Over the millennia the tree of mathematics has branched in dozens of different directions, arching out from a base that looks like a high school transcript: geometry, algebra, analytic geometry, calculus. In some ways mathematicians have been working there ever since, extending those basic concepts to more and more sophisticated ideas, building mathematical objects (like the set of all positive integers, 1, 2, 3, ...), and constructing ever more complicated beasts (like the set of all fractions, to give a trivial example,). Mathematicians often find this beauty, this truth, in their efforts to unite previously disparate areas of their field, to tame the unruly beasts they have unleashed.

Lafforgue made his mark in such unification. Decades ago a young Princeton mathematician named Robert Langlands conjectured that two very different animals are intimately connected. Roughly speaking, as Charles Seife described in Science magazine, these animals are mathematical objects that can be distorted in certain ways and still retain their original shape [such as the fundamental equivalence between a rubber coffee cup and a doughnut -- one can be stretched into the other, as long as you respect the hole] and objects that reveal the relations between solutions of equations."

Langlands' conjecture, described as a "Rosetta stone" of mathematics, was formalized into the Langlands Program, a quest that has happily occupied scores of mathematicians for more than 30 years. Andrew Wiles, the Princeton mathematician who a few years ago announced a heralded proof of Fermat's Last Theorem, established a important ingredient of the conjecture in his own work.

In general, mathematicians believed that Langlands' conjecture was true, but proving it was extremely difficult. Parts of the proof had already garnered two Fields Medals, and in 1999 Lafforgue made his mark with a 300-page handwritten proof of the conjecture in the case of what are called "function fields."

This is where the writer, fearing that he is losing the reader, must bring the discussion back to earth. He'll begin by pointing out that, the foregoing remarks notwithstanding, mathematicians are as human as you or me, even if they often have funny-looking hair or peculiar habits. The genius Paul Erdos called little children "epsilon," which is humorous if you're a mathematician (the Greek letter epsilon is often used as a symbol to express the concept of something approaching zero) but probably irritating if you are not.

Sometimes mathematicians make mistakes. After his proof of Fermat's Last Theorem was announced in the New York Times ("At Last, Shout of 'Eureka!' in Age-Old Math Mystery") Wiles discovered a mistake in his work and presumably just about had a bird. It took him a year to fix the problem, and Fermat was put to bed at last.

In 2000 Lafforgue was awarded the Clay Research Award by the Clay Mathematical Institute in Massachusetts. Just five days later he found a mistake in his own work. Lafforgue contacted Arthur Jaffe, the Clay Institute's president and a mathematics professor at Harvard, and offered to return his prize.

"Andrew Wiles and I convinced him that the award would be, under the circumstances, even more valuable to him," said Jaffe. "He could travel or collaborate however he liked in order to repair his proof."

Lafforgue said he worked day and night and fixed the flaw a few months later, proving that two very different-looking things are the same. At the same time it gave mathematicians confidence that the Langlands Program would succeed in other areas where work proceeds on the conjecture.

Why does Lafforgue's proof matter to me or to you? For a moment let's set aside the part about beauty.

In 1960 the physicist Eugene Wigner spoke of "the unreasonable effectiveness of mathematics." Mathematics is useful, and not just in the spreadsheet that is going to be part of the report you have due in three hours. Scientists and engineers have constructed our world on it, from Newton's calculus -- which he invented to describe the laws of motion -- to the quantum mechanics that describe the workings of the chip inside your personal computer.

Mathematics, amazingly enough, works; that is why it has been called the queen of the sciences. It works in the real world, and not just in the airy heights of the mathematician's imagination. It's an ugly kind of thing, in a way, in the minds of some pure mathematicians. "Real mathematics has no effect on war," wrote Hardy in "A Mathematician's Apology," a book intended to justify his existence as a pure mathematician. "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity."

Hardy wrote this in 1940, five years before the nuclear bomb was built on Einstein's fundamental ideas in relativity and well before today's cryptography built on prime numbers. These days no one is clean. That is one thing of which we can all be sure.

Recent Stories