To mathematicians, Andrew Wiles is a household name. Wiles burst into the public eye this last week when it was announced all over the media that he won the Abel Prize.
In 1995, Wiles published a proof of Fermat’s Last Theorem, quoted by many as one of the deepest open problems in number theory, if not all of mathematics. His proof was long and profound, and sent ripples through the mathematical community. I had just begun my doctoral studies in 1994, and was following developments about Wiles’s proof in the media. The internet was very young back then, and there certainly was no social media. What a thrill when I finally heard that the proof was completed.
On March 14, 2016, the Norwegian Academy of Science and Letters awarded the Abel Prize for to Wiles, who is now at Oxford University as a Royal Society Professor (he was at Princeton when he published his proof). The prize comes with a monetary award equivalent to $700,000 US dollars. Think of it as the Nobel Prize for Mathematics (mathematicians are not eligible for the Nobel). Past winners have included mathematical luminaries such as Jean-Pierre Serre, Endre Szemerédi, and John Nash, Jr.
Wiles is a brilliant, quiet man who does not seek out the spotlight. Here is an interview with him after he learned that he received the award.
Fermat’s Last Theorem is a statement about a certain kinds of equations with integer solutions. The theorem says that certain equations have no solutions.
Consider an equation of the form:
an + bn = cn ,
where a, b, and c are integers, and n is an integer greater than 2. If n = 2, then we have the familiar Pythagorean equation:
a2 + b2 = c2
which has infinitely many integer solutions. For example, a = 3, b =4, and c= 5 is one such solution.
Pierre de Fermat in 1637 conjectured that for all n > 2, there is no integer solution to his equation. This was his last unproven assertion, and so the name followed suit. Fermat settled the problem for n = 4, and from that it reduces to the odd cases for n > 2. Despite the work of legions of mathematicians through the centuries, the full proof resisted all attempts at its resolution for 358 years.
Fermat claimed to have a solution for his problem, but couldn’t fit into the margin of the text written by the 3rd century Alexandrian scholar Diophantus. Given what we know now, it is unlikely Fermat had the correct solution.
While Wile’s proof is highly technical, his fundamental ideas relate to properties of curves in the plane. His work lies in the area of something called arithmetic geometry. This is not the arithmetic you learn in elementary school.
Wiles proved an assertion about elliptic curves that gives Fermat’s theorem as a corollary. Elliptic curves are certain cubic curves in two variables, and they play a major role in number theory, and its application in cryptography. Wiles proved the Taniyama–Shimura–Weil conjecture, which asserts the modularity of semi-stable elliptic curves over the rationals.
Wile’s proof is a very modern one. He uses many advanced tools and techniques that you wouldn’t see even at a university undergraduate level of mathematics education: Hecke algebras, class number formulas, Galois cohomology, and Galois representations, to name a few. It is hard to imagine that Fermat would have any idea of the scope of the mathematics involved to finally prove the theorem with his name.
Wiles submitted the proof for publication in 1993, and it was reviewed by University of Berkeley mathematician Ken Ribet. In a slightly unusual turn, the pair were in close contact with Wiles during the long review process (most reviews are done anonymously). The stakes were too high to get this wrong! Ribet would get stuck and ask Wiles a question. Wiles would respond with enough details until Ribet was satisfied, and he would continue reading.
One day Ribet noticed a problem that Wiles couldn’t solve. He found a gap in the proposed proof.
After months of additional effort, Wiles finally closed the gap and grasped the correct proof. He pulled in earlier, discarded ideas from so-called horizontal Iwasawa theory, and theorem of Barry Mazur’s called the 3/5 switch.
The proof appeared in 1995 in the Annals of Mathematics, taking up 150 pages. Wiles described in a BBC documentary the moment he found the proof as the most important moment of his working life. Fast forward to 00:55 below to see Wile’s emotional reaction describing the moment he proved the theorem. Mathematicians are often stereotyped as unemotional, which is a shame since we are truly passionate about our work.
Wiles completed his historic proof after the age of 40 making him ineligible for the Fields Medal. Even though the Abel Prize comes decades after the proof first appeared, it is a fitting award for such an historic work of genius.