MARCH OF THE MONSTERS: MONSTER GROUP
BY SWAP
Right when March of the Monsters started, I talked Swap into writing a mathematical monster story in honor of Pi Day. I told him i had wanted to write a story about a mathematician being driven slowly mad by a number or an equation that was somehow alive, only i didn't have the math background to do it so i was drafting him, and in that inimitable Swappy way that makes it impossible to hate him he pointed out that my premise was way cliched (tell me something i don't know, asshole!) BUT that he would love to do it. To which i said !!! because i'm the dude that always knows what to say.
And in the meantime, he left E2, at least temporarily. But like any good programmer would do, he left me a note with a link to the story he had written, and told me to post it for him. (!!!)
This is what he wrote. He asked me to edit it, said he viewed it as a collaboration, but i haven't. The only thing i've done to it is clean up typos.
--DM
"Don't you think it's weird that the Monster exists?"
I looked up from my computation of a certain group cohomology, and
peered impatiently at Brown. The grad lounge where I liked to spend
evenings in case that some interested young student would miraculously
come by with ideas that I had not yet considered was getting dusky as
the sun set upon the city, for which the lounge afforded us a
magnificent view. Brown was just talking almost to himself, glancing
out the window at the aforementioned spectacle of the city skyline.
"What do you mean?" I indulged him. Brown was a group theorist,
getting old, out of ideas, tenured and mostly teaching, so he seemed
to just get mystical about the basic stuff. I was younger than him,
but not by much, so perhaps a sense of future vicarious sympathy
for myself is what makes me listen to his old fogey cogitations.
"Well, I mean, just think about it for a while. I know it's common to
us by now. I mean, the damn thing has been known for almost thirty
years now, old hat, but just stop to think about it for a while. Here
we are, groups, confound it, just groups. The things have
even fewer assumptions than numbers. I mean, in a sense, of course it
makes sense, numbers satisfy lots more properties than just groups,
they're much more restricted, so perhaps it's not so surprising that
groups hold deeper magic in a sense than numbers."
There was that aimless rambling of Brown's I was fearing. I decided to
listen to him, since I wasn't getting much of anywhere with my
computations.
"The building blocks of numbers," he continued, "prime
numbers, well, of course there's all of number theory and the
Riemann zeta function and the prime number theorem, and so on, but
in a sense, primes are much simpler than their cousins, the simple groupsin group theory. And there are an infinite number of primes, and most primes are
alike; the only interesting thing is where they happen and how often,
but in the end, infinitely many primes, QED. Of course, for these
groups, these quantifications of symmetry, there are also infinitely
many simple groups, and also but then we have these aberrations that
don't fit into any of the three families. And there's nothing quite
like that anywhere else. If you believe the proofs spread out over 50
years of group theory in various journals, all of the building blocks
of finite groups are in one of three families, and then these 26
aberrations. There are no aberrations in the prime numbers, except
if you want to be funny and say that 2 is an odd prime because
it's even, but we have 26 or 27 aberrant groups. Crowning these
aberrations is the Monster, that group of an unimaginable size, the
symmetry group of a 196883-dimensional algebra, an aberration that
contains almost all but 6 of the other aberrations, a giant group, a
freak of nature, and a giant one at that, indivisible, encompassing,
supersymmetric, too large to fit into our most powerful computers."
"What are you getting at?" I interrupted, my patience again waning.
"Have you been working with the Monster's Griess algebra or
something?"
"No. Not really. I've been doing the usual, teaching half-interested
undergrads about the easy stuff. You don't think much of it, but it
does keep me sharp, in a way, to teach them elementary subjects.
Anyways, I'm not getting at anything... just... oh, damn, Nielsen,
don't lose your sense of awe for the folklore and the common things
either. The Monster exists, there is a largest aberration, and not
even that, but the physicists have found it fitting to use it to
describe the symmetries of one their string theories.
That's all. It's strange that we have sporadic simple groups, and it's
strange that we have a largest one, and even if some primes are
special in there way, there's no sensible way in which you could say
that any prime is generally aberrant. It's strange. You should think
so too."
After reiterating his thesis, Brown again became silent and returned
to staring out the window that served as the lounge's south wall. A
few flakes of snow were already forming, an apt seasonal decoration
for our city. I knew better than to believe that Brown wasn't doing
anything with his staring. Mathematicians work is to think, and to
disturb him now would be to disturb a man at work, even if he had been
unable to show much for his work over the past years. There are those
rare cases, aberrations, perhaps, like Brown describes them, those
rare profs who at age fifty or even sixty and beyond still are
creative and productive, but Brown wasn't one of those. As ErdÃ's Pal
would have said, Brown was dead, not publishing anymore.
After a 30 second pause that I understood to mean that our
conversation was over, I returned to the computation I was undergoing.
