[impendia]

> The Langlands Programme was an effort in group theory to develop a representation (a way to generate all the elements in a group as far as I understand it) for all possible groups.

As a professional number theorist, I would not say that this is accurate.

A representation of a group, formally, is a homomorphism of that group to the group of automorphisms of some vector space. More informally, it is a way to make the group "act on something".

Here is an example. Consider the group generated by the symbols a, b, and their inverses a^{-1} and b^{-1}. As usual write a^2 = aa, a^0 = 1, etc. And impose the relations

a^4 = b^2 = 1, a^3 b = ba.

At first, this is very hard to understand. Which group elements are equivalent to which others? Is the group finite or infinite? If finite, how many elements does it have?

To construct a "representation", make a square out of cardboard, mark the edges and sides, and interpret the symbols in the following way: a means rotate clockwise 90 degrees, and b means flip across its vertical axis. a^{-1} and b^{-1} mean do the same thing in reverse.

Now you can see, for example, that this group has exactly eight elements -- corresponding to the positions of the cardboard you can reach.

For any group, there are always representations which you can easily construct, such as this one:

https://en.wikipedia.org/wiki/Regular_representation

But there are representations which are "hidden" in some sense, where it wasn't clear initially that they should exist at all. That's where the magic happens.

The Langlands program is extremely technical; I specialize in another area of number theory, and I only vaguely understand it myself. But very very roughly speaking, the Langlands program describes multiple ways of constructing certain kinds of group representations, and says that you end up constructing the same representations.

The modularity theorem, which was the linchpin in the proof of Fermat's Last Theorem, is an example of a theorem along these lines.

https://en.wikipedia.org/wiki/Modularity_theorem

And the Langlands program is very far from complete.

[doonesbury]

Thank you for bringing smarter, better information. I stand corrected then.

>a^4 = b^2 = 1, a^3 b = ba >has exactly eight elements

I suppose the missing issue (or at least the one I feel could have had more sunshine on it) is how and why elements of the monster groups are represented as this would more subtly capture the symmetry that keeps the group finite. It's not so much symmetry as a subject unto itself, as the subtle interplay between group operations on the form of elements with symmetry that keeps the group finite.

Your term "a^4=1" has exactly that missing emphasis: take element 'a' (i.e. a square) and rotate it four times and it's the same as multiplying a by 1, which is the identity element. Ditto b^2: perform the mirror operation twice and again it's the same as doing b*1 etc.

[onecommentman]

“Thank you for bringing smarter, better information.”

Thank you for being modest, but the information isn’t “better” than yours, just more precise and more “messages from the front”. Both are equally valuable.

There has been an ongoing issue in mathematics between documentation and communications.

Documentation is rampant. It is a necessary part of the process of determining whether something is correct, rather than almost correct. It’s a technical requirement of the process, much like desks and chairs.

Communications is almost an afterthought to the community, but is math’s basic value to society. Here I define “communications” as imparting the basic insights to anyone who isn’t either developing the proof or reviewing the proof for correctness, I.e. the entire planet minus 10 or 20 people.

Communications has relatively little to do with documentation, although documentation can and has masqueraded as communications. The current mathematical enterprise has put 95% (99%?) of the effort on proofs to expand the mathematical universe, and precious little on communications.

I’d argue that should be reversed at this stage in the development of mathematics. But, as a nod to the established order, I’d propose that 50% of the effort of professional mathematicians be spent in the development and improvement of the communications side of mathematics. No better time, given the maturity of media technologies.

My two cents...