James Waddell Alexander 1910 *15 was the scion of an old and distinguished Princeton family. He was also a world-class mathematician and a mountaineer who loved climbing in the French Alps and the Rockies, where to this day the perilous east face of Longs Peak is called “Alexander’s Chimney” in his honor. Back home in Princeton, where he was a math professor, Alexander kept in shape by clambering over campus rooftops like an agile gargoyle come to life. His office was on the top floor of the old Fine Hall — now Jones Hall — and he often left the window open at night so that in the morning he could scramble up the wall and in through the window to his desk, ready for a good day’s work.

It has not always been quite that hard to get into the Princeton math department, but it has never been easy. Ever since its founding in 1905, by Henry Burchard Fine 1880, it has been one of the University’s most luminous departments, a world leader in a variety of mathematical fields, and the cradle of several of them, notably topology, algebraic geometry, and game theory, which now underpins an ever-multiplying variety of other disciplines. If there was a peak in all this excellence, it surely came in the 1930s, when Fine Hall was the Olympus of world mathematics, with the likes of Alexander, topologists Solomon Lefschetz and Oswald Veblen, logician Alonzo Church ’24 *27, the mathematical physicists Hermann Weyl and Eugene Wigner, and even Albert Einstein — a faculty member of the Institute for Advanced Study — all turning up for the afternoon teas that were a key feature of departmental culture.

The math world has changed tremendously in the intervening 60-plus years, with more fields of study for more mathematicians. “When I came as a grad student, there was a small number of books; you mastered them and you knew the subject,” says Professor Robert Gunning *55. “Now, there are a small number of books coming out each month.” At the same time the Internet has shrunk the mathematical world, crumpling it up like some topological model. Today’s seminar in Tokyo can be discussed in Princeton tonight.

For the recordThe Nov. 7, 2007, story on Princeton’s math department incorrectly described Fermat’s Last Theorem. Fermat claimed to have found a “truly marvelous proof” that there are no triplets of positive integers that satisfied x^n + y^n = z^n for any n larger than 2. Andrew Wiles, chairman of the department, won international acclaim when he did find such a proof. The article also incorrectly described a game invented by John Nash *50 when he was a graduate student (the game also was invented independently by a Danish mathematician). The game is played on a board with hexagonal cells.

One thing that has not changed is the preeminence of Princeton in that world. And lately there seem to be more reasons than ever to celebrate in Fine Hall:

• Two of the four winners of the most recent Fields Medals — the math world’s answer to the Nobel Prize — have Princeton connections: Professor Andrei Okounkov, the department’s co-director of graduate studies, and Terence Tao *96, who is now a professor at UCLA. (Tao is called the “Mozart of Math” for his facility at working problems in just about every area he tries.) Counting past and present faculty alone, Princeton has won six Fields Medals, more than any other institution. Throw in the five won by mathematicians at the Institute — who remain tied to Fine Hall — and you’ve got a breathtaking concentration of mathematical brilliance in one small town.

• Last spring, for the first time ever, Princeton won the Putnam Prize, the prestigious undergraduate math competition in which some 500 North American schools take part annually. Princetonians previously have won as individuals but never as a team. Princeton’s winning team included Andrei Negut ’08, Aaron Pixton ’08, and Ana Caraiani ’07, who had won the individual Putnam in both her freshman and sophomore years.

• Caraiani also won the 2007 Alice T. Schafer award, given annually to the nation’s top female undergrad in math. Tying for second was Tamara Broderick ’07. That one-two finish matched the 2006 results, when a pair of Princeton roommates, Alexandra Ovetsky ’06 and Allison Bishop ’06, took the top two spots.

“I’ve been brainwashed to believe it, but I *do* believe it,” says Charles Fefferman *69, who joined the department as a tenured professor in 1973, at the ripe old age of 23. “This is the top math department in the world.”

How did it get that way? The answer is a combination of determination, generous funding, and the gravitational pull top mathematicians exert on one another, for they are far more sociable than the rest of us might suppose.

In a way, Princeton math began with Woodrow Wilson 1879, not that he particularly cared for the subject himself. Wilson once said that it was possible only with “a painful process of drill” to “insert” mathematics into “the natural, carnal man.” But one of Wilson’s great friends as an undergrad had been Fine, whose love of the classics had given way to a fascination with science and math. After graduation, Fine became that very unusual 19th-century American interested enough in math to pursue a doctorate in the subject. He earned his at one of the great centers of European math, the University of Leipzig, in 1885, and then returned to Princeton as an assistant professor of mathematics. Fine proved to be an invaluable adviser to Wilson as he set about shaping the modern Princeton, and so when he requested permission to hire a small math faculty, Wilson agreed.

