# Charles Fefferman extras

We give below five short pieces either by Charles Fefferman or about him.

**Click on the link to go to that piece.****1. Fefferman's own account of his upbringing and education recounted in his HLF interview. (18 February 2019).**

In this interview, Fefferman responded to various questions. In the version we give below, we have omitted the questions and just give Fefferman's words, with minor changes to edit out pauses and interruptions by the interviewer.

My father was an economist. He had a dual career alternating between the insurance industry in which he was treated very well but was somewhat bored, and the government in which he was treated pretty badly, but was fascinated by the work. His last job was as an economist on the Joint Committee of Congress that dealt with taxation. So if a tax bill was introduced, a lot of lawyers would focus on how to phrase it and my father tried to explain what might happen if the bill got passed.

My mother was a housewife. She was born in Germany and fled the Nazis in 1939. A confident family - my grandfather was a successful businessman who felt that when things got bad he could bribe his way out, and so at the last possible opportunity he managed to do that.

I was born in Silver Spring. I have a brother [Robert Fefferman] who for a long time was Dean of Physical Sciences at the University of Chicago. When not an administrator he is a Professor of Mathematics, so that is the family business. He is younger than me.

I think I was about nine when I first became interested in mathematics. I was very interested in science and I wanted to know how rockets work. The kids explanation didn't satisfy me. I took out a freshman college book from the local public library but I understood nothing. My father explained to me, "Of course you can't understand this, it uses mathematics." Well, I asked him if I could study mathematics. He didn't say that was ridiculous - I'm extremely grateful to him for that. He asked what level I was on at school. I told him the grade and he said he was buy me a 4th grade maths textbook. I zipped through it, maybe t took me a day or two. I than asked my father for the next maths textbook. He said, "You can't have read it already." I said I had so he asked me a few questions, then said "OK, I'll buy you the next one." This kept going on until I got through calculus. By that time I was about eleven. Then there was the problem of what to do after that. My father could test my knowledge and could help me with questions through calculus, but not after that. What could we do?

Well, my father had a friend whose son was interested in mathematics - he was way ahead and was being tutored by a professor at the University of Maryland. We were in driving distance of the University of Maryland. My father investigated and found a professor who was willing to work with me. This was Jim Hummel, and I took private lessons with him from the age of eleven. Jim introduced me to some of the other professors at the University of Maryland, in particular to the chairman of the department Leon Cohen - a wonderful fellow - who took me under his wing. For example, he arranged for me to take the first year maths courses, and then the maths and physics courses, at the University of Maryland. After a couple of years of that, he arranged for me to skip high school and enter the University of Maryland as a freshman when I was fourteen years old.

My parents and my teachers were very concerned lest I be pushed or feel extraordinary pressure - even self-generated. I think they slowed down my education a little. My fellow students were nice to me and I didn't feel like an alien creature. My father would occasionally give me a history book or an economics book or a novel, and ask me to read it. I would dutifully read whatever he asked and then put it aside and continue studying mathematics and physics.

I took art lessons as a kid and at some point I was trained very well. I like the best two or three things I produced at that time. If you had asked little Charlie what he was going to do, he would have said be would become either a painted or a scientist. I became a scientist - thank God - my talent in visual arts is not the same as my talent in mathematics.

When I entered university at age fourteen, my programme was heavily concentrated on mathematics and physics but it contained a normal range of subjects - I fulfilled the university requirements. I took two English courses intended for English majors - that was an interesting experience. I gather that most students consider mathematics and physics to be the most difficult subjects to study, but I can remember opening my report card and seeing the lower half of a C and being delighted that I had succeeded- thank God - in getting a C for Shakespeare in an English major. I got a lot more needed humility when I got to Graduate School. Having been a child prodigy in undergraduate school, I was used to being the best and the brightest. People were sensitive and nice so I was not obnoxious but in Graduate SchoolI was hearing many times in my first year sentences beginning with "you don't know ..."and that taught me considerable humility.

