Correction Appended

LOS ANGELES —Four hundred people packed into an auditorium atU.C.L.A. in January to listen to a public lecture on prime numbers,one of the rare occasions that the topic has drawn astanding-room-only audience.

Mr. Tao earned his Ph.D. at 20 and gained broadacclaim. At age 7, he was taking high school math classes.

Another 35 people watched on a video screen in a classroom nextdoor. Eighty people were turned away.

The speaker, Terence Tao, a professor of mathematics at theuniversity, promised “a whirlwind tour, the equivalent to goingthrough Paris and just seeing the Eiffel Tower and the Arc deTriomphe.”

His words were polite, unassuming and tinged with the accent ofAustralia, his homeland. Even though prime numbers have beenstudied for 2,000 years, “There’s still a lot that needs to bedone,” Dr. Tao said. “And it’s still a very excitingfield.”

After Dr. Tao finished his one-hour talk, which was broadcastlive on the Internet, several students came down to the front andasked for autographs.

Dr. Tao has drawn attention and curiosity throughout his lifefor his prodigious abilities. By age 2, he had learned to read. At9, he attended college math classes. At 20, he finished hisPh.D.

Now 31, he has grown from prodigy to one of the world’s topmathematicians, tackling an unusually broad range of problems,including ones involving prime numbers and the compression ofimages. Last summer, he won a Fields Medal, often considered theNobel Prize of mathematics, and a MacArthur Fellowship, the“genius” award that comes with a half-million dollars and nostrings.

“He’s wonderful,” said Charles Fefferman of Princeton University, himself a former child prodigy and aFields Medalist. “He’s as good as they come. There are a few in ageneration, and he’s one of the few.”

Colleagues have teasingly called Dr. Tao a rock star and theMozart of Math. Two museums in Australia have requested hisphotograph for their permanent exhibits. And he was a finalist forthe 2007 Australian of the Year award.

“You start getting famous for being famous,” Dr. Tao said.“The Paris Hilton effect.”

Not that any of that has noticeably affected him. His campusoffice is adorned with a poster of “Ranma ,” a Japanese comicbook. As he walks the halls of the math building, he might bewearing an Adidas sweatshirt, blue jeans and scruffy sneakers,looking much like one of his graduate students. He said he did notknow how he would spend the MacArthur money, though he mentionedthe mortgage on the house that he and his wife, Laura, an engineerat the NASA Jet Propulsion Laboratory, bought last year.

After a childhood in Adelaide, Australia, and graduate school atPrinceton, Dr. Tao has settled into sunny Southern California.

“I love it a lot,” he said. But not necessarily for what thearea offers.

“It’s sort of the absence of things I like,” he said. No snowto shovel, for instance.

A deluge of media attention following his Fields Medal lastsummer has slowed to a trickle, and Dr. Tao said he was happy thathis fame might be fleeting so that he could again concentrate onmath.

One area of his research — compressed sensing — could havereal-world use. Digital cameras use millions of sensors to recordan image, and then a computer chip in the camera compresses thedata.

“Compressed sensing is a different strategy,” Dr. Tao said.“You also compress the data, but you try to do it in a very dumbway, one that doesn’t require much computer power at the sensorend.”

With Emmanuel Candès, a professor of applied and computationalmathematics at the California Institute of Technology, Dr. Tao showed that even ifmost of the information were immediately discarded, the use ofpowerful algorithms could still reconstruct the original image.

By useful coincidence, Dr. Tao’s son, William, and Dr.Candès’s son attended the same preschool, so dropping off theirchildren turned into useful work time.

“We’d meet each other every morning at preschool,” Dr. Taosaid, “and we’d catch up on what we had done.”

The military is interested in using the work for reconnaissance:blanket a battlefield with simple, cheap cameras that might eachrecord a single pixel of data. Each camera would transmit the datato a central computer that, using the mathematical techniquedeveloped by Dr. Tao and Dr. Candès, would construct acomprehensive view. Engineers at Rice University have made a prototype of just such acamera.

Dr. Tao’s best-known mathematical work involves prime numbers— positive whole numbers that can be divided evenly only bythemselves and 1. The first few prime numbers are 2, 3, 5, 7, 11and 13 (1 is excluded).

As numbers get larger, prime numbers become sparser, but theGreek mathematician Euclid proved sometime around 300 B.C. thatthere is nonetheless an infinite number of primes.

Many questions about prime numbers continue to elude answers.Euclid also believed that there was an infinite number of “twinprimes” — pairs of prime numbers separated by 2, like 3 and 5 or11 and 13 — but he was unable to prove his conjecture. Nor hasanyone else in the succeeding 2,300 years.

A larger unknown question is whether hidden patterns exist inthe sequence of prime numbers or whether they appear randomly.

In 2004, Dr. Tao, along with Ben Green, a mathematician now atthe University of Cambridge in England, solved a problem related tothe Twin Prime Conjecture by looking at prime number progressions— series of numbers equally spaced. (For example, 3, 7 and 11constitute a progression of prime numbers with a spacing of 4; thenext number in the sequence, 15, is not prime.) Dr. Tao and Dr.Green proved that it is always possible to find, somewhere in theinfinity of integers, a progression of any length of equally spacedprime numbers.

