Volume 4, Issue 33 November/December
Making History in Math
UC Berkeley professor of mathematics Ken Ribet Photo credit: Jenny Shi Wen
Talk to Ken Ribet about mathematics, and it's clear he has a great regard for history as well as numbers. A Berkeley professor of mathematics and expert in number theory, he discusses the work of scholars from ancient Rome and Enlightenment France as easily as he would contemporary research. But this respect for his professional forebears is actually quite fitting: Ribet is headed for a place in math's hall of fame himself. A number of important theorems and mathematical objects bear his name.
Ribet is best known, however, for his role in solving modern mathematics' most famous problem. Known as Fermat's Last Theorem, it is based on a brief note scribbled by a seventeenth-century French lawyer in the margins of an ancient math text. In the centuries that followed, attempts to solve it have frustrated legions of brilliant mathematicians. Ribet's work ultimately sparked a solution to the problem in 1993.
As math problems go, Fermat's Last Theorem is deceptively easy to describe. For one thing, it bears a great resemblance to the Pythagorean theorem so familiar from basic geometry. According to Pythagorus, the squares of two sides of a right triangle, added together, should equal the square of its hypoteneuse, or x2 + y2 = z2. The great mathematician Diophantus of Alexandria discussed this problem in detail in his seminal third century work Arithmetica.
More than 1,300 years later, Pierre de Fermat, a distinguished mathematician in his own right, took this idea one step further. He asked, in his copy of Arithmetica, what happens if the exponent is an integer of three or greater instead of two? Can the equation be solved if neither x nor y is zero? Fermat then claimed he could prove the problem has no answer-but left behind no further explanation.
His tantalizing statement fueled more than three centuries of effort to develop a proof. But the puzzle's seeming simplicity belied the fiendish difficulty of the solution. "A whole cottage industry of people who were invested in the theorem developed," Ribet says. "They developed more and more intricate and sophisticated techniques to chip away at the problem."
A 2001 French postage stamp commemorated the 400th anniversary of mathematician Pierre de Fermat's birth. Image credit: La Poste Francaise
A crucial insight to the problem came in the early 1980s. German mathematician Gerhard Frey proposed that Fermat's Last Theorem could be linked to problems involving elliptic curves. Previous research had suggested that such curves were, surprisingly, related to a completely different branch of mathematics called modular forms.
"I and others stepped in because we were experts in both and wanted to see if it was really possible to link these two together," Ribet says.
But Ribet was the one who moved the problem forward. He proved that Frey's prediction was true. In doing so, he provided the link needed to show that Fermat's Last Theorem was true.
Ribet's work galvanized Princeton mathematician Andrew Wiles into action. Wiles saw Ribet's finding as a way to transform Fermat's Last Theorem from an insurmountable enigma into a solvable problem. In 1993, after seven years of laboring in secrecy, Wiles announced he had a proof of Fermat's longstanding puzzle.
By linking elliptic curves to modular curves, Ribet laid the cornerstone for the proof of Fermat's Last Theorem. Image credit: Wikipedia
The achievement, Ribet says, "turned me overnight into a historical figure, which is an amusing situation to be in."
All of the practice he gained fielding press calls about the theorem, Ribet says, has turned him in to a much better communicator about math. He has since participated in a number of forums designed to make high-level math more accessible to the public. These include a sold-out event at San Francisco's Palace of Fine Arts celebrating the proof of Fermat's Last Theorem, a BBC documentary on the same topic, and symposia on campus and elsewhere on other aspects of number theory.
"It's been incredibly validating that mathematics was able to have this concrete success," Ribet says. "It was really a big shot in the arm for mathematics as a whole."