Tuesday, August 30, 2011

Monstrous Moonshine

In Stanisław Lem’s famous philosophical-SF novel, ‘His Master’s Voice’, an alien message is received from the stars. A Manhattan Project-style research establishment is set up to decode it, the world’s best minds from all disciplines are recruited - but the years pass while no progress is made. The conclusion is stark – whatever the message is, it’s just ... well, too difficult for humans ever to understand it, the aliens remain forever inaccessible in their esoteric strangeness.

Lem’s point is well-taken: the alien is likely to be more different from us than we can possibly imagine but we don’t need to wait for strange symbols from the sky. We have an existence proof both of extraordinary conceptual difficulty and astonishing personal idiosyncrasy right in front of us.

Let’s talk about Monstrous Moonshine.

Monstrous Moonshine is so incomprehensible that I can’t even give you a summary here – if you’re up for it there’s a note at the end. Like that message from space, the basic concepts are impossibly-hard to comprehend (except to a small handful of specialist mathematicians). The two leaders in research on the Monster and the Monstrous Moonshine conjectures were the famous Professor John Conway (also the inventor of the Game of Life) and the little-known Dr Simon Norton. While Professor Conway continues to work at Princeton University, Dr Norton has just been the subject of an amiable biography which I highlight below: he is a most unusual person.

Today in his late fifties, the genius Dr Norton is ‘unemployed and unemployable’. He owns a Victorian town houses fronting Jesus’ Green in Cambridge, England where he lives by himself in the basement surrounded by deeply-piled squalor and piles of bus-timetables. He dines exclusively on tinned mackerel and boiled rice.

Norton spends his days taking bus rides around the United Kingdom as part of his campaign to bolster public transport (he is fanatically anti-car). Occasionally he is invited to a Mathematical Research Colloquium where he blows away that small fraction of his listeners who can follow him with penetrating analysis of ‘The Monster’.

At age three and half, Simon’s IQ was rated at 178. A prodigy, he represented England three years in succession at the International Mathematics Olympiad where he scored a perfect 100% one year and 95% the next. While in his mid-teens at Eton he took a first class degree in mathematics with London University and then continued to Cambridge, where after his PhD he joined the Maths faculty, working with his doctoral supervisor Professor John Conway on the classification of simple finite groups.

His colleagues bewail the almost complete loss of Simon’s mathematical productivity since Conway left for America 25 years ago (‘the collapse of genius’) but Norton doesn’t see it that way at all: he has no sense that one part of his life is more or less important than any other: the crafting together of optimal bus routes is as interesting and fulfilling to him as the deepest problems of mathematics.

The interesting thing for me is the uniqueness of Dr Norton’s interior life: which is how I think Stanisław Lem imagined it would be with the alien.


The Hard Stuff

The Monster and Monstrous Moonshine are inaccessible unless you took Group Theory in your math MSc. If, however, you have some background then it is possible to get a sense of what it’s all about (if you’ve forgotten what a group is, check here).

It turns out that there are some special groups which can be used to create all possible groups, rather like the way prime numbers can be multiplied together to create all other numbers. These special groups are called the simple groups, and in the third quarter of the twentieth century it was found that they could be systematically classified. The multi-year international campaign to document the complete Atlas of Finite Groups was led by a five-man team at Cambridge University, headed by Professor John Conway (also the inventor of the Game of Life) and Dr Simon Norton, discussed above.

Wikipedia tells us that ‘The list of finite simple groups consists of 18 countably-infinite families, plus 26 sporadic groups that do not follow such a systematic pattern. The Monster group is the largest of these sporadic groups and contains all but six of the other sporadic groups as subquotients.’

Clear enough?

Reading on we discover that ‘The Monster’ is not just large; it’s enormous, with 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements. The Monster has a 196,883-dimensional representation, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.

As they were coming to the end of their task, Conway and Norton were alerted to a strange coincidence between the 196,883-dimensional representation of ‘The Monster’ and the coefficients of the Fourier expansion of a particular modular function, j. They made some conjectures concerning this unlikely relationship which they dubbed ‘Monstrous Moonshine’ – (these were later proven by Fields Medallist Richard Ewen Borcherds in 1992 ‘using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac-Moody superalgebras’).

Monstrous Moonshine has been the subject of several books. Symmetry and the Monster, by Mark Ronan, tells you a lot about elementary symmetry and the history of group theory, but nothing much about the topic under consideration; Finding Moonshine by the better-known Marcus Du Sautoy tells you quite a bit about elementary group theory and some biographical details of Professor Sautoy himself, but little of substance about the central mathematical topic.

Like I said, this stuff is inaccessible unless you’re a post-graduate mathematician specialising in the area. But see this great cartoon.


Originally written for sciencefiction.com but decided here was better.