box, the surface of the balloon would behave differently when it is free than when it presses against the wall (a strongly nonlinear differential equation) creating a separation curve (the free boundary) between both regions. In this area, I investigated extensively the mathematical problems associated with solid-liquid interphases, jet and cavitational flows, and gas and liquid flow in a porous media.

Caffarelli's paper Surfaces of minimum capacity for a knot was published in 1975 but it was the following year that proved a remarkable one for Caffarelli in terms of publications for in that year six of his papers were published: Certain multiple valued harmonic function; On the Hölder continuity of multiple valued harmonic functions; The regularity of elliptic and parabolic free boundaries; (with Néstor M Rivière) On the rectifiability of domains with finite perimeter; (with Néstor M Rivière) Smoothness and analyticity of free boundaries in variational inequalities; and The smoothness of the free surface in a filtration problem. He continued his remarkable publication record with three papers in 1977, two of which were written jointly with Néstor M Rivière, and five papers in 1978, four of which were written jointly with Avner Friedman.

Although Caffarelli continued to hold his professorship at the University of Minnesota until 1983, he spent the two years 1980-82 as a Professor at the Courant Institute. This brought about a change in his research interests as he explained in [15]:-

In 1980 I was invited to join the faculty at the Courant Institute, where I developed new interests: fluid dynamics and fully nonlinear equations under the advice and in collaboration with Louis Nirenberg. A standing area of research we pursued was the three-dimensional Navier-Stokes flows (a model for the evolution of viscous, incompressible fluid flows) where we showed that the speed of the flow could become infinite at most on a set of zero one-dimensional measure (that is less than a curve) in space and in time. (A nearly optimal result, according to the recent examples of Shefer). We also extensively investigated the properties of hypersurfaces in n-dimensional Euclidean space for which elliptic relations among their principal curvatures are prescribed (i.e. mean curvature equation, Monge-Ampère equation in real or complex space, etc.).

In 1982 Caffarelli was awarded the Guido Stampacchia Prize by the Scuola Normale Superiore di Pisa (see [62] for details). This prize was jointly awarded to six mathematicians, Caffarelli and his co-authors, for the series of six papers: Brownian Motion and Harnack inequality for Schrödinger operators; Existence and regularity for a minimum problem with free boundary; Axially symmetric jet flows; Asymmetric jet flows; Jet flows with gravity; and Asymptotic behaviour of minimum problems with bilateral obstacles. It was the first of many awards and prizes given to Caffarelli.

In 1983 he was appointed as a professor at the University of Chicago, a position he held for three years. During these years he continued to receive major awards for his outstanding contributions. In 1984 he was awarded the Bôcher Prize by the American Mathematical Society [44]:-

... for his deep and fundamental work in nonlinear partial differential equations, in particular his work on free boundary problems, vortex theory and regularity theory.

In 1986 Caffarelli left Chicago when he was appointed Professor at the Institute for Advanced Study in Princeton. On 31 October 1988 Pope John Paul II presented Caffarelli with the Pontifical Academy of Sciences Pius XI Gold Medal. We have quoted several times above from the speech that Caffarelli delivered on that occasion. Let us quote here what he said about his current research interests (in 1988) [15]:-

My current interests include uniform estimates on singular perturbations when approaching a singular limit, for instance for the level surfaces approaching a surface of discontinuity and its implications on the stability of both the surface of discontinuity and numerical methods employed to simulate the limiting problem.

For ten years Caffarelli worked at the Institute for Advanced Study, then in 1994, because he missed working with graduate students, he returned to the Courant Institute where he spent three years. In 1997 he moved to the University of Texas at Austin where he was appointed to the Sid W Richardson Foundation Regents Chair. Awards continued to acknowledge the extremely high quality of his research contributions. In 2003 he was awarded the Premio Konex, Platino y Brillantes, Argentina [43]:-

... as the most important scientist of his country in the last decade.

The Swedish Academy of Sciences awarded him their Rolf Schock Prize in 2005 [52]:-

... for his important contributions to the theory of nonlinear partial differential equations.

The American Mathematical Society awarded him their Leroy P Steele Prize for Lifetime Achievement in Mathematics in 2009. The citation begins [1]:-

Luis Caffarelli is one of the world's greatest mathematicians studying nonlinear partial differential equations (PDE). This is a difficult field: there are rarely exact formulas for solutions of nonlinear PDEs, and rarely will exact algebraic calculations yield useful expressions. Instead researchers must typically invoke functional analysis to build "generalised" solutions for many important equations. What remains is the profound and profoundly technical problem of proving regularity for these weak solutions and, by universal acclaim, the greatest authority on regularity theory is Luis Caffarelli. His breakthroughs are so many, and yet so technical, that they defy any simple recounting here. But it was certainly Caffarelli's work on "free boundary" problems that first showed his deep insights. Free boundary problems entail finding not only the solution of some PDEs, but also the very region within which the equation holds. Luis Caffarelli's vast work totally dominates this field, starting with his early papers on the obstacle problem. ...

In 2012, he was made a Fellow of the American Mathematical Society. In the same year Caffarelli received the prestigious Wolf Prize in Mathematics [23]:-

... for his work on partial differential equations.

