The Guardian, 28 August 1998
According to a probably apocryphal story, Alfred Nobel’s wife had an affair with a mathematician, and as an act of revenge he omitted mathematics from the list of Nobel Prizes. Instead, mathematicians have the Fields Medal. Awarded every four years rather than annually, it is, if anything, even more prestigious than a Nobel Prize. On 18th August Professor Richard Borcherds received the Fields Medal. Unlike a Nobel Prize winner, it is difficult to difficult to write a single sentence which explains why he is an intellectual giant. He has not discovered a black hole, he has not found the origin of life, and he has not invented a new vaccine. Instead, Borcherds has proved the so-called “moonshine conjecture”, one of the most abstract and esoteric achievements imaginable.
Borcherds is a 38-year-old, bearded, spectacled, slightly nervous genius. His work virtually defies explanation, which means that everybody knows that he is brilliant, but nobody understands why. Only one other person in Cambridge really comprehends his calculation, and the two gentlemen rarely meet. It seems natural that somebody might become bitter and frustrated at the failure of others to understand his work, but as far as Borcherds is concerned, it is not a problem. He has proved the moonshine conjecture, those that need to know have acknowledged it, and nothing else matters.
All the newspapers which 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 in Isaac Newton’s study. Instead, I wandered to the maths department which occupies a decrepit building, with metallic industrial staircases, and corridors which have the drab feel of a deprived inner city school. 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 to the “Curry Mahal” opposite.
As we talked, the professor reminded me of a youthful Captain Haddock, with an unnerving penchant for balancing on the hind legs of his chair. On several occasions he began to topple backwards and grabbed 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 cards which say “Congratulations” in large letters. Apparently newsagents do not carry cards which specifically say “Well done on winning a Fields Medal”.
Borcherds was born in South Africa, but left at the age of six months, and spent his childhood in Birmingham. 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 are able to understand established mathematics, but few are able to 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 … great film. I thought the critics were absolutely wrong, because it delivered exactly what it promised – a two hundred foot monster stomping all over New York. I am currently waiting for the next Star Wars movie to come out.”
Then 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 calculate 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. I remember I used to give talks on vertex algebras, and usually nobody turned up. Then there was this one time when I got a really big audience. But there had been a misprint, and the title read “vortex algebras”, not “vertex algebras”. The audience was made up of fluid physicists, and when they realised it was a misprint, they weren’t interested either in what I had to say.”
Borcherds admits that being ignored was partly his own fault. He finds it difficult to communicate, and tends to avoid discussions with others. For example, he prefers 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 which is characterised by introversion and a lack of emotion. Borcherds considers it possible, but doesn’t seem to be too bothered. “I’ve got a hell of a lot of the symptoms. I once read something in a newspaper and it said there are six signs of Asperger’s Syndrome, and I said to myself, ‘Hey, I’ve got five of those.’”
It would take several years before Borcherds’s vertex algebras would be accepted by the community at large, and so in the meantime he concentrated on something else which had caught his attention, a problem which would ultimately bring him the recognition he sought. At this point it was lunch time, and so we left the department and headed to the sandwich bar. Typical of an absentminded professor, he forgot to collect his change. After the shopkeeper reminded him, he was quick to work out that she had given him too much. We sat on a fallen tree near the river, munched our sandwiches and continued our conversation.
He explained that the problem that attracted him throughout the 1980s 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 such that it apparently remains unchanged. In fact, there are 24 distinct symmetries for a cube, which is quite a few, 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, i.e., 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 the spring of 1989, he had an insight which essentially proved the conjecture. “I was in Kashmir. I had been traveling 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 just toying with some calculations on this bus journey and finally I found an idea which made everything work.”
Borcherds had solved one the most intractable problems in maths. However, his traveling 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, Richard can not fully comprehend her work.
He 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 important compared to completing an immense calculation, and his reaction to the news was lukewarm at best. “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.”