“Terry has a style that very few have,” Dr. Fefferman said.“When he solves the problem, you think to yourself, ‘This is soobvious and why didn’t I see it? Why didn’t the 100 distinguishedpeople who thought about this before not think of it?’ ”

Dr. Tao’s proficiency with numbers appeared at a very youngage. “I always liked numbers,” he said.

A 2-year-old Terry Tao used toy blocks to show older childrenhow to count. He was quick with language and used the blocks tospell words like “dog” and “cat.”

“He probably was quietly learning these things from watching‘Sesame Street,’ ” said his father, Dr. Billy Tao, apediatrician who immigrated to Australia from Hong Kong in 1972.“We basically used ‘Sesame Street’ as a babysitter.”

The blocks had been bought as toys, not learning tools. “Youexpect them to throw them around,” said the elder Dr. Tao, whoseaccent swings between Australian and Chinese.

Terry’s parents placed him in a private school when he was 3 .They pulled him out six weeks later because he was not ready tospend that much time in a classroom, and the teacher was not readyto teach someone like him.

At age 5, he was enrolled in a public school, and his parents,administrators and teachers set up an individualized program forhim. He proceeded through each subject at his own pace, quicklyaccelerating through several grades in math and science whileremaining closer to his age group in other subjects. In Englishclasses, for instance, he became flustered when he had to writeessays.

“I never really got the hang of that,” he said. “These veryvague, undefined questions. I always liked situations where therewere very clear rules of what to do.”

Assigned to write a story about what was going on at home, Terrywent from room to room and made detailed lists of the contents.

When he was 7 , he began attending math classes at the localhigh school.

Billy Tao knew the trajectories of child prodigies like Jay Luo,who graduated with a mathematics degree from Boise State Universityin 1982 at the age of 12, but who has since vanished from the worldof mathematics.

“I initially thought Terry would be just like one of them, tograduate as early as possible,” he said. But after talking toexperts on education for gifted children, he changed his mind.

“To get a degree at a young age, to be a record-breaker, meansnothing,” he said. “I had a pyramid model of knowledge, that is,a very broad base and then the pyramid can go higher. If you justvery quickly move up like a column, then you’re more likely towobble at the top and then collapse.”

Billy Tao also arranged for math professors to mentor Terry.

A couple of years later, Terry was taking university-level mathand physics classes. He excelled in international mathcompetitions. His parents decided not to push him into college fulltime, so he split his time between high school and FlindersUniversity, the local university in Adelaide. He finally enrolledas a full-time college student at Flinders when he was 14, twoyears after he would have graduated had his parents pushed him onlyaccording to his academic abilities.

The Taos had different challenges in raising their other twosons, although all three excelled in math. Trevor, two yearsyounger than Terry, is autistic with top-level chess skills and themusical savant gift to play back on the piano a musical piece —even one played by an entire orchestra — after hearing it justonce. He completed a Ph.D. in mathematics and now works for theDefense Science and Technology Organization in Australia.

The youngest, Nigel, told his father that he was “not anotherTerry,” and his parents let him learn at a less accelerated pace.Nigel, with degrees in economics, math and computer science, nowworks as a computer engineer for Google Australia.

“All along, we tend to emphasize the joy of learning,” BillyTao said. “The fun is doing something, not winningsomething.”

Terry completed his undergraduate degree in two years, earned amaster’s degree a year after that, then moved to Princeton for hisdoctoral studies. While he said he never felt out of place in aclass of much older students, Princeton was where he finally felthe fit among a group of peers. He was still younger, but was notnecessarily the brightest student all the time.

His attitude toward math also matured. Until then, math had beencompetitions, problem sets, exams. “That’s more like a sprint,”he said.

Dr. Tao recalled that as a child, “I remember having this vagueidea that what mathematicians did was that, some authority, someonegave them problems to solve and they just sort of solvedthem.”

In the real academic world, “Math research is more like amarathon,” he said.

As a parent and a professor, Dr. Tao now has to think about howto teach math in addition to learning it.

An evening snack provided him an opportunity to question hisson, who is 4. If there are 10 cookies, how many does each of thefive people in the living room get?

William asked his father to tell him. “I don’t know howmany,” Dr. Tao replied. “You tell me.”

With a little more prodding, William divided the cookies intofive stacks of two each.

Dr. Tao said a future project would be to try to teach morenon-mathematicians how to think mathematically — a skill thatwould be useful in everyday tasks like comparing mortgages.

“I believe you can teach this to almost anybody,” he said.

But for now, his research is where his focus is.

“In many ways, my work is my hobby,” he said. “I alwayswanted to learn another language, but that’s not going to happenfor a while. Those things can wait.”

Correction: March 13, 2007

A profile of Terence Tao, a world-renownedmathematician, in Science Times yesterday referred incorrectly towork he did with another mathematician on prime numbers. Theyproved that it is always possible to find, somewhere in theinfinity of integers, a progression of any length of *equallyspaced* prime numbers — not a progression of prime numbers ofany spacing and any length.

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