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An example of why mathematics is so appealing

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Erica Klarreich writes in Quanta:

In 1978, the mathematician John McKay noticed what seemed like an odd coincidence. He had been studying the different ways of representing the structure of a mysterious entity called the monster group, a gargantuan algebraic object that, mathematicians believed, captured a new kind of symmetry. Mathematicians weren’t sure that the monster group actually existed, but they knew that if it did exist, it acted in special ways in particular dimensions, the first two of which were 1 and 196,883.

McKay, of Concordia University in Montreal, happened to pick up a mathematics paper in a completely different field, involving something called the j-function, one of the most fundamental objects in number theory. Strangely enough, this function’s first important coefficient is 196,884, which McKay instantly recognized as the sum of the monster’s first two special dimensions.

Most mathematicians dismissed the finding as a fluke, since there was no reason to expect the monster and the j-function to be even remotely related. However, the connection caught the attention of John Thompson, a Fields medalist now at the University of Florida in Gainsville, who made an additional discovery. The j-function’s second coefficient, 21,493,760, is the sum of the first three special dimensions of the monster: 1 + 196,883 + 21,296,876. It seemed as if the j-function was somehow controlling the structure of the elusive monster group.

Soon, two other mathematicians had demonstrated so many of these numerical relationships that it no longer seemed possible that they were mere coincidences. In a 1979 paper called “Monstrous Moonshine,” the pair — John Conway, now of Princeton University, and Simon Norton — conjectured that these relationships must result from some deep connection between the monster group and thej-function. “They called it moonshine because it appeared so far-fetched,” said Don Zagier, a director of the Max Planck Institute for Mathematics in Bonn, Germany. “They were such wild ideas that it seemed like wishful thinking to imagine anyone could ever prove them.”

It took several more years before mathematicians succeeded in even constructing the monster group, but they had a good excuse: The monster has more than 1053 elements, which is more than the number of atoms in a thousand Earths. In 1992, a decade after Robert Griess of the University of Michigan constructed the monster, Richard Borcherds tamed the wild ideas of monstrous moonshine, eventually earning a Fields Medal for this work. Borcherds, of the University of California, Berkeley, proved that there was a bridge between the two distant realms of mathematics in which the monster and the j-function live: namely, string theory, the counterintuitive idea that the universe has tiny hidden dimensions, too small to measure, in which strings vibrate to produce the physical effects we experience at the macroscopic scale.

Borcherds’ discovery touched off a revolution in pure mathematics, leading to a new field known as generalized Kac-Moody algebras. But from a string theory point of view, it was something of a backwater. The 24-dimensional string theory model that linked the j-function and the monster was far removed from the models string theorists were most excited about. “It seemed like just an esoteric corner of the theory, without much physical interest, although the math results were startling,” said Shamit Kachru, a string theorist at Stanford University.

But now moonshine is undergoing a renaissance, one that may eventually have deep implications for string theory. Over the past five years, starting with a discovery analogous to McKay’s, mathematicians and physicists have come to realize that monstrous moonshine is just the start of the story.

Last week, . . .

Continue reading. Fascinating. The last two paragraphs:

Physicists are also excited about a highly conjectural connection between moonshine and quantum gravity, the as-yet-undiscovered theory that will unite general relativity and quantum mechanics. In 2007, the physicist Edward Witten, of the Institute for Advanced Study in Princeton, N.J., speculated that the string theory in monstrous moonshine should offer a way to construct a model of three-dimensional quantum gravity, in which 194 natural categories of elements in the monster group correspond to 194 classes of black holes. Umbral moonshine may lead physicists to similar conjectures, giving hints of where to look for a quantum gravity theory. “That is a big hope for the field,” Duncan said.

The new numerical proof of the Umbral Moonshine Conjecture is “like looking for an animal on Mars and seeing its footprint, so we know it’s there,” Zagier said. Now, researchers have to find the animal — the string theory that would illuminate all these deep connections. “We really want to get our hands on it,” Zagier said.

Written by LeisureGuy

12 March 2015 at 5:15 pm

Posted in Math, Science

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