History
Specifically, Conway and Norton, following an initial observation by John McKay, found that the Fourier expansion of j(τ) (sequence A000521 in OEIS), with τ denoting the half-period ratio, could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (sequence A001379 in OEIS):
where, and
Conway and Norton formulated conjectures concerning the functions obtained by replacing the traces on the identity by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus zero. In other words, if Gg is the subgroup of SL2(R) which fixes, then the quotient of the upper half of the complex plane by Gg is a sphere with a finite number of points removed, corresponding to the cusps of Gg.
It turns out that lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by 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. Borcherds won the Fields medal for his work, and more connections between M and the j-function were subsequently discovered.
Read more about this topic: Monstrous Moonshine
Famous quotes containing the word history:
“To care for the quarrels of the past, to identify oneself passionately with a cause that became, politically speaking, a losing cause with the birth of the modern world, is to experience a kind of straining against reality, a rebellious nonconformity that, again, is rare in America, where children are instructed in the virtues of the system they live under, as though history had achieved a happy ending in American civics.”
—Mary McCarthy (1912–1989)
“No event in American history is more misunderstood than the Vietnam War. It was misreported then, and it is misremembered now.”
—Richard M. Nixon (b. 1913)
“All history attests that man has subjected woman to his will, used her as a means to promote his selfish gratification, to minister to his sensual pleasures, to be instrumental in promoting his comfort; but never has he desired to elevate her to that rank she was created to fill. He has done all he could to debase and enslave her mind; and now he looks triumphantly on the ruin he has wrought, and say, the being he has thus deeply injured is his inferior.”
—Sarah M. Grimke (1792–1873)