Fine seems to have had a remarkable eye for young talent. Among his first hires were Veblen, Gilbert Ames Bliss, and Luther Eisenhart, who years later would succeed him as chairman. He lured the great algebraist J.H.M. Wedderburn from Scotland and hired James Jeans from Cambridge University to teach applied math. Within the University he cultivated Alexander, who would discover the algebraic tool known as the Alexander Polynomial, which is useful in the area of math known as knot theory.

It did not take long for Fine’s influence to extend well beyond Princeton. He helped found the New York Mathematical Society, which morphed into the American Mathematical Society, and served as its president. As Prince-ton’s dean of sciences, Fine oversaw the creation in 1928 of a $3 million scientific research fund that would provide essential financial support to European mathematicians, many of whom were Jewish and were nervously watching anti-semitism grow in Germany. To honor Fine, his great friend Thomas D. Jones 1876 established the Henry Burchard Fine Professorship, one of the first such chairs in the U.S. and an inspiration to other universities.

When Fine died in 1928, struck down by a car while riding his bike on Nassau Street, his Princeton admirers cast about for an appropriate way to memorialize him. He had wanted a new building for his math department that, as late as 1924, occupied just two rooms in Palmer. Jones and his niece donated $600,000 for a new building that was to be dedicated exclusively to mathematics. Their guiding principle was, as Jones put it, that “nothing is too good for Henry Fine.”

The finished building was truly sumptuous. Each of the large professorial studies boasted carved oak paneling and a splendid fireplace. Built into the leaded stained glass windows in the common rooms were some of the great formulas of math history. There was even a shower for those who wished to use the nearby tennis courts: In the words of the Faculty Song, “He’s built a country-club for Math/Where you can even take a bath.” Speaking in 1970 at the opening of the bigger, blander Fine Hall in use today, longtime departmental secretary Agnes Henry described the beautiful old building as a “Grand Hotel of Mathematics,” a description some grad students took a bit too literally. They had to be reminded gently to find more permanent lodging elsewhere.

Veblen, whose enthusiasm for math would earn him a reputation as an ambassador for the subject, played a key role in the building’s design. At his urging, a comfortable, Oxbridge-style common room occupied the second floor, between the classrooms on the first and the library on the third. That, Veblen believed, would force the building’s occupants to meet and mix and talk, which they did, round the clock.

“[Fine Hall] was open all the time,” recalls Gunning. “And everyone going to and from the library or classes went right by the common room and stopped and talked. When tea was served at 4 o’clock the room really filled up.”

Students played Go or chess or Kriesgspiel, a chess variant played partly blind. When tired of those games, they invented their own, which is what a brilliant young grad student from West Virginia named John Nash *50 did soon after arriving in 1948. “Nash,” as his game was known, required players to build a bridge of octagonal tiles from one side of the board to the other, while his opponent tried to do the same for the other two opposing sides.

Then, as now, the department was wonderfully laissez faire. There were no formal graduate classes and few re-quirements. At some point in the 1940s or 1950s, the dean of the Graduate School noticed that the math department was not giving any grades. He called Albert Tucker *32, the professor who handled most of the department’s administrative chores (Tucker is perhaps best known for formalizing the famed “Prisoner’s Dilemma”), and told him he must assign grades. So Tucker invented the grade “N,” which stood for “no grade.” (Today the department gives graduate students an asterisk, which, for official purposes, is recorded as “other.”)

That did not — and does not — mean that students weren’t learning. Grad students were left pretty much on their own to conduct what Lefschetz called “baby seminars,” small groups of students working through various problems that interested them. “We collaborated, quizzing one another,” recalls Gunning. “We took turns working through the proofs in order to really understand them. You’d have to convince people that the proof was correct. Together, you learned a great deal.”

It did not hurt that the Institute for Advanced Study (IAS) was little more than a mile away. Indeed, of the first five faculty members at the IAS, four came straight from the Princeton math department; the fifth was Einstein. This illustrious group did not have far to travel: Until its main building was completed in 1939, the Institute rented space in Fine Hall. That meant that in the ’30s Einstein and John von Neumann had offices there (Einstein was soon driven from the ground floor by the parade of gawkers at his window). As professor emeritus Harold Kuhn *50 puts it: If you sat long enough in the Fine Hall common room, you eventually would have tea with 98 percent of the world’s great mathematicians.

“As a grad student, I think I already knew half of the great mathematicians in the world,” says Kuhn, who at 82 is craggily handsome beneath a shock of snowy white hair.

Lefschetz, who took over as chairman from Eisenhart in 1945 and held the post until ’53, may have had an even greater impact on Princeton math than Fine. He had arrived at Princeton in 1924 as one of the first Jewish professors on campus, a distinction that earned him more cold shoulders than respect. But along with being one of the most immensely powerful mathematicians of the last century, Lefschetz was a tough and determined man who had overcome much in his life. After losing both his hands in a transformer explosion while working at Westinghouse, he’d been fitted with prosthetics, into which students had to place chalk at the beginning of lectures.

