**Charles Fefferman**'s parents were Arthur S Fefferman and Liselott Stern. 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 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.

Charles Fefferman, known to his friends and colleagues as Charlie, was a child prodigy. He said [11]:-

In fact he had mastered the calculus before the age of twelve. He explained how he got so rapidly to that level [11]:-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 a4^{th}grade Mathematics textbook. That was the beginning.

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 [4]:-I read the45^{th}grade Mathematics textbook in a day or two. My father didn't believe me, so he asked me a few questions and realized that I understood. Then he bought me a^{th}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 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 [11]:-Jim Hummel, an authority on complex variables, was sitting in his tiny office one morning in July1961. He got a phone call from the Feffermans. They had this avid6^{th}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 realized 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.

This is the paperAt 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 generalized. 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.

*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 [4]:-

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.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.

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 [11]:-

He was awarded his PhD in 1969 for his thesis entitledMy 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.

*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:-

These papers were not his first publications for, in addition to the paper mentioned above, he had already publishedI 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.

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

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.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.

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 [4]):-

Fefferman described two topics that he worked on in a way that might be understood by non-mathematicians in [2]:-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.

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 [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 a1-inch-long needle through a full360degrees, 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 the1970s. 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 his reply Fefferman said:-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 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 Univesity (1985), and the University of Madrid (1990). In 2008 he was awarded the Bôcher Memorial Prize from the American Mathematical Society:-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 his reply Fefferman said:-... 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 C164^{m}', Annals of Math.(2006),313-359.

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

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 bookProfessor 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.

*The Ambient Metric*. The publisher's description is as follows:-

Michael G Eastwood writes in a review:-This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2dimensions 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+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics.

As of September 2015, Fefferman has nearly 200 published items listed in Math.SciNet. 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.This monograph has been keenly anticipated for more than25years! 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!

**Article by:** *J J O'Connor* and *E F Robertson*

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