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Monstrous Moonshine conjecture

The Monstrous Moonshine conjecture is an outrageous idea that stemmed from an observation made by John McKay of Concordia University in 1978. McKay was leafing through a table of abstruse mathematical data, giving possible values for coefficients of the j-function of certain elliptic curves, when he noticed the number 196,884 in the expression j(q) = q - 1 + 196884q + 21493760q2 + .... In a moment of inspiration, he recognized this number as being one more than the number of dimensions in which the Monster group can be most simply represented. Looking into this "coincidence" more closely, he found that it was no coincidence at all. In fact, all the coefficients of the j-function were simple combinations of the degrees of representations of the Monster. This pointed to some deep connection between two seemingly unrelated areas of mathematics. On the one had were the coefficients of what is called an elliptic modular function – exactly the kind of function that would play a key role in the proof of Fermat's last theorem. On the other was the number of dimensions, and combinations of degrees, of a crystal lattice whose symmetry rotations and reflections formed the Monster.

Subsequently, McKay and a few other mathematicians, including John Conway and Simon Norton, drew out the link between elliptic modular functions and the Monster in a proposition christened, because of its fantastic nature, the Monstrous Moonshine.1 In 1998, this conjecture was proved by Richard Borcherds (a former student of Conway's) at the University of California at Berkeley. Astonishingly, Borcherds proof reveals a deep relationship between elliptic curves, the Monster Group, and string theory – the most promising theory on offer to unify our understanding of nature at the subatomic level. Borcherds showed that the Monster is the group of symmetries of 26-dimensional strings expressed in a form known as vertex algebra. Some people believe the connection may run even deeper and that Monstrous Moonshine may hold clues to the very existence of the reality in which we live.

Reference

1. Conway, J. H., and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc., 11: 308-339 (1979).