Lefschetz kept a tight grip on departmental policies and hiring. Far more than his predecessors, he made research the principal goal of the department. But Lefschetz’s preference for pure math created a schism that would mark the department for years. This divide — between “pure” and “applied” math — certainly was not restricted to Princeton. G.H. Hardy, the great Cambridge University mathematician in the first half of the 20th century, once argued that pure math was the only kind worth pursuing; the rest meant nothing.

At Princeton, that sort of elitism was felt for years. Adam Rosenberg ’78 says that his interest in subjects like combinatorics, networks, and graph theories made him an outsider in a department where algebraic topology was king. “These areas, which tend to be more applied, clearly didn’t interest them,” says Rosenberg. “People in the department called what I did ‘toy mathematics.’”

Today Rosenberg works for the business software giant SAP in Scottsdale, Ariz., as an “industrial mathematician,” using math to solve practical problems such as finding optimal routing plans for airlines and pricing schemes for chain stores. He would surely feel more at home in the department today. “There’s a long history of stories like that,” says Ingrid Daubechies, the acting head of applied math. “I know people who are fantastically smart engineers, members of the National Academy of Engineering, who came to Princeton and felt they couldn’t do math because they weren’t at home in the math department.”

So, about seven years ago, when Fefferman was serving as chairman, he asked Daubechies to head up a committee to look into how to make the math major more attractive to a broader slice of students. “We wanted to target students who were bright but were not intending to become professional pure mathematicians,” says Daubechies, noting that her own degree is in physics, not math. Following her recommendations, the department now offers two new courses, Math 199 (“The Magic of Numbers”) and Math 211 (“Models, Proofs, and Applications: Introduction to Applied Mathematics”) and lets it be known that, within the major, there is an applied-math track. One sign of their success is the fact that 34 juniors became math majors this year, which is thought to be a departmental record.

It’s not the only way the department has changed for the better. “I think the culture is more warm and fuzzy,” says Fefferman. “When I came here as a grad student [in 1966], there were individual outstanding teachers, but the culture of the department didn’t encourage the view that teaching was really important. Teaching is now taken quite seriously.”

Mathematicians have an image problem, and they know it. To many of us, they seem a bit, well, different — smarter, but also, it often seems, less securely tethered to the real world. Consider one of Princeton’s greatest homegrown mathematicians, Alonzo Church, who is remembered today as the father of mathematical logic. Church was once voted “the most unconscious man on campus” by his fellow students, though unconsciousness seems to have been the least of his oddities. According to the late Gian-Carlo Rota ’53, who became a professor of math and philosophy at MIT, Church looked like a “cross between a panda and a large owl” and spoke in long, logically impeccable sentences.

“He would not say, ‘It is raining,’ ” Rota recalled in one of the many entertaining interviews about the department to be found on the math department Web site. “He would say instead, ‘I must postpone my departure for Nassau Street, inasmuch as it is raining, a fact which I can verify by looking out the window.’ ” (Probably it was sunny by the time he’d finished.) Church made a habit of arriving at the end of afternoon tea, when he would pour the contents of the cream pitcher into the dregs of the teapot and suck it all down, thus fortifying himself for a good night’s work.

Face it: Most of us really haven’t the vaguest idea what mathematicians are up to, but we know it’s probably well beyond us. The blackboards in Fine are covered with what look like hieroglyphics. One Septem-ber seminar was called “Asymptotic Optimality in Sequential Quickest Change-Point Detection for General Stochastic Models.”

Our fear frustrates mathematicians like Daubechies, who is a tireless proselytizer for math, eager to lend a visitor books until she realizes they’ve already been lent out. “What I worry about is people who believe from the word ‘go’ that they can’t do math,” she says. “That is something we should worry about because it’s not true. Math, to me, is the name we give to our talent for constructing logical reasoning, logical ways of deducing things. In order to become a very good professional mathematician, maybe talent is needed, just like to become an Olympic athlete. But to enjoy mathematics, like to enjoy sports, you can do that without having any extraordinary talents.”

Math looks so specialized, so difficult, that it’s easy to forget that we live in a world that increasingly is based on incredibly sophisticated math that we pretty much take for granted. Cell phones, iPods, medical imaging — all use math. One of Gunning’s former grad students works for General Motors, applying very sophisticated geometry to the shape of auto bodies. Daubechies is working with specialists in an increasing number of fields — neuroscientists, geophysicists, and even art historians who are beginning to use medical imaging techniques to detect counterfeit paintings.

