While swimming laps, Pavel Etingof thinks about math. The crowds and noise on a city bus do little to distract Allen Knutson from the equations he scribbles on a notepad he keeps handy. Francis E. Su gave up his songwriting hobby to spend more time on his proofs.
The constant devotion of these and other mathematicians to their work has allowed them to produce seminal proofs and impressive results that have won them high praise early in their careers -- all are under 35.
But many of them have something other than math to worry about. According to one common belief, these young researchers may already have passed their prime.
Mr. Su, a 31-year-old assistant professor of mathematics at Harvey Mudd College, says that idea can create pressure. "Sometimes I think, 'Gosh, my best work is done when I'm young. Maybe I'm over the hill,' " he says.
Like many other outstanding young mathematicians, Mr. Su doesn't buy into the folklore entirely, though. And his skepticism may be justified. On closer inspection, the question of when mathematicians peak is not as simple as the myth makes it seem. Though great mathematical discoveries are often made at a young age, certain factors feed the illusion that mathematical superstars flame out early, even if their creative fires still burn brightly later in life.
The notion that youth is a necessary ingredient for great work has held sway in mathematics for decades, if not centuries. A famous mathematician, G. H. Hardy, wrote in his 1940 memoir, A Mathematician's Apology, "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game."
Certainly math, as well as its sister field theoretical physics, boasts many examples of the phenomenon. Evariste Galois, a 19th-century French mathematician whose contributions to a branch of algebra called group theory are now taught to all students of mathematics, developed his ideas as a teenager. He wrote a manuscript spelling them out when he was 20, the night before he was killed in a duel. A Norwegian mathematician and contemporary of Galois's named Niels H. Abel died of tuberculosis at age 26 after solving a 300-year-old problem and discovering what are now known as Abelian functions. Although death cut short the careers of those two men, Albert Einstein lived for 50 years after formulating his most famous equation, E=mc2, when he was 26.
Why might many great mathematicians make their most important contributions at such a young age? The legend consists of two parts: Mathematical researchers make a splash early in their lives and then do less-significant work as they grow older. The first half of that premise has some truth behind it, for many top mathematicians demonstrate their promise quite early, often as children. Terence Tao, a professor of mathematics who received tenure this year at the University of California at Los Angeles before his 25th birthday, remembers that his "favorite pastime" when he was 3 or 4 years old was doing math problems in workbooks. In grade school, he took math courses with students four years older than he and had to sit on a special cushion to reach the desk. Like many mathematicians of all ages at American universities, Mr. Tao was born elsewhere, in this case, Australia. He now works on widely varying problems, but his major field involves studying mathematical functions that describe waves.
Mr. Etingof, a 31-year-old mathematician at the Massachusetts Institute of Technology and Columbia University, recalls knowing even before he started grade school that he wanted a career in mathematics. "I did not ask for pieces of candy, but I asked for math problems." He currently studies a physics-inspired area of mathematics called quantum groups.
Many of these superstars graduated from high school quite young. Mr. Su, who now studies an area of game theory that deals with dividing goods fairly, was called "the Brain" by other students at his high school because he was three years younger than his classmates. He says he didn't reveal his age when he went to college at the University of Texas at Austin to avoid the social pigeonholing.
Ruth E. Lawrence started college at the University of Oxford when she was 12 and received a bachelor's degree in mathematics at age 13. She earned her doctorate at 17, went on to prestigious postdoctoral positions at Harvard, and landed her first faculty job at 22. Now 29, she is an associate professor on leave from the University of Michigan at Ann Arbor, working at the Hebrew University of Jerusalem. She studies knot theory, an area of mathematics that describes whether two knots in separate loops of rope, each twisted and entangled upon itself, can be transformed to look the same without cutting either rope. With no hint of irony she says, "I did things a little bit earlier than usual."