I visited the four Graduate Schools to which I applied. I was given advice by my teachers at the University f Maryland as to where to apply. I had interesting interviews. When I arrived at Princeton, I got confused and missed the little local train to go from Princeton Junction to the town of Princeton and that played havoc with the schedule for my interview. I was seventeen at this point. I can't remember anymore why I picked Princeton. I had fantastic guidance from the University of Maryland all along the way from my first contact as a little kid through undergraduate school. Although Maryland was a big State school, it felt like an army of private tutors so I had, perhaps, half a dozen mentors who took on exactly how my education should be handled. There was no required undergraduate thesis but I published a couple of papers.

**2. The Philadelphia Inquirer article:**

*A prodigy keeps young by just thinking*by Edward Schumacher published in*The Mathematics Teacher*in 1979.One of the curiosities about mathematicians, practitioners of the purest and most arcane of sciences, is that they share a critical trait with athletes: Both peak young.

Many of the greatest mathematicians developed their most brilliant theories by the time they were in their 20s, when their minds were still clear and facile and able to grasp abstractions at the limits of man's conceptual ability. Einstein, a physicist, discovered his very abstract theory of general relativity, for example, when he was only 27.

So, why isn't Charlie Fefferman worried?

Fefferman, a Princeton University professor, is considered by his peers to be one of this century's greatest mathematicians, a man who has developed formulas so brilliant and so abstract that there is not, as yet, any practical use for some of them.

And already he is an ageing 29.

"You can never tell when you're doing mathematics that you've still got it," Fefferman, a slight, full-bearded man, said recently in an interview in his house here. "The question is something that you have to live with. You have to live with the constant tension. Every mathematician feels it."

But Fefferman figures he has an angle. Although he is considered a true genius, he in fact reads little - keeping his mind clear of others' thoughts - but he thinks a lot.

"Perhaps I'm extreme," he said. "Compared to other mathematicians, I probably know less than most." The approach apparently has worked. Fefferman's accomplishments in mathematics, the father of all sciences, have been astounding.

Charles L Fefferman, the son of a federal government economist, was an honours student at the University of Maryland at age 12. At 22, he was a full professor at the University of Chicago, the youngest full professor in the history of any college, according to the Chronicle of Higher Education.

Last year, in Helsinki, he became one of the first Americans and one of the youngest mathematicians to receive the Fields Medal, which is awarded only once every four years and is the most coveted international prize among mathematicians.

Today, he is in his third year of a three-year leave of absence, courtesy of Congress, which is giving him $50,000 for each of the three years as its first Waterman Award winner. All he has had to do in return is what he likes most - think.

He has spent months at a time concentrating on just one problem. For most of the day and well into the night, he paces, sits, fidgets and lies on his belly, lost in intense thought. During particularly frustrating periods, his wife, Julie, drags him to Altman's department store in Paramus, N.J., where, for reasons beyond both of them, some of his greatest inspirations have hit him and he begins scribbling formulas on brown paper bags.

But if Fefferman is a genius, he is no eccentric.

As a child, his parents protected him from publicity and encouraged him to socialise with children his own age, he said. Today, he is an open and warm man, and the centre of attention in his life is his 2-month-old daughter.

He works mostly at home, in a comfortable ranch-style home. His wife, an outgoing person, said she "wouldn't know what it was like not having him around all the time."

Fefferman. admits that the acclaim he has been receiving is "ego-gratifying," but he is, nonetheless, extremely modest.

"I've made a contribution, but if I hadn't been around, someone else would probably do the work," he said. "There are a lot of others in the field who do good work."

"Already, a 29-year-old prodigy is a little shop-worn," he said.

Most of Fefferman's work has been in three fields: the relation among complex variables, Fourier analysis (the study of vibrations) and partial differential equations (the relation between time and space).

One of his most significant breakthroughs was his development of a new formula that relieved the strictures of analysing complex variables only in a two-dimensional sense. Fefferman's formula allows three-dimensional analysis of such questions as how to make an accurate flat map when the earth is round and how electrical charges in space will react to each other.

For him, mathematics is like art and takes a special temperament, one he cannot describe. Considering that both fields deal heavily with the abstract, the comparison is probably apt.

"After months, you look at a problem from a different point of view and suddenly it breaks apart and seems that it wasn't difficult at all," he said. "There's something elegant about that, something awe-inspiring. You aren't creating. You're discovering what was there all the time, and that is much more beautiful than anything man can create."

