Proving the `moonshine conjecture’ won Professor Richard Borcherds the maths equivalent of the Nobel prize. He tells Simon Singh about the trials of living in a world few can enter
There is a – probably – apocryphal explanation for why no Nobel prize has ever been awarded for mathematics. The story goes that Alfred Nobel’s wife had an affair with a mathematician, and, as an act of revenge, Nobel omitted mathematics from the list of his prizes. Instead, mathematicians have had to make do with the less glamorous Fields Medal, awarded just once every four years.
Two weeks ago, Professor Richard Borcherds received one of these coveted medals. But how does one begin to explain why the South African-born scientist is an intellectual giant? He has not discovered a black hole, he has not found the origin of life, nor has he invented a new vaccine. Instead, he has proved the so-called “moonshine conjecture”, one of the most abstract and esoteric achievements imaginable.
Borcherds is 38, bearded, spectacled, slightly nervous and a genius. Everybody knows that he is brilliant, the problem is that nobody understands why. Only one other person in Cambridge really comprehends his calculation, and the two men rarely meet. It seems natural that somebody might become bitter and frustrated at the failure of others to understand his work, but for Borcherds it is not a problem. He has proved the moonshine conjecture, those who need to know have acknowledged it, and nothing else matters.
All the newspapers that announced Borcherds’s award linked him with Trinity College, Cambridge, but I was not to meet him in the Great Court, in the Wren Library, or Isaac Newton’s study. Instead, I wander into the maths department, which occupies a decrepit building. Although there are plans to build a shiny new department, with large open spaces designed to encourage brainstorming and collaboration, Borcherds is happy where he is. He does not like to collaborate, and is content to spend most of the day in his spartan office, scribbling at his desk or staring out of his window.
As we talk, the professor reminds me of a youthful Captain Haddock, with an unnerving penchant for balancing on the hind legs of his chair. On several occasions, he begins to topple backwards and grabs the desk just in time to save himself. Other than the hundreds of books on mathematical group theory, the office contains nothing but cycling paraphernalia, a lego dinosaur and two congratulations cards.
Borcherds was born in South Africa, but left at six months and spent his childhood in Birmingham, United Kingdom. He recalls being top of the class at school, but is quick to draw a distinction between being good at maths and being great. Many can understand established mathematics, but few can create new ideas, develop original proofs and solve long- standing problems.
Even as a young researcher at Cambridge, he suffered from insecurity. “I wasn’t getting very far. Most of the time I was struggling to keep my job. I’d see other people my age, such as Simon Donaldson [1986 Fields Medallist], being considerably more successful and I thought I’m obviously not all that good. There were times when I thought of dropping out.”
Mathematics was all that had ever captivated him. Even today, he has no real interests outside his work. Visits to the cinema are merely opportunities to relax, periods when his subconscious can take over the calculating. “My idea of a good film is Godzilla. I thought the critics were absolutely wrong, because it delivered exactly what it promised – a 200-foot monster stomping all over New York. I am currently waiting for the next Star Wars movie to come out.”
In the early 1980s, Borcherds created his first significant and original piece of maths. He had been reading some physics papers, which used a simple cross (vertex) to represent interacting particles. Borcherds was intrigued because physicists had used new calculations to predict what would happen at a particular vertex, but he was also annoyed by the sloppiness of the mathematics.
Physicists are notorious for their lack of rigour in comparison with mathematicians, and we simultaneously recall an old joke which highlights the difference: a meticulous mathematician, a sloppy physicist and an even sloppier astronomer are on their way to Scotland. They cross the border and observe a black sheep in the middle of a field. “Look,” exclaims the astronomer, “all Scottish sheep are black!” The physicist responds: “No, no! Some Scottish sheep are black!” The mathematician shakes his head, takes a breath and proclaims, “Gentlemen, all we can truly say is that in Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.”
Borcherds applied mathematical rigour to what he had uncovered and created a new, rich area of research, which he called vertex algebras. The significance of his discovery was clear to him immediately, but others were slow to appreciate the work of a young researcher with no reputation. “I was pretty pleased with it at the time,” he remembers. “But after a few years, I got a bit disillusioned, because it was obvious that nobody else was really interested in it. There is no point in having an idea that is so complicated that nobody can understand it.”
Borcherds admits that being ignored was partly his fault. He finds it difficult to communicate, preferring to read a published paper, rather than talk to the author, and he no longer teaches or gives tutorials.
His wife sometimes claims that he has Asperger’s Syndrome, a very mild form of autism characterised by introversion and lack of emotion. He doesn’t seem too bothered. “I’ve got a hell of a lot of the symptoms. I once read that there are six signs of Asperger’s Syndrome, and I said to myself, `Hey, I’ve got five of those.'”
It took several years for Borcherds’s vertex algebras to be accepted by the community at large, and in the meantime he concentrated on something else. The problem that had attracted him throughout the 1980s, and that would ultimately bring him recognition, was related to the strangely named moonshine conjecture, which concerns the idea of symmetry.
A cube can be reflected and rotated in a number of ways, in such a way that it apparently remains unchanged. In fact, there are 24 distinct symmetries for a cube, which is quite a lot, but nothing compared to the number of symmetries possessed by the Monster.
The Monster is a purely mathematical and unimaginable object which lives in 196 883 dimensions, and it has 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 symmetries.
Some mathematicians had spotted that numbers associated with the Monster group appeared in an apparently unrelated area of mathematics called number theory. Initially, it was considered nothing more than a coincidence because it seemed impossible that two such diverse areas could have something in common. It was the mathematical equivalent of suggesting that there is a direct artistic link between Beethoven’s symphonies and Aqua’s Barbie Girl.
The idea of a link gradually gained a modicum of respectability and was formally called a conjecture; that is, an interesting but unproven theory. The “moonshine” was added because the term has long been used to describe absurd scientific ideas. Ernest Rutherford once said that it was moonshine to suggest that we could ever obtain energy from atoms.
The challenge for mathematicians was to prove that the moonshine conjecture was true. It is worth noting that the proof would be of no practical use whatsoever. The motivation for such problems is merely curiosity. Borcherds worked on the conjecture for eight years without making any real progress, and throughout this period he was still worried that he had not established his reputation as a mathematician.
Then, in 1989, he had an insight which essentially proved the conjecture. “I was in Kashmir. I had been travelling around northern India and there was one really long, tiresome bus journey, which lasted about 24 hours. Then the bus had to stop because there was a landslide and we couldn’t go any further. It was all pretty darn unpleasant. Anyway, I was toying with some calculations on this bus journey and finally I found an idea which made everything work.”
Borcherds had solved one of the most intractable problems in maths. However, his travelling companion was not a mathematician and could not appreciate what he had done. As a pure mathematician, he has had to get used to the fact that nobody understands what he does. Specialisation means that even his mathematical wife Ursula (a tall, slim, cheerful topologist) has not been able to fully grasp his proof of the moonshine conjecture. Similarly, Borcherds cannot fully comprehend her work.
But Borcherds claims that lack of understanding from others does not bother him, and that what really matters is the satisfaction of solving a great problem. Even the award of a Fields Medal is not as important to him as completing an immense calculation – indeed, his reaction to the news was decidedly lukewarm.
“I didn’t really feel anything,” he says. “Before the award, I used to think it was terribly important, but now I realise that it’s meaningless. However, I was over the moon when I proved the moonshine conjecture. If I get a good result, I spend several days feeling really happy about it. I sometimes wonder if this is the feeling you get when you take certain drugs. I don’t actually know, as I have not tested this theory of mine.”
Simon Singh is the author of Fermat’s Last Theorem