Noam D. Elkies wasn't quite as precocious as Ms. Lawrence, graduating from college at Columbia University at age 19 and finishing his Ph.D. at Harvard when he was 21. Now 34, Mr. Elkies became the youngest person ever granted tenure at Harvard when he became a full professor at 26. He was 21 when he solved a problem first proposed in 1769. Mr. Elkies proved it possible to find three numbers that, when raised to the fourth power and added together, the result is another number that is raised to the fourth power. (The numbers he found are 2,682,440; 15,365,639; 18,796,760; and 20,615,673.)
Early professional success is only part of the story, however. Many researchers in other fields show early promise but typically take more time to make important contributions because of the nature of their work.
Mathematics, Mr. Elkies says, is one of a few fields "in which one can do top-level work without a lot of life experience," something that might be key in the arts or humanities. "One does not have to have experience raising children through school, dealing with family tragedies, and so forth, to be able to find three numbers whose fourth powers add up to another one."
David A. Vogan Jr., the chairman of M.I.T.'s math department, says that experience also means more in the other sciences than it does in mathematics. "In a lot of the sciences, there's a tremendous value that comes from experience and building up familiarity with thousands and thousands of complicated special cases," he says. "Whereas mathematics tends to be concerned with simpler things... . And the people who do it best are the ones who understand nothing about how it was understood before and bring some completely new perspective."
Not only can mathematicians come up with important ideas without spending years learning the work that came before them, but the time between idea and publication is much shorter than in fields that require laborious experiments or reams of documentation, says Dean K. Simonton, a professor of psychology at the University of California at Davis, who has studied how age is related to scientific discovery.
Mr. Knutson, a 31-year-old assistant professor at the University of California at Berkeley, explains why many graduate students in mathematics complete their dissertations so quickly: "I think it's typical for theses that the great majority [of the work] happens over a period of a month or two. False start, false start, false start -- aha! Then you write up the stuff in the 'aha!' with a bunch of prefatory material saying what other people have done.
"My thesis was much like this. My thesis was 25 pages. The ones that scare me are the history ones, where you have to accumulate evidence for 800 pages. In math, all the evidence you need may take up a paragraph, and everyone says, 'Yup, it's true.'"
The young mathematicians' experiences are representative of a larger trend, according to Mr. Simonton. In a study of nearly 2,000 famous scientists throughout history, he found that mathematicians were the youngest when they made their first important contribution. The average age at which they accomplished something important enough to land in history books was 27.3. By contrast, biologists were 29.4 years old, physicists were 29.7, and chemists were 30.5.
But starting at a young age doesn't necessarily mean one's career will end early or that later contributions will pale in importance -- the second half of the legend. In fact, Mr. Simonton found that mathematicians make their best research contributions (which he defined as the ones mentioned most often by historians and biographers in reference books) at what many might consider doddering old age: 38.8. That age is very similar to those he found in other sciences: 40.5 in biology, 38.2 in physics, and 38.0 in chemistry.
In fact, although "mathematicians do wring their hands a lot" about becoming too old to do great work, according to John H. Ewing, the executive director of the American Mathematical Society, numerous counterexamples show that the rule, if true, doesn't hold for everyone. Carl Friedrich Gauss, a 19th-century mathematician sometimes called the "prince of mathematics," continued to produce important results in both math and physics late in life, and died at age 76. Paul Erdos, the most prolific mathematician ever, having published 1,500 papers, tried to prove as many theorems as possible as he aged, working essentially constantly until he died at age 83 in 1996.
As a contemporary example, many mathematicians mention Charles L. Fefferman. As a young man, "he was a real star," says Mr. Ewing. Mr. Fefferman got an early start, receiving his Ph.D. when he was 20 and becoming a full professor at the University of Chicago when he was 22. His research on Fourier analysis, which looks at complicated vibrations -- as of a violin's string -- and breaks them down into simpler ones, led to a Fields Medal, often referred to as math's Nobel Prize, when he was 29. Mr. Fefferman, now 51 and chairman of the mathematics department at Princeton University, is still a leading mathematician.
Mr. Fefferman is not sure whether his career diverges from the well-known pattern. "I did some work that I'm very proud of between the ages of 19 and 25," he says quietly. "I've stayed productive, and whether I've gotten better or worse or stayed about the same -- it's not so clear."
