Thursday, March 09, 2006

(153) Messrs Borcherds, the Monster and other Moonshine beings

On August 28th 1998 I found myself thinking of someone from the Midlands who had been a talented teenage chess player back in the seventies; R.E.Borcherds.

The above photograph from January 1975 shows he and I seated next to each other as two of thirty or so opponents that Soviet Grandmaster, Alexander Belyavsky played simultaneously, in London.
My hands are clasped and I look upwards. Borcherds is on my left.

That afternoon I noticed a letter in The Times from a Midlands academic.
His name was Peter Borcherds, and I reasoned that this must be a relative, perhaps his father.

On November 14th 1998 I picked up a discarded copy of Scientific American that lay on a table in the Cherry Orchard Hotel in Port Erin, Isle Of Man.
It contained this article -

Monstrous Moonshine Is True - Richard Borcherds proved it - and discovered spooky connections between the smallest objects imagined by physics and one of the most complex objects known to mathematics.
It was the same chap, now a thirty-eight year old Cambridge mathematician. I had come across no reference to him in over twenty years.
Borcherds had gained a Fields Medal, the highest honour in maths.
It carries a Latin inscription urging its bearer to "transcend human limitations and grasp the universe."

A greatly simplified version of his achievement might go something like this -

In 1978 the mathematician J.McKay of Concordia University noticed a rather bizarre coincidence between a particular elliptic modular function known as the i function and the first number of a massive group known as the Monster.
An object, e.g a cube, has a finite number of ways in which it can be twisted around and still retain its shape. With a cube there are 24 symmetries. A tetrahedon has 12.

The Monster group, the largest sporadic, finite, simple group, is comprised of the symmetries of geometric objects and other mathematical constructs. The Monster was predicted to exist for quite a few years before it was actually constructed. It represents the symmetries of - well, what exactly the mathematicians had not a clue. Something that is a bit too complex to call a mere geometric object because the Monster lives not in 3 dimensions but in 196,883. And in 21,296,876 dimensions and in all the higher dimensions listed in the first column of elliptic modular functions.

McKay noticed that the third coefficient of the i function, when it is written as an infinitely long sum, is 196,884.

This is the sum of the first two numbers in the Monster Group, 1 and 196,883.

So far removed are modular functions from finite groups that there was no connection that anyone could imagine, but eventually it was realised that the coincidences ran too deep for them to be ignored, for it transpires that every coefficient of the modular function is a simple sum of the numbers of this list of dimensions in which the Monster lives.

Thus arose the conviction that these could not be just coincidences, but rather they were aspects of some deeper unity.
They dubbed the wild conjecture ‘Moonshine’ and a new speciality arose in mathematics to try to prove it.

Borcherds then linked in a third, and quite separate, area.

One Physical theory of how matter may operate at its smallest levels is called String Theory. A particular string theory, when applied to a particular shape (a folded doughnut in 26 dimensions!) has more than 10 to the power 53 symmetries and produces the Monster group.

Borcherds showed that the Monster is simply the group of all the symmetries of this particular string theory - a theory that almost certainly has nothing to do with the universe we inhabit.

Some scientists did not think that it all ended there.
The existence of so many spooky coincidences prompted the suspicion that there is something even larger and deeper beneath it, something that hasn’t been found yet.

John C. Baez of the University of California commented
"Borcherds has begun to uncover it. But there are still a lot of mysteries left."

The discovery of this profound and mysterious truth arose from the recognition that certain coincidences ran too deep to be just coincidences.

... ... ...
On an evening in early December 1998 I happened to think back to a lady I met at the BBC in 1985.
Her memorable surname was Moonshine and she had reimbursed my travel expenses when I had appeared on a chess programme.
I remembered that I had asked if she had heard of the Moonshine Uwarie, and then told her that this was a native name for a South American possum, which it was believed only came out in the moonshine. I had read of it in Gerald Durrell’s book Three Singles To Adventure.

The next day in December 1998 I saw her name mentioned on the credits to a 1998 BBC programme.

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