1980

# UW Advances Aid in Cracking One of Math's "Toughest Nuts"

One of the most exciting events in mathematics in recent years was the proof of "Fermat's Last Theorem" by Andrew Wiles of Princeton University. The proof drew, in part, on the work of UW mathematician Ralph Greenberg.

This famous conjecture of the seventeenth century French mathematician Pierre Fermat takes the following form:

If n is an integer greater than 2, then the equation x + y = z cannot have any solutions where x, y, and z are positive integers (positive whole numbers).

Fermat scribbled a note in the margin of his copy of a math book he was studying, saying that he had discovered a marvelous proof of this theorem, but the margin was too small to describe it. He never did write it down and the secret went with him to the grave. Over the centuries, this problem intrigued mathematicians and served as a catalyst in mathematics. As an example, the development in the 19th century of the very important and useful branch of mathematics called algebraic number theory was prompted by the search for a tool to prove Fermat's conjecture.

Algebraic number theory was used by some mathematicians to prove Fermat's conjecture for many values of n, but this approach never succeeded in proving the conjecture completely, that is, for all n. In the mid-1980s, researchers proposed attacking the problem using the theory of elliptic curves, which had been studied extensively since the 1920s. This approach seemed promising because it was soon proved that Fermat's conjecture would follow as a corollary of an important conjecture about elliptic curves.

These developments in the 1980s spurred Wiles to rekindle his childhood dream of proving Fermat's conjecture. He came up with an ingenious strategy to prove the conjecture about elliptic curves.

To carry out his strategy, Wiles made crucial use of ideas and methods of a number of other mathematicians, including the work of UW mathematician Ralph Greenberg, which played a significant role in Wiles's approach.

In the 1980s, Greenberg had studied the notion of "Selmer Groups." Originally, the term referred to a certain group of equations for elliptic curves introduced in the 1950s by the mathematician Selmer. Greenberg saw that he could unify the work of many mathematicians in a much broader framework by using the notion of Selmer Groups. Greenberg says he was "quite surprised" to learn that Wiles's strategy was dependent on a certain mathematical object which turned out to be a Selmer group.

This pathbreaking achievement in mathematics stands as an example of how multiple steps by different researchers often converge to yield a major discovery. The importance of each part to the whole is sometimes under-appreciated until the evolution of the discovery can be traced.