Borcherds’ Fields Medalist accomplishment

Source: Simon Singh website, Aug 1998

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.

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