Despite such counterexamples, the idea persists that not only do young mathematicians make early breakthroughs, they make more than their share. "This myth, if you wish to call it a myth, is so prevalent that it's quite probable that there's some truth to it," says Christopher M. Skinner, a 28-year-old associate professor of mathematics at the University of Michigan at Ann Arbor, who describes his work as trying to establish a glossary to translate concepts between certain areas of algebra and analysis.
Many mathematicians explain the phenomenon in terms common to any academic field: With increasing seniority and age comes a heavy load of responsibilities that can distract mathematicians from their research. These demands include serving on committees, teaching and overseeing graduate students, and attending to family affairs.
"Life takes a lot of time and effort," Mr. Fefferman says. "I think the big jump there came with taking care of babies, taking night shifts. There's nothing like sleep deprivation to make one less than brilliant."
"Doing the great mathematical work requires a hell of a lot of energy," says Mr. Etingof, of M.I.T. and Columbia, suggesting that older mathematicians may not be able to keep up that pace. "Doing mathematics at a very high level is really as exhausting as any sport."
Mr. Knutson says, "There have been times when I've been thinking about something so intensely when I lie down [to sleep at night, that] after half an hour, I have to get up and start writing again because I'd made too many advances and I was afraid I'd lose them if I didn't write them down. I'll go to sleep at midnight and I'll wake up at 6, desperate to be working again." Though that happens only rarely, he admits that "from an external viewpoint, it could look like a dangerous addiction."
But some of the perception that mathematicians slow down as they age may be based more on illusion than on reality. "There's a demographic fallacy," says Spencer R. Weart, a historian at the American Institute of Physics. Because the ranks of mathematicians and other researchers have expanded extremely rapidly until recently, most active researchers are very young. Since more people in the field are young, it stands to reason that more discoveries are made by young people, Mr. Weart says.
What's more, because mathematicians can make great discoveries at a young age, they may receive awards and become highly visible as young people. In fact, the Fields Medal stipulates that winners must be 40 or younger as a result of the wishes of John C. Fields, who left the money for the medal to both honor existing work and encourage future achievement.
Many of these forms of public recognition are given only once to a researcher, so "there's an impression that [older mathematicians] have run out of steam," says Mr. Simonton, even if their work continues at the same level.
Understanding such factors has not stopped mathematicians from worrying about whether they will soon be -- or already are -- over the hill. Several, though not all, mid-career and older mathematicians contacted by The Chronicle say they think their best work is behind them. The younger mathematicians, in general, have a sunnier outlook. "I don't really think that one can make an argument that over all the stuff I did in my early 20's is significantly better than what I am doing now," says Mr. Elkies, of Harvard. In fact, he thinks that what he has learned in the intervening years has improved his work.
Everyone has a prescription for avoiding dormancy as they age. Many suggest that older mathematicians can work more effectively than their younger colleagues on problems that may take a long time and a great deal of patience and confidence. Mr. Tao, for instance, says he works "obsessively" for two weeks at a time on a problem. "But if I'm not getting anywhere, I tend to give up and try something different."
By contrast, Andrew J. Wiles, a Princeton mathematician, solved math's most famous problem by working for seven years to prove Fermat's Last Theorem. He finished his proof when he was 40, in 1993, but due to a subtle but crucial error, the proof was not complete for another two years. "This requires a great amount of courage and stamina," says Mr. Etingof.
Doing significant work late in one's career involves seeking out problems that require more knowledge than young mathematicians can have accumulated, according to George W. Mackey, 84, an emeritus professor of mathematics at Harvard. That often means learning about several different areas of math and looking for ways to tie them together, he says. Princeton's Mr. Fefferman agrees, adding that picking up new specialties, while risky, is the best way to avoid going stale.
Mr. Mackey says that by connecting disparate fields, he has gained a deeper understanding of group theory. A few years ago, he wrote a summary of his ideas for a publication at Rice University, his alma mater. "I was in a constant state of euphoria because all these things fit together," he says. "There's a huge amount of unity within mathematics."
