Monstrous Moonshine Conjecture

Date: 11/12/98 at 00:30:20
From: Daniel
Subject: Monster moonshine conjecture

I've been reading a lot on the mystery of the monster moonshine 
conjecture. Quite frankly I don't exactly understand it.  

I do know that it involves some sort of correlation between the 
symmetries of a Hilbert dimensional object and the j function of 
ellipses. 

First, I don't entirely understand the j function or the monster.  
Second, I don't understand how it was proven using the string theory.  

I know this is a very advanced question and that the answer will 
probobly be over my head. I would like to research this more, to a 
point of moderate understanding at least. I thank you for your time and 
any information you can give to me on this subject.

Daniel Hayes

Date: 11/12/98 at 13:10:01
From: Doctor Rob
Subject: Re: Monster moonshine conjecture

Quite frankly, very few mathematicians even understand it!

There is an article in the current issue of Scientific American that
would be a good place to start. It is online at:

   Monstrous Moonshine Is True - W. Wayt Gibbs
  http://www.sciam.com/1998/1198issue/1198profile.html   

First of all, the j-function is associated with elliptic curves, not
ellipses. This is a confusing name, I know. An elliptic curve is a
nonsingular curve whose equation in coordinates x and y has degree 3.
An ellipse has an equation of degree 2. The reason for the name has
to do with elliptic integrals, which arise when trying to find the arc 
length of an ellipse. These integrals evaluated from 0 to x give a 
function of x whose inverse function is called an elliptic function. 
All these functions are linear combinations of a certain function P
and its derivative P'. It turns out that this function satisfies the
differential equation:

   P'^2 = 4*P^3 - g4*P - g6

If you put y = P', x = P, you get the cubic equation of an elliptic
curve. The quantities g4 and g6 are parameters which vary as you vary 
the parameters of the ellipse. It also turns out that if you let x and 
y have complex values, the graph of this curve can be stretched into a 
torus. A torus is just a parallelogram with the opposite sides glued 
together. These parallelograms can be specified with a single complex 
number tau (in the complex plane, one corner is at the origin, one 
corner is at 1, one corner is at tau, and the last corner is at 1+tau). 
Then the function j(tau) is defined as the sum of a convergent infinite 
series. It is also definable in terms of g4 and g6 above. 

The j-function has lots of interesting properties which I can't 
possibly describe briefly!

The Monster Group is a finite simple group. That means that it doesn't 
have any normal subgroups. There are a few infinite classes of finite 
simple groups that are well known. There is also a finite number of 
other, so-called "sporadic" finite simple groups, of which it is the 
largest. It does have lots of non-normal subgroups, however, and they 
include all the smaller sporadic simple groups. It was a great 
achievement of the last quarter of the 20th century that the Monster 
Group was shown to exist, and the complete list of sporadic finite 
simple groups was given and proven complete. Now the Monster is very 
large, so we seldom deal with it explicitly. Instead we talk about 
homomorphic images of it in square matrices, called its group 
representations. The size of the matrices is called the degree of the 
representation.

It is the degrees of the representations of the Monster Group which
seemed to coincide with coefficients in the power series expansion of
the j-function. That is a very unlikely pair of things to be connected, 
yet that is just what the Monstrous Moonshine conjecture stated, and 
what has recently been proven.

I cannot say anything at all about string theory, I regret to say.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/