Let me finish off the story of that busy week a few weeks ago.
One of the 5 books that arrived that week was “An Introduction to the Langlands Program”, ISBN 9780817632113. (I may refer to it in this post as “the book”.) The First 8 chapters were written by one of its 6 authors… and his writing style was interesting. But let me cut to the chase. From the editors’ preface – yes, plural, there were 2 editors. (That says something about the complexity of the topic: 2 editors and 6 authors for just under 300 pages of math.)
“The Langlands program roughly states that, among other things, any L-function defined number-theoretically is the same as the one which can be defined as the automorphic L-function of some GL(n).”
That’s probably not exactly clear, is it? It certainly isn’t to me. (I can’t say it’s Greek to me… I’ve had a couple of years of classical Greek.)
Anyway, having skimmed the First 8 chapters and then some on a Friday evening, I went searching the Internet Saturday morning when I should have been drafting a happenings post. I found some very surprising links.
I was much more comfortable searching the Internet after I had looked through a book. Does that seem strange? The Internet seems to me to be a good place to find tidbits – but I like having more context before I go searching it… I like having compartments to drop the tidbits into. Of course, there are entire books available out there… but while I may take the time to download them, I don’t read them then… I generally keep searching.
It may be that having looked through the beginning of the book, I was hoping that I might recognize a particularly good short summary if I found one.
I have no hope of explaining what the Langlands program is… out on the Internet, in response to the question, “can someone explain it in layman’s terms?”, the answer was basically “I don’t think I can explain it to someone with just a bachelor’s degree in math.”
But before I say anymore about the Langlands program, let me provide some background of a different kind. Before I embarked on that search, there were 2 other things in the back of my mind.
One was the report on the evening news that CERN had experimental results which, if correct, showed neutrinos moving faster than light. I had checked the CERN site, but hadn’t found anything useful about superluminal neutrinos.
The other was some postings out on the sci.math newsgroup about someone named Ed Nelson showing that the Peano axioms for arithmetic were not consistent. I would have ignored this, except that Ed Nelson turned out not to be a kook, but a professor of mathematics at Princeton. He might be wrong, but he isn’t a kook, someone pretending to understand – and rewrite, of course – all of mathematics and to publish his earth-shaking discoveries on the newsgroup.
Now let me return to the Langlands program.I think the best I can do is to locate it in mathematics – where did it come from?
Analytic number theory. Think Riemann zeta function. Which is an L-function. The first L-function according to the book.
Think quadratic reciprocity. Which I really ought to understand someday. After all, it’s in any first course in number theory. It’s undergraduate material.
Rather than spit out definitions, I’ll let you look up the zeta function and quadratic reciprocity if you want. This post is more about where the search led me, rather than about what it told me.
There is, of course, a Wikipedia article about the Langlands program. It mentions L-functions and quadratic reciprocity:
“The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin’s [generalization of quadratic reciprocity] in this more general setting.”
So what happened that Saturday morning? My simple search on the Langlands program yielded 4 different fruits.
- First off, it led to some interesting attempts to explain the Langlands program in elementary terms.
Second, it led to some very reassuring confessions of ignorance on the part of mathematicians.
Third, it led me to a website that gave me a satisfactory response to those faster than light neutrinos that CERN is asking for help explaining.
Fourth, it led me to a website that gave me a satisfactory answer to Edward Nelson’s purported proof that the Peano axioms are inconsistent.
Let me take these in order.
Here are the simplest explanations I have encountered so far as to what the Langlands program is.
There is a rather simple talk. One of the Fields Medals awarded last year was for work on the Langlands program, so it became necessary to describe to non-experts what it was.
There is a pro explaining it to his friends:
What fascinated me, however, was this link about the Langlands program, which I recognized as soon as I saw the name, “Not Even Wrong”: Peter Woit’s blog. He is the particle physicist who switched to mathematics, who wrote the book “Not Even Wrong” criticizing string theory.
I didn’t read much of what he had to say about the Langlands program and quantum field theory. But, I said to myself, surely he has something about those faster than light neutrinos.