Sometimes it’s a practical problem that opens a door to abstract work. Consider Fefferman, yet another Princeton winner of the Fields Medal. He was staggeringly precocious, even in the world of math, where genius often shows itself quite early. His interest in mathematics began when, after experimenting with toy rockets, he decided he wanted to know how real rockets worked. He went to the library and took out a physics book and understood nothing, which is not surprising since he was in the third grade. His father suggested he read some math books, and he quickly worked his way up to calculus. He published his first academic paper at 15 and had completed his Ph.D. at 20, simultaneously amazing and irking his older department mates. At the moment he’s working with assistant professor Bo’az Klartag on “a problem of finding a smooth surface that passes through or near a whole lot of dots in space,” he explains, comjuring up the phrase “computational geometry” to describe the field. “I would love for it to be useful, but it’s too early to be useful.”

Fefferman insists that 38 years after earning his doctorate, he’s still in love with his subject. “I found it wonderful and infinitely challenging then, and I find it wonderful and infinitely challenging now,” he says. “Mixed in with the joy there’s a lot of pain because typically if you are working on a hard problem, you sit in front of a blank page and you can’t think of a thing to write. You go home at the end of the day and you’ve accomplished nothing.

“Then, if you are lucky, you think of an idea and it turns out to be wrong, but you don’t find that out immediately and it leads to some interesting other ideas and pretty soon you’re cooking with enough ingredients that something can happen. And then there’s a stage in which you are confused and finally, at the very, very end, with luck, you understand something and solve the problem. I’ve had a few instant successes, by sheer luck. Those are very sweet.”

Fefferman wishes more people today would experience this. “Unfortunately, I think that the experience of almost any kid learning math is negative,” he says. “They’re not exposed to interesting stuff. The number of kids who are filtered out is 99 percent. It’s terrible.”

In 1991 Fefferman authored a federal study called “Renewing U.S. Mathematics — A Plan for the Nineties.” The study pointed to a crisis that many people were worried about, a shortage of mathematicians. “But the world evolved differently,” says Fefferman. “What actually happened was that the former Soviet Union collapsed and there was a big immigration [to the United States] of the cream of the cream of Soviet scientists and mathematicians.”

One was Okounkov, who came to Princeton in 2002 after a stint at Berkeley. A Muscovite tall and lean enough to be a volleyball player, Okounkov began his studies as a mathematical economist. (“It was much freer than being a political economist in the Soviet Union,” he says with a wry laugh.) Upon his discharge from the military in 1989, he switched to mathematics and now he works on the fruitful boundary of math and physics, exploring “a certain interplay between probability and algebraic geometry that might be considered natural from a physical point of view, but from a mathematical point of view is very surprising.”

Like Fefferman, Okounkov worries about his beloved field. He feels obliged to be something of an ambassador for math. “It’s occasionally scary how little people know,” he says, “not in terms of knowledge — more in terms of this really widespread impression that mathematics is some kind of arcane game that has nothing to do with anything around it. That just couldn’t be more wrong.” And he’s concerned that so few people choose math as their major. “I’m actually very worried about this. Science and engineering are the backbone of Western civilization. Math is the blood that flows through them.”

One mathematician who is not quite as worried is department chairman Andrew Wiles, famous for his proof of Fermat’s Theorem, which for years was the math world’s Holy Grail. He admits to not being rooted in the “real world of applications.” But, he explains, “applications come without one expecting them. G.H. Hardy said one of the things about number theory is you’re sure it’s never, ever going to be used. But now number theory is the basis of all of our Internet and banking security. It’s one of the most useful branches of mathematics. No one could have anticipated that.”

As a boy growing up in Cambridge, England, Wiles always was fascinated by math. He read about the 17th-century French mathematician Pierre de Fermat and his claim to have found a “marvelous proof” of Pythagoras’ theorem (that the sum of the square of the two short sides of a right triangle will always equal the square of the hypotenuse). Believing that a bright 10-year-old in the mid-20th century just might know as much math as a 17th-century Frenchman, Wiles set about trying to discover such a proof. He failed then but succeeded magnificently many years later, in 1994. To obtain that proof took Wiles eight years, including the discovery of a gap in his proof that other number theorists had predicted would take 100 years to close.

Wiles in person is soft-spoken, slender, and modest. Despite his having been knighted for his work, he makes it sound as if his only real talent is for persevering. At the moment he’s trying to extend one area of his work on Fermat.

Like Okounkov, Wiles has concerns about the public’s relationship to math, noting that in England universities recently added a fourth year to what had always been a three-year math course. “And it’s not a fourth year at the end,” he notes. “It’s at the beginning, because the high school students just weren’t prepared for what previous generations had done.” But Wiles says he sees no signs of such weakness at Princeton. And he is confident that math’s deep truths will always inspire bright young students, just as they did a 10-year-old boy in Cambridge.

“There are certain math problems that everyone can appreciate,” he says — like the Fermat proof, which landed him on the front pages of newspapers around the world. “These are really deep questions about our world. It’s not ephemeral. It’s part of the human culture that’s been here for three or four thousand years.”