**3. Fefferman's 1983 article:**

*An account of a mathematician's education*in*The Mathematics Teacher*.As a child, I read everything I could find on "children's science" - dinosaurs, how rockets work, and so forth. I became dis satisfied with laymen's explanations and tried at age eight or nine to read a physics textbook. I couldn't understand a word, be cause it assumed a mathematical back ground. When I asked my parents for help, my father understood the problem and began to teach me basic mathematics. I loved it immediately. After a while, I brought mathematics books to school and studied them while I pretended to listen to my teachers. My parents worried that I was bored in school, but neither teachers nor school board officials offered any real help. Once I had learned calculus, my father brought me to James Hummel at the University of Maryland for tutoring. Hummel's lessons were much more exciting than the textbooks I had been reading. We would talk about elementary number theory, and he led me to some conjectures that I tried to settle between sessions. Usually I failed, but I felt joy at every small success; and I learned a great deal.

Meanwhile, I still had to plod through the usual course of school mathematics. By seventh grade the situation had become in tolerable. Thanks partly to a mathematics teacher with whom I didn't get along, Hummel and his colleagues arranged for me to take mathematics classes at the University of Maryland in place of eighth grade algebra. The following year I took physics at Maryland as well, and in 1963 I entered Maryland as a freshman, skipping grades 10-12. The mathematicians at Maryland supervised my education very carefully. When I graduated from Mary land in 1966, I had acquired a strong undergraduate background without feeling either pressured or bored.

From 1966 to 1969 I studied mathematics at the Princeton graduate school. The Princeton mathematics department is a very special place for a graduate student. Many of the students are brilliant, and they often learn from one another as well as from the distinguished faculty. It is essential for a mathematics student to be pointed in a fruitful direction by a good advisor. For my advisor, I was fortunate to have had a teacher of advanced mathematics without peer, Elias Stern. His influence on my mathematical development was decisive. After graduating from Princeton in 1969, I remained at Princeton one more year as an instructor, then moved to the University of Chicago. Chicago is a major centre for my field of mathematics, and I learned a great deal from my colleagues. My four years in Chicago were mathematically very active for me. Since 1974 I have been back at Princeton. I continue to work at my re search, and I have never stopped enjoying it

**4. The citation for Fefferman winning the Wolf prize in 2017.**

Affiliation at the time of the award: Princeton University, USA

Award was presented jointly to Charles Fefferman and Richard Schoem

... for their striking contributions to analysis and geometry.

Charles Fefferman (born in 1949) evidenced an affinity for mathematics at an early age. As a 9-year-old in suburban Maryland, his interest in science fiction and rocketry led to a study of physics, and with it the desire to master the mathematics necessary to understand the physics textbooks he read. At age 12, he was being tutored by professors at the University of Maryland, where he matriculated at 14. He graduated in 1966 at the age of 17 with bachelor's degrees in both mathematics and physics, earned with the highest distinction, having already published a paper on symbolic logic.

Fefferman earned his doctorate in mathematics in 1969 at Princeton University, supervised by Elias M. Stein, for a thesis entitled Inequalities for Strongly Regular Convolution Operators. After serving as a lecturer at Princeton for the 1969-1970 academic year, he took a position at the University of Chicago; one year later, in 1971, he was promoted to full professor there, making him the youngest full professor in the United States. In 1973 Fefferman returned to Princeton as a full professor, and was appointed the Herbert E Jones '43 Professor. He served as Chair of the Mathematics Department from 1999 to 2002.

Fefferman has been a dominant figure in analysis in the last 40 years, having made dramatic advances in a number of major new directions. It is no exaggeration to say that he is the most influential analyst of his generation. His major achievements include:

- He revolutionised our understanding of fundamental parts or real analysis by proving the duality of BMO with Hardy space $H^{1}$, whose results nowadays have innumerable applications.

- Fefferman made the first major breakthroughs in the areas related to Bôchner-Riesz summability and restriction theorems, in finding a counter-example for the disc multiplier, and proving the first important positive results in this area.

- Fefferman's study of the Bergman kernel and mapping properties of strongly pseudo-convex domains is a landmark in the field of several complex variables and has also opened the way for important geometric consequences.