Well, okay, a tiny bit; two replies to comments by readers:
“I’m afraid I think the “No News” heading is quite appropriate for the neutrino story. Even if I weren’t traveling I’d probably not write a posting about it here. It seems nearly certain that there is some subtle problem with the experiment, so the only story here is what that problem might be, and virtually no one submitting comments here appears to have the kind of expertise necessary to make a good guess about that (I certainly don’t).”
“To make this absolutely clear: I’m convinced this has to be due to some sort of error in the experimental analysis, and I’m pretty sure this is the majority opinion of knowledgeable people in this subject. To find the error though requires a real expert, and I’m certainly not that.”
I think that some of the comments in response to his post pointed out that a burst of neutrinos is generated in a supernova a few hours before the visible light output changes. If neutrinos moved even a tiny bit faster than light, the neutrino burst would arrive months ahead of the light, because of how far away supernovae have been.
(Hmm. Since we now know that neutrinos have mass, and therefore must move slower than light, just when do they get here? Months after the light?)
Now back to the Langlands program and the confessions of ignorance. There was a wonderful, long discussion here. If you look at only one link in this post, let it be this one. It starts with one mathematician asking what the Langlands program is – and then, in response to the submitted answers, he details exactly what he doesn’t understand.
“I don’t know what a maximal torus or a weight is. I can look up maximal torus and learn that it’s a compact, connected, abelian Lie subgroup that is maximal among all such subgroups.”
Me raises hand. Hell, me jumps out of chair and waves my whole body: I know what a maximal torus is! (Thanks to Stillwell’s “Naive Lie Theory”. Let’s see. You know what a torus looks like, it’s just a regular old donut. But it’s also a surface of revolution, obtained by rotating a circle in a circular path… that says that the old familar torus is the direct product of two circles, i.e. S1 x S1. Well, in general, an n-torus is a direct product of n circles. And once we decide that, then we should go the other way and include the 1D torus, the case n =1, namely the circle itself. Oh, and the circle is equivalent to the set of 2D rotations. Then if we’re looking at say, the Lie group SO(3) of 3D rotations, then the set of 2D rotations about a fixed axis – the z-axis, for example – is an SO(2)… is a circle… is a torus – and is in fact a maximal torus in SO(3). No, they’re not unique. There should be a maximal torus for each axis of rotation.)
It turns out there was another mathematician who freely confessed his ignorance – and he’s a Fields Medallist, therefore certifiably one of the best mathematicians in the world. Let me grab a couple of sentences from him (here on Tim Gowers’ blog):.
I don’t understand what an automorphic form is, but there are levels of non-understanding (I would be enjoying several deeper ones later in the talk) and Jim Arthur lifted me to a slightly higher one — by which I mean that I had a slightly better idea what automorphic forms were after the next section of his talk. Before, I just thought of them as particularly nice kinds of functions that number theorists liked, and often mentioned in the same breath as modular forms (which I understand slightly better but still by no means fully).
Finally, somehow, I discovered that the same site – the n-category Cafe – which had the long discussion of the Langlands, also had a post about Nelson’s attempts to show that Peano arithmetic was inconsistent.. Ah, the opening line – “Faster-than-light neutrinos? Boring… let’s see something really revolutionary.” – gives me some idea why a blog post about FTL neutrinos would have led to Ed Nelson. The last thing I was expecting… but it’s a small world sometimes.
I was delighted to see that Terry Tao – another Fields medallist – had weighed in, and that while he had an opinion as to where the error in the purported proof was – it wasn’t a trivial error, and Ed Nelson has some grounds for pursuing the effort.
So, the Langlands program… some mathematicians letting down their hair… and two topics I wasn’t expecting: faster than light neutrinos, and a possible inconsistency in the foundations of mathematics.
It was good – it was great – that I felt comfortable “wasting” that Saturday morning…. free to spend my time as I wished, chasing a butterfly.
Let me close by saying that this post was drafted last Saturday… but I didn’t want to rush it.. and so there was no diary post last week. And the same thing happened to the technical post for Monday – drafted but not coming together – so there was no technical post either. You knew that, but now you know why.