Symmetry, Algebra, and the Monster
quantamagazine.org
quantamagazine.org
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. Groups depend both on the operation and the elements it operates over. Groups satisfy the four definitional requirements. Assume addition here:
- If 'a' is an element in the group there must be another element 'e' such that a+e=a. e.g. 2+0=2
- If 'a' is in the group there must be another element in the group 'b' such that a+b=e e.g. 2 + (-2) = 0. This is the inverse requirement.
- The group operation must be associative a+(b+c) = (a+b)+c
- If 'a,b' is in the group then c=a+b and 'c' must be in the group. This is called closure
- If, optionally, a+b = b+c then the group is called Abelian. Notably when a,b are matrices and instead of addition we do multiplication ab <> ba generally. So being Abelian is hardly automatic.
If you try to make a group on the addition operator using only whole numbers you'll soon see you'll break the inverse requirement e.g. for some whole number 'a' there's no other whole number 'b' s.t. a+b=0. You'll need the negative of the whole numbers too. Therefore grade school arithmetic addition is a group on (+,Z) where Z is all integers.
Note that the number of elements in Z is infinite. You'll get the same sort of infinite sets trying addition over complex numbers, or reals, or fractions some being countably infinite and some being uncountably infinite.
Returning to the monster groups, how do we get a group with a finite number of elements in it? Well, there's a pseudo way to do this: define addition (or your operation in general) to work over a modulus. So we could define Abelian addition over the three elements 0,1,2 like this:
0+0 = 0
0+1 = 1
0+2 = 2
1+0 = 0
1+1 = 2
1+2 = 0 we wrap mod 3
2+0 = 2
2+1 = 0
2+2 = 1 e.g. 2+1+1 = 0 + 1 = 1
Ok, what do we add to 1 to get 0 for inverse? What's 1's inverse? It's 2 since 1+2=0 and 0 is the identity element. Other examples can be worked out from above. And because it's Abelian we can reverse the order of the elements in the example and get the same thing.
Typically these kind of mod groups like this mod-3 example can be done mod-N where N is a prime. Granted, it's a trick to make closure work without having one's hand forced through in an infinite number of more elements.
Thus most of the possible groups are either of the (+,Z) form addition over Z, (.,R) multiplication over reals, (+,C) addition of complex numbers kind etc. And the second major swath are groups operating over a prime modulus.
Is that it? No! The Langlands programme found there are indeed other groups which have a finite number of elements but in a way that's not contrived like the modulus example. To get a finite number of elements in a non-contrived way requires symmetry. You start with 'e' and two other elements you like a,b. As you try a+e, b+e, a+b check to see if you get resultant elements in your group. If no, throw that in. And repeat.
Now, for a while all you do is get more and more elements until suddenly because of symmetry, you don't. There's something in the nature of the elements and the operations whereby they only transform themselves into unique elements up to some point then they don't.
This is why the article plays up symmetry without really saying why. Imagine a+b means rotate a by b. Then you can do line symmetry of (a+b)+c through c and on and on an on and on ... until at some point you've played through all the symmetry on all the permutation and choices of elements until it goes back into something you started with earlier. Extending the analogy monster groups have rotation, translate, mirror, and zillion other kinds of symmetries ...
The monster groups thus are finite groups where symmetry is in play except that takes a huge number of elements -- the article mentions 10^53, a monster sized number --- at which point there's no new uniqueness. Evidently then (Z+) lacks the symmetry of these monster groups since you need an infinite number of elements there.
I can't answer in any detail how that carries over to physics except to repeat that physics people have seen the same math forms in monster groups in their theories. I also cannot answer what operation is in play in the monster groups --- well in most cases in algebra on these kinds of questions --- the group operation gets less focus. More focus goes on the form of the elements.
I don't know what an element of the monster group looks like. It almost certainly is not a single number ... it's a more complex thing like a matrix. I do know there's some ~25 monster groups ... so this process of it gets bigger until it doesn't happens in a few variations collectively called the sporadic groups. Sporadic in the sense there are but a few of them and they differ tremendously in the number of elements in their respective groups.
As far as I know the Langlands programme is done: all possible forms have been found. Thus since Mother Nature likes, for example Linear algebra, since it follows it in QM, and since sporadic groups are applicable to linear algebra maybe there's new physics there. The Standard Model is built-on Linear Algebra well in the case of non-countable infinite groups ... so the tantalizing question is: are there hidden gems in algebra of sporadic groups which we are less familiar?
For multiplication the identity element is 1, and therefore inverses relate to 1 not 0 like addition.
So what 'a' makes a*0 = 1? There is no such element and it has to be excluded as per other comments.
"then you can do line symmetry of (a+b)+c through c and on and on an on and on ... until at some point you've played through all the symmetry on all the permutation and choices of elements until it goes back into something you started with earlier. Extending the analogy monster groups have rotation, translate, mirror, and zillion other kinds of symmetries ..."
That's slightly misleading. What I should have wrote is starting with two random elements in the set a,b form c=a+b. Almost always this will yield a new, distinct element in the set c. So throw c into the set and repeat trying all permutations and combinations getting d,e,f,...z.
At some point the symmetry in the monster group will overtake insuring that, because of symmetry, a subsequent operation on z will transform it equal an earlier element. Thus the monster group maintains an interesting ultimate boundary where all elements eventually through symmetry revert back to an earlier form.
In the article's 90deg rotation analogy on 2D squares, we can rotate a to b to c thus making 3 distinct squares (if we label their corners), however, upon rotating c again we get back to a. In monster groups it takes a longer run to revert.
I suppose it's worth emphasizing monster groups of finite size can only exist in a few variations. And between two monster groups M,O there is no in-between group N. It's definitely discrete in that sense.
My background is in algebra; these things excite me as well, but it's quite challenging to share the drama, beauty, and excitement with non-mathematicians.
Regarding the cyclic groups, they don't need to be constructed from a "trick" any more than the other finite groups. Think about the rotational symmetries of an n-sided polygon!
A lot of these finite simple groups have similar visualizations as symmetries of some geometric object. One of the tantalizing things about the monster group is that we haven't found anything like that for it. We don't know of what the monster group is a symmetry group.
> I'm also a bit jealous of the writers here: they seems to have fun job and I'd like to interview the same guys/gals and I'd love to type something up too.
you could have simply googled to make sure you weren’t outright wrong.
So (explained in a manner pleasantly understandable by someone with an order of magnitude more credentials and experience than you), what is the Langlands program?
I’ve never heard much of the formal Langlands program, but vaguely read some material on it over the decades. I’d be in a position to have tripped over a good, mathematically sophisticated introduction/summary, and I haven’t read one. The poster’s description is in line with the general tone of what I’ve read in the past, and it is well-written and thoughtful. Your comment is low effort.
I’d propose, if you actually understand the program well enough to critique, to contribute something to convince the rest of us that it is a worthwhile intellectual endeavor. By doing that, making one small step in addressing the issue of the non-existence of communications, as contrasted with documentation, as a core element of the profession of mathematics.
What if the commenter doesn't have time or expertise to do that? Isn't it better to at least state that the OP wrote something very inaccurate? Or should we just give up and let the misconceptions propogate?
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.
>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.
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...
As a result of this confusion, instead of describing the Langlands program, you described something like the classification of finite groups.