Here is how a great mathematician works: He lies down on the sofa for hours at a stretch, thinking intently about shapes, relationships and change—rarely about numbers as such. He explores idea after idea in his mind, discarding most. When a concept finally seems promising, he's ready to try it out on paper. But first he gets up and changes the baby's diaper.

Though the last step is a new one for Charles Louis Fefferman, courtesy of 9-month-old Nina Heidi, he has been perfecting the others for most of his 30 years.

Probably because he is so young, everyone calls him "Charlie." Such informality cannot obscure his towering reputation as a mathematician at Princeton, an institution that is to math what Hollywood is to movies. Few professors have stirred such intellectual acclaim since Einstein lived and worked in the New Jersey town.

As a 9-year-old in Silver Spring, Md., Fefferman developed a precocious interest in science fiction and rocketry. That led him to read about physics, a subject whose textbooks, he discovered, were filled with mathematics. He began studying it and by 12 had mastered all levels up through calculus. Charlie's father, a government economist, took the boy to the nearby University of Maryland to be tutored.

Charlie never got to high school. He jumped straight from junior high to the university at 14, graduating with high honors at 17, already the author of a published paper on symbolic logic. He earned a Ph.D. at Princeton in 1969 and at 22 was teaching at the University of Chicago, the youngest full professor in the nation. After three years, an offer came from Princeton. At 25, Fefferman became the youngest full professor in its history.

In 1976 he won the first Waterman award given by the National Science Foundation: $50,000 a year for three years just to think about mathematics. In 1978 he was one of four recipients of the Fields Medal, the Nobel Prize of math. This year he was elected to the National Academy of Sciences.

Fefferman's ideas are so complex they are beyond the comprehension of many other mathematicians. Laymen are at a total loss, though he patiently tries to simplify his explanations. His research is focused in three general areas. The first is Fourier analysis, which he describes as "the study of vibrations, any vibrations." Another is partial differential equations "in which you are not trying to find just a number, but something more complicated, some quantity that depends on something else." The third area, called equations with several complex variables, involves "studying problems from physics that become more readily understood when relationships between them are discovered.

"It is very rough," Fefferman adds gently, "to describe mathematics to someone who hasn't studied it for years."

Surprisingly, he rarely uses a computer. "If your job is to come up with original ideas," he says, "no one so far knows how to use the computer for that purpose." That is not to say he doesn't welcome help. "New ideas are not easy to find. If you are lucky enough to be working on a good 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."

He does not apologize for the fact that his work has no direct practical application, though, as he points out, "Mathematicians have solved some practical problems by reflecting on 'pure' problems. The idea of making a computer, for example, was suggested by trying to answer questions like, 'Is mathematics consistent?' "

Despite his ability to concentrate so intensely, Fefferman has never lived in a cloister. In graduate student days, he dabbled in the campaigns of Eugene McCarthy and George McGovern. He plays chess ("I'm very bad—I know lots of distinguished mathematicians who are bad chess players"). He also loves listening to—and singing along with—classical music.

In October 1974, a fellow student introduced him to Julie Anne Albert, now 28, a child prodigy who had studied violin at the Juilliard School when she was only 9. They were married in 1975, whereupon she gave up a job teaching music in a New York high school. Since then she has accustomed herself to the demands of Charlie's work. "He may get mad at himself for a day or so," she admits. "Usually it passes when he gets his next idea. Then he's excited again."

Fefferman is teaching two courses this semester. When he started in the classroom, he says, he felt uneasy, because most of his students looked older than he did. He also quickly developed a reputation as an absent-minded professor, who once inadvertently divulged answers to an upcoming test when questioned persistently. Time has eased his self-consciousness about his age—"My beard has helped too," he says.

In fact, he may be over the hill intellectually according to one theory; it holds that mathematicians do their most creative work before 30. Considering that notion, Fefferman says, "Einstein's papers in 1905 [when he was 26] were superb. But I would most highly recommend his 1915 work. It was magnificent."