His work leading to the award of the Wolf Prize is described in [23]:-

Cafferelli has repeatedly made very deep breakthroughs. His early work on free boundary problems was the first place where his extraordinary talent and intuition began to show. Free boundary problems are about finding the solution to an equation and the region where the equation holds. In a series of pioneering papers, Caffarelli put forward a novel methodology which eventually leads, after several truly amazing technical estimates that step by step improve the regularity of the solutions and the boundary, to full regularity under very mild assumptions. Although the theory is complicated, the arguments are elementary and full of beautiful geometric intuition and mastery of analytic technique. A second fundamental contribution by Caffarelli is the study of fully nonlinear elliptic partial differential equations (including the famous Monge-Ampère equation), which he revolutionised. The upshot is that, although the equations are nonlinear, they behave for purposes of regularity as if they were linear. ... Another fundamental contribution by Caffarelli is his joint work with Kohn and Nirenberg on partial regularity of solutions of the incompressible Navier-Stokes equation in 3 space dimensions. Although the full regularity of solutions is still unknown and likely very hard, Caffarelli-Kohn-Nirenberg showed that the singular set must have parabolic Hausdorff dimension strictly less than one. In particular, singular fibres cannot occur. ... Caffarelli has also produced deep work on homogenisation and on equations with nonlocal dissipation. The list could be continued. Caffarelli is the world's leading expert on regularity of solutions of partial differential equations.

In 2018 Cafferelli was made a fellow of the Society for Industrial and Applied Mathematics (SIAM). The citation states that [47]:-

Luis Angel Caffarelli, The University of Texas at Austin, is being recognised for seminal contributions in regularity theory of nonlinear partial differential equations, free boundary problems, fully nonlinear equations, and nonlocal diffusion.

In the same year of 2018 he was awarded the Shaw Prize in Mathematics by the Hong Kong-based Shaw Foundation [8]:-

... for his ground-breaking work on partial differential equations, including creating a theory of regularity for nonlinear equations such as the Monge-Ampère equation, and free-boundary problems such as the obstacle problem, work that has influenced a whole generation of researchers in the field.

This prize is one of the world's largest mathematics prizes with a monetary value of US$1.2 million.

Caffarelli has received many honours in addition to those mentioned above. He has been elected to the American Academy of Arts and Sciences (1986); the National Academy of Sciences (1991); the Pontifical Academy of Sciences; the Argentina Mathematical Union; the Accademia Nazionale dei Lincei; and several others. He was invited to deliver the Fermi Lectures at the Scuola Normale di Pisa (1998) and was the American Mathematical Society Colloquium Lecturer in 1993. A summary of his research by the Pontifical Academy of Sciences is as follows [32]:-

Luis Caffarelli works in non linear analysis, mainly on non linear partial differential equations arising from geometry and mechanics. He has conducted extensive research into free boundary and singular perturbation problems. He has worked on free boundary problems that arise naturally when a constitutive relation or a conserved quantity (a temperature, a pressure, a density) changes discontinuously its behaviour across some value of the variables under consideration. Typical examples are solid-liquid interphases, burnt-unburnt mixtures in flame propagation, and flow in porous media. Understanding of the geometry and stability of the solution and its interphase is important in selecting and evaluating simulation methods, as well as understanding the models themselves. Another area of research is fully non linear equations and optimal transportation. Fully non linear equations arise in optimisation and optimal control. They have also been recently studied in relation to optimal transportation and optimal antenna design. Other areas of interest are incompressible flows, harmonic maps, and minimal surface theory and more recently, on non linear random homogenisation.

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented to Luis Caffarelli in 2023 [16]:-

... for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge-Ampère equation.

For the Press Release, the Citation and other information about the award of this prize to Caffarelli, see THIS LINK.

In the interview [57], Louis Nirenberg was asked, "What is Caffarelli like as a mathematician?" He replied:-

Fantastic intuition, just remarkable. We haven't worked together for several years now, but when we worked together, I had a hard time keeping up with him. He somehow immediately sees things that other people don't see, but he has trouble explaining them. He says things and writes very little, so when we were working at the board, I would always say, "Luis, please write more, write down more." Once I said to him, "Luis, to use a Biblical expression, 'Where is it written?'" Somebody said he once heard a talk in which Luis proved something in partial differential equations - using nothing! Just somehow out of thin air, he can come up with ideas. He's really fantastic - and a very nice person.

Both the quantity and quality of Caffarelli's work is astounding. We give data up to the end of September 2023. MathSciNet lists 322 publications by Caffarelli, three of which appeared in 2022 and one in 2023. These papers have 20,200 citations in 11,499 publications by 7,236 distinct authors. He has advised more than 30 PhD students. While giving this data, we note that his wife, Irene Martínez Gamba, had 99 publications listed in MathSciNet with the earliest publication being in 1986. These 99 papers have 1,715 citations in 954 publications by 975 by distinct authors. Her Ph.D. was from the University of Chicago in 1989 with the thesis Asymptotic behavior at the boundary of a semiconductor device in two space dimensions. She holds the W A Tex Moncrief, Jr. Chair in Computational Engineering and Sciences III at the University of Texas.

Given Caffarelli's incredible output of mathematics it is hard to see how he has much time for hobbies. When asked about these in the interview [19] he said:-

I like to cook, but my wife is a better chef than I am. I also like to play some soccer if I have time. I play the piano. I prefer mostly classical music.

Let us end with a quote by Caffarelli from [1]:-

Through the years, I have had the opportunity to belong to wonderful institutions and to befriend and collaborate with extraordinary scientists all over the world. This led to further opportunities to mentor very talented young people who have invigorated my research with new ideas.