I cursed my stupidity when I found a simple mistake, flipping a sign,
and the error had already propagated for two pages of calculation. I
retraced my steps, but I didn't finish computing the cohomology by the
time I left the grad lounge and the math department for my comfortable
but solitary apartment in the fashionable end of town.
Back home, getting ready to sleep, sipping on a warm mug of herbal
tea, no caffeine. Most of my colleagues lived on that drug, a few of
them actually went for the amphetamines, believing that this would
keep them sharp, anabolic steroids for the mind, but they never worked
for me. Dreams are my own stimulation. I often worked out problems in
my dreams too, often producing nonsense that worked in dream logic,
but the occasional breakthrough did come to me in slumber. For that
reason, I was content with staying off all further stimulation. A
dream-induced paper now and then was sufficient to satisfy my need.
I looked out my apartment window, to the people strolling by, in their
winter clothes, their breath visible. Polychromatic lovers in embrace,
youngsters, people my own age, workers, hucksters, artistes, buskers.
Tonight I was aloof. I like to watch, but on nights like these, I
prefer to stay a safe distance away from society. I didn't have
teaching duties tomorrow, so I would be able to do work from the
moment I woke up. This kept me content. My work is the only friend I
need for now.
Thinking of Brown's ramblings brought the Monster back to mind. It's
folklore, like he said. For us, this means that it's part of what
every mathematician "knows", a popular bit of trivia, but not
something most of us ever work with. I suppose on some days I wear a
group theorist hat, and I've never found occasion to work with the
Monster either. The giant supersymmetric beast holding 20
other smaller beasts in its belly. Hah. Like Sophie Germain said,
every good mathematician is also something of a poet.
I prodded a bit more at the computations I was working on earlier, but
I wasn't getting anywhere. I finished my tea, got into bed, and didn't
bother setting an alarm. My brain has a mind of its own, and would
wake me up when it felt like it had an idea it needed to work out. I
turned off the light and welcomed Morpheus' embrace.
Plains. Desolate. Vast. Brown, chiding me again, "Nielsen, don't lose
awe for the simple things", simple, simple, finite and simple, finite
simple groups. Three families, three happy families, 26 sporadic
simple groups, the aberrations, 6 pariahs, 20 simple groups inside
another, a huge group, an unimaginable group.
The plains aren't alright. No, wait. These aren't desolate plains.
They're desolate planes, nothing in them but the abstractions
of my mind. I'm in the centre of them all, at the origin. A set of one
hundred ninety-six thousand, eight hundred eighty-three mutually
perpendicular lines skewer my stomach, right where my centre of mass would be. I am spinning. I don't feel well. I
might lose my lunch. Or dinner. Or anything at all, if I had a body
here, if food made any sense. Spinning, gyrating, tossing, turning, in
all dimensions, in all at once, in only some at a time. In the
distance I see strings, chords, vibrating tiny loops, mimicking my
movements. My motion gets restricted to only a subset of the mutually
orthogonal lines. I hear a thunderous growl. From
everywhere at once, from inside me, echoed at various volumes
by twenty other voices.
Oh god. I'm it.
The Twenty move inside me. None of them divide me, for I am
indivisible. I divide others. I am Monster, devourer of worlds,
constituent of others. I should not exist. I am multiplied, reflected,
rotated, into more directions than my mind can actually conceive.
The number that describes the number of ways in which I can change is
an eight followed by fifty three more digits. I move through the
planes, the reflections provide me with motion, too much motion, my
mind can't bear the thought of it. The Twenty laugh, laugh at
themselves and at the echidna I am myself. They are my children,
residing in my belly, a litter that I shall never release, for I am
timeless, immaterial, immutable.
Now rotating on a different subplane. And I see Her. My baby. The Baby
Monster. My successor, my proudest offspring, part of me, one of the
Twenty, her symmetries almost rich enough to rival mine. She kisses me
from the inside, another freak of Nature, the one who will understand
me the best.
Light shines upon us. Afar, in a different universe
altogether, I see the Six, the pariahs, the ones that would not find
refuge in my belly. They are in darkness. Light is upon me only, not
them. But not any light. Silver light. Moonlight. Moonshine.
Monstrous moonshine, on me, part of me, meromorphic modular forms
begin swarming around me. I can't understand it yet, but we're somehow
the same. Somehow we're related. The nibble playfully at my
extremities.
But I cast them away. I am ONE, the largest, the Sporadic, the
Unexpected, the Irregular. I have no time for this hogwash.
And then I'm alone again. Back to familiar three dimensions. I'm me.
In the distance, in that other universe, I see It now. We've become
separate again. I see that almost amorphous mass of endlessly spinning
symmetry and the Twenty in it, forming the rest of the universe, all
possible universes, describing everything. Suddenly, the Monster
doesn't seem so monstrous. What's 196883 compared to infinity?
Nothing.
Cold sweat. Only three A.M. I've got to stop listening to
Brown before midnight.