- Fefferman obtained among the best results in the theory of pseudo-differential operators in the last 40 years:

Resolving with Beals a fundamental open problem of local stability.

Obtaining with Phong the optimal results for positive operators and sharp Garding inequalities.

Achieving the decisive results for second-order non-negative sub-elliptic operators.

- In the last ten years he has obtained dramatic breakthroughs and important generalisations in the area of the Whitney extension problem. This fundamental work is contained in a series of works beginning with five papers in the Annals of Mathematics from 2005 to 2009, and including more recently collaborations with B Klartag, A Israel, and K Luli. It deals with the general problem of given a set $E$ in $\mathbb{R}^{n}$ and a function$f_{0}$ on $E$, when (and how) does it extend to a function $f$ on $\mathbb{R}^{n}$ that belongs to a given Banach space $X$ of "smooth" functions (e.g. when $X = C^{k}$).

**5. The Introduction to the 2017**

*Notices*of the*American Mathematical Society*article*Ad Honorem Charles Fefferman*.**Introduction**

The prestigious Wolf Prize of 2017 has been awarded to Charles Fefferman, ex aequo with Richard Schoen. Charles Fefferman (Charlie) is a mathematician of the first rank whose outstanding findings, both classical and revolutionary, have inspired further research by many others. He is one of the most accomplished and versatile mathematicians of all time, having so far contributed with fundamental results to harmonic analysis, linear PDEs, several complex variables, conformal geometry, quantum mechanics, fluid mechanics, and Whitney's theory, together with more sporadic incursions into other subjects such as neural networks, financial mathematics, and crystallography.

I have requested the help of a distinguished group of his friends and collaborators to provide reflections on Fefferman's contributions to their respective fields. Before reading their remarks, it will be interesting to hear from Charlie himself:

Problems seem to select me! It's just so exciting. A problem sort of chooses you, and you can't stop thinking about it. At first, you try something, and it doesn't work. You get clobbered! You try something else and get clobbered again! Eventually you get some insights and things begin to come together. Everything starts to move. Everyday things can look different. It's very exciting. Eventually you manage to solve it all, and that's a great feeling!Born on April 18, 1949, Fefferman was a child prodigy who at the age of seventeen graduated from the University of Maryland, where he received a joint bachelor's degree in mathematics and physics. In 1969 he gained his PhD at Princeton University under the supervision of Eli Stein. In 1971, at the University of Chicago, he became the youngest full professor at any US college or university, a fact that merited his appearance in Time and Newsweek magazines in that same year.

Charlie returned to Princeton University in the fall of 1974, where since then he has pursued his mathematical career. In 1975 he and his wife, Julie, got married and went on to have two daughters, Nina and Lainie. Nina is a computational biologist who applies mathematical models to complex biological systems, while Lainie is a composer and holds a PhD in musical composition from Princeton University. Charlie has a brother, Robert, who is also a mathematician and professor at the University of Chicago. Julie has this to say about her husband:

When Charlie was young, he fell in love: (1) with painting and art and (2) with math. He says that for a while the two were tied, but painting was never in the lead. Eventually he realised that he was much better at math. After knowing him for thirty-five years, if I had to account for his choice, I'd say that his passion for the beauty of math was what overtook everything else. I think that for him it's almost an addiction to the art of beautiful mathematics ... Whenever I ask him to try to explain his work to me, his eyes sparkle and his voice and gestures are infused with an animation that is not present at any other time.I met Charlie at the University of Chicago during the academic year 1971–1972. He was then a recently appointed full professor and I was a first-year graduate student. The Calderón-Zygmund seminar was probably the place where we first got acquainted. At the end of that academic year Charlie agreed to be my thesis advisor. Let me add that for me it was a fantastic experience; we are of the same age, and at that time we became close friends. We played ping-pong together in the Eckhard Hall basement and had long conversations about science, art, movies, music, politics, and, of course, mathematics.

I had the privilege of being his first graduate student, thereby initiating a set that now contains more than twenty elements. There is no doubt in my mind that the opportunity of enjoying Charlie's advice and friendship is an experience we all will treasure.

Last Updated December 2023