"In mathematics, it's not a game where the fastest wins," says Edward V. Frenkel, a 32-year-old professor at Berkeley. "But rather, it's more like who can see farther, who can see deeper. That's the one who achieves more."
Noam D. Elkies, 34, a professor at Harvard University, currently on sabbatical at the Mathematical Sciences Research Institute in Berkeley, Calif.
He earned a Ph.D. at 21, rising to prominence in his field at about the same time. Barry Mazur, his graduate adviser at Harvard, says that in his work on number theory as a graduate student, Mr. Elkies "seemed to have enormous insight in a field that you would otherwise imagine would take years." The summer after he finished graduate school, Mr. Elkies went on to solve a problem that had stood unconquered for more than 200 years.
Pavel Etingof, 31, an assistant professor at Massachusetts Institute of Technology, currently on leave and working as an associate professor at Columbia University.
He was 24 years old when he received his Ph.D. At his first academic job, at Harvard, he did important work on quantum groups, a new, physics-inspired area in group theory. David A. Vogan Jr., chairman of M.I.T.'s math department, says Mr. Etingof is already a leading figure in the study of quantum groups, and adds that "he can do anything at all."
Edward V. Frenkel, 32, a professor at the University of California at Berkeley.
He completed his thesis in just a year, receiving his Ph.D. when he was 23. He has worked to find and exploit commonalities between two seemingly unrelated fields in mathematics, representation theory and geometry. Nicolai Reshetikhin, a colleague and collaborator at Berkeley, calls the connections "unexpected" and describes Mr. Frenkel's results as "fascinating and important."
Allen Knutson, 31, an assistant professor at the University of California at Berkeley.
He was 26 when he received his Ph.D. In 1998, he completed his best-known work, in collaboration with Terence Tao. It answers a question in linear algebra that had remained open since it was first posed, in 1962. Calvin C. Moore, chairman of Berkeley's math department, describes the proof as "outstanding work on a classical problem."
Ruth E. Lawrence, 29, an associate professor at the University of Michigan at Ann Arbor, currently on leave and working at the Hebrew University of Jerusalem.
She attracted attention as a child prodigy, entering the University of Oxford at age 12, receiving a bachelor's degree at 13 and a Ph.D. at 17. She studies knot theory to find connections to other areas of mathematics, including topology, geometry, and mathematical physics. Igor Dolgachev, a professor at Michigan, says Ms. Lawrence's doctoral work "found some very beautiful relationships between theoretical physics and mathematics."
Christopher M. Skinner, 28, an associate professor at the University of Michigan at Ann Arbor.
He was 25 when he received his Ph.D., but was making waves well before that. According to Donald J. Lewis, who has retired from the math department at Michigan, even when Mr. Skinner was an undergraduate, journals "were already aware of him and competing for his papers." Mr. Lewis calls Mr. Skinner's undergraduate thesis "a major attack" in the field of algebraic number theory.
Francis E. Su, 31, an assistant professor at Harvey Mudd College.
He received a Ph.D. when he was 25, working on probability. But he changed his field when he started teaching at at Harvey Mudd, and now studies problems of fair division. His best-known work solved the problem of how to divide rent in an envy-free way in an apartment with any number of roommates and bedrooms of varying sizes. Steven J. Brams, a political scientist at New York University, says Mr. Su's work uses "good mathematical theory [that] is truly applicable to real-world problems."
Terence Tao, 25, a professor at the University of California at Los Angeles, currently teaching at the University of New South Wales, in Sydney, Australia.
He received his Ph.D. when he was 20, doing work on harmonic analysis to demonstrate that a certain equation describes waves that never break, in contrast to ocean waves hitting the beach. He has received high praise for subsequent work in two other areas of mathematics as well -- partial differential equations and linear algebra, collaborating with Mr. Knutson on the latter. "If you just looked at his work and didn't know anything about him," says John B. Garnett, a math professor at U.C.L.A., "you'd probably say he was 50 years old and an extremely productive mathematician."