Andrew Wiles: 60th birthday

Andrew Wiles, the man who proved Fermat's Last Theorem, is 60 tomorrow. Congratulations.



In 1996, Simon Singh produced this 50-minute documentary for the BBC Horizon series. And I just watched it again and liked it a lot.




Wiles' father was a chaplain in Cambridge – and a professor of divinity at Oxford. The program convinced me – without saying it – that this religious background was important for Wiles to know how to get concentrated for the years that were needed to complete the proof. Note that he first presented a proof in 1993, one that was seen to have a gap, and he was able to fill the gap within 15 months.




Singh's documentary is composed of testimonies of mathematicians. I personally know Barry Mazur – as a kind companion from the Harvard Society of Fellows – and he was clearly a pretty important puppet master behind some key developments although this modest man wouldn't claim credit for that.

Recall that Fermat conjectured, among other things, that\[

x^n+y^n=z^n,\quad \{x,y,z,(n-2)\}\subseteq \NN^+

\] has no solutions. If the exponent were \(n=1\), there would surely be solutions. For example, you may remember from your college that \(1+1=2\). Similarly, there are lots of solutions for \(n=2\); we know them as simple examples of the Pythagorean theorem. For example, \(7^2+24^2=25^2\) if I avoid the two most notorious examples.

But for third powers and higher powers, the identity just can't hold, we know today and people have suspected for centuries. It can't hold for any positive integers. Equivalently (as you may see by multiplying the equation by a common denominator), it can't hold for any positive rational numbers. Most of the famous mathematicians in recent centuries focused on the problem. At most, they achieved partial results, most typically a proof for some value of \(n\).

Note that it's enough to prove the theorem for \(n=4\) – which is particularly simple – and for prime integers \(n\gt 2\). It's because for other numbers factorized as \(n=pq\), one may see that a counterexample with the exponent \(n\) would also imply the existence of counterexamples with the exponents \(p,q\). Those exist for \(n=2\) which is why \(n=2\times 2\) requires a special treatment but otherwise the pattern simplifies things in the most natural way you can think of.

By the early 1990s, the proof had been known for \(n\) up to a very large value. But no general proof existed. In fact, the conjecture was even closer to "fringe maths" in the 1970s when the impatient, industrialized world of state-funded mathematics nearly decided that there couldn't be a proof of the general theorem because the tons of state-funded mathematicians would have already found it.

Well, that was a wrong expectation. The proof existed and was ultimately found by Wiles but it required some modern mathematical techniques that were probably unavailable to Fermat. It seems virtually impossible for Fermat to possess a proof that is nearly equivalent to Wiles': the type of mathematical technology these two men could use differed as much as nuclear reactors differ from steam engines. You just don't expect James Watt to play with similar devices as Robert Oppenheimer.

However, it's somewhat more imaginable – although still insanely sounding - that Fermat had a more elementary proof, one which remains unknown to us. Most likely, Fermat either made a mistake or he deliberately wanted to present himself as a super-genius by a false claim that "he has a wonderful proof that doesn't fit to this small margin". However, he didn't quite fool us because we have rather good reasons to think that Fermat didn't have a proof. ;-)

In the 1970s, the conjecture was returning to mainstream maths because it was realized that Fermat's Last Theorem followed from a modularity theorem ("Taniyama-Shimura"). This theorem says that every elliptic curve – a torus written using complex variables as\[

y^2 = x^3+ax+b

\] and I may publish a crash course in F-theory where I explain these matters in a near future – is a modular curve, one written using the \(j\)-invariant and enjoying lots of special mathematical properties.

These elliptic curves were relevant for the validity of Fermat's Last ex-Conjecture because if there were a counterexample to Fermat's negative claim, you could also construct an associated elliptic curve that isn't modular (weird!), in contradiction with the modularity theorem. These ideas provided the mathematicians with a "sketch" of proofs and the actual proofs were gradually found, by the 1980s. It was firmly proven that the modularity conjecture, if true, implied Fermat's Last Theorem, and the so-called \(\varepsilon\)-theorem was one of the last pieces needed to establish these links.

So the remaining task – one that Wiles solved – was to prove the modularity conjecture for some curves (equations). He did so. The original strategy was to "count and match" the elliptic curves and the modular forms. The minimal implementation of this strategy didn't work so he decided to match the Galois representations instead.

At the general level, the proof of the Riemann Hypothesis will follow the same strategy. A counterexample to the Riemann Hypothesis – a non-trivial root of the zeta-function away from the critical axis – would probably allow you to construct some weird mathematical object that can't exist, either. We may even know what the next step – the rough type of this "weird object" – is. It's probably a weird, non-real eigenvalue of a Hermitian operator or (my strategy) a non-existent representation of \(SL(2,\ZZ)\). Still, we need to know the third step to be closer to a solution.

I've spent hundreds of hours with the Riemann Hypothesis in my life and yes, I repeatedly thought that I essentially had it. ;-)