https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/

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.[…]

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 the j-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.”[…]

Monstrous MoonshineThe symmetries of any given shape have a natural sort of arithmetic to them. For example, rotating a square 90 degrees and then flipping it horizontally is the same as flipping it across a diagonal — in other words, “90-degree rotation + horizontal flip = diagonal flip.” During the 19th century, mathematicians realized that they could distill this type of arithmetic into an algebraic entity called a group. The same abstract group can represent the symmetries of many different shapes, giving mathematicians a tidy way to understand the commonalities in different shapes.

Over much of the 20th century, mathematicians worked to classify all possible groups, and they gradually discovered something strange: While most simple finite groups fell into natural categories, there were 26 oddballs that defied categorization. Of these, the biggest, and the last to be discovered, was the monster.

Before McKay’s serendipitous discovery nearly four decades ago, there was no reason to think the monster group had anything to do with the j-function, the second protagonist of the monstrous-moonshine story. The j-function belongs to a special class of functions whose graphs have repeating patterns similar to

M. C. Escher’s tessellation of a disk with angels and devils, which shrink ever smaller as they approach the outer boundary. These “modular” functions are the heroes of number theory, playing a crucial role, for instance, in Andrew Wiles’ 1994 proof of Fermat’s Last Theorem. “Any time you hear about a striking result in number theory, there’s a high chance that it’s really a statement about modular forms,” Kachru said.[…]

Moonshine’s ShadowsIn 1913, the English mathematician G. H. Hardy received a letter from an accounting clerk in Madras, India, describing some mathematical formulas he had discovered. Many of them were old hat, and some were flat-out wrong, but on the final page were three formulas that blew Hardy’s mind. “They must be true,” wrote Hardy, who promptly invited the clerk, Srinivasa Ramanujan, to England, “because, if they were not true, no one would have the imagination to invent them.”

Ramanujan became famous for seemingly pulling mathematical relationships out of thin air, and he credited many of his discoveries to the goddess Namagiri, who appeared to him in visions, he said. His mathematical career was tragically brief, and in 1920, as he lay dying in India at age 32, he wrote Hardy another letter saying that he had discovered what he called “mock theta” functions, which entered into mathematics “beautifully.” Ramanujan listed 17 examples of these functions, but didn’t explain what they had in common. The question remained open for more than eight decades, until Sander Zwegers, then a graduate student of Zagier’s and now a professor at the University of Cologne in Germany, figured out in 2002 that they are all examples of what came to be known as mock modular forms.

After the Zurich moonshine conference, Cheng, Duncan and Harvey gradually figured out that M24 moonshine is one of 23 different moonshines, each making a connection between the special dimensions of a group and the coefficients of a mock modular form — just as monstrous moonshine made a connection between the monster group and the j-function. For each of these moonshines, the researchers conjectured, there is a string theory like the one in monstrous moonshine, in which the mock modular form counts the string states and the group captures the model’s symmetry. A mock modular form always has an associated modular function called its “shadow,” so they named their hypothesis the Umbral Moonshine Conjecture — umbra is Latin for “shadow.”

Many of the mock modular forms that appear in the conjecture are among the 17 special examples Ramanujan listed in his prophetic letter.[…]

.

.

.

Love it. It’s all stringed together…and coming back to me again. I remember studying Ramanujan for a math class, and picking Escher apart (Deconstructionism) while designing *A House For Escher* for one of my architecture design studio projects. And of course, this song from the growing years.

**Moonlight Shadow (1983):**

The last that ever she saw him

Carried away by a moonlight shadow

He passed on, worried and warning

Carried away by a moonlight shadowLost in a riddle that Saturday night

Far away on the other side

He was caught in the middle of a desperate fight

And she couldn’t find how to push throughThe trees that whisper in the evening

Carried away by a moonlight shadow

Sing a song of sorrow and grieving

Carried away by a moonlight shadowAll she saw was a silhouette of a gun

Far away on the other side

He was shot six times by a man on the run

And she couldn’t find how to push throughI stay, I pray

See you in Heaven, far away

I stay, I pray

See you in Heaven one day4 am in the morning

Carried away by a moonlight shadow

I watched your vision forming

Carried away by a moonlight shadowStars move slowly in a silvery night

Far away on the other side

Will you come to talk to me this night

But she couldn’t find how to push throughI stay, I pray

See you in Heaven, far away

I stay, I pray

See you in Heaven one dayCaught in the middle of a hundred and five

Far away on the other side

The night was heavy and the air was alive

But she couldn’t find how to push throughCarried away by a moonlight shadow

Carried away by a moonlight shadow

Far away on the other side