van Ekeren, Möller, Scheithauer on holomorphic orbifolds

There aren’t many blog posts about vertex operator algebras, so I thought I’d help fill this gap by mentioning a substantial advance by Jethro van Ekeren, Sven Möller, and Nils Scheithauer that appeared on the ArXiv last month. The most important feature is that this paper resolves several folklore conjectures that have been around since near the beginning of vertex operator algebra theory. This was good for me, since I was able to use some of these results to prove the Generalized Moonshine Conjecture much more quickly than I had expected. I won’t say much about moonshine here, as I think it deserves its own post.

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What makes the Monster Lie Algebra special?

This is a post I’d been meaning to write for several years, but I was finally prompted to action after talking to some confused physicists. The Monster Lie Algebra, as a Lie algebra, has very little structure – it (or rather, its positive subalgebra) is quite close to being free on countably infinitely many generators. In addition to its Lie algebra structure, it has a faithful action of the monster simple group by Lie algebra automorphisms. However, the bare fact that the monster acts faithfully on the Lie algebra by diagram automorphisms is not very interesting: the almost-freeness means that the diagram automorphism group is more or less the direct product of a sequence of general linear groups of unbounded rank, and the monster embeds in any such group very easily.

The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

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ABC conjecture rumor

There is a rumor circulating here in Japan, to the effect that S. Mochizuki has proved the ABC conjecture. My understanding is that blogs are for spreading such things.

Apparently, he had predicted some years ago that he would finish a proof in 2012, so I suppose this is an “on-time delivery”. It is certainly no secret that his research program has been aiming at the conjecture for several years.

Our very own Noah Snyder did some original work on the function field version of this conjecture, when he was a high-school student.

Update (Sept 4, 2012): This rumor seems to be true! You can find the four “Inter-universal Teichmuller Theory” papers on the very bottom of his papers page.

Farey fractions, Ford circles, and SL_2.

The topic of this post came up during a conversation with some physicists about the fractional quantum Hall effect (which is quite fascinating, but I don’t feel particularly qualified to discuss).  I have decided to set it down here in the hope that, as long as I have an internet-capable device with me, I won’t have to rederive it in front of people again.  Some of this material appears in Apostol’s Modular functions and Dirichlet series in number theory and Conway’s The sensual form. I’d be happy to hear about other good treatments.

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Lattices and their invariants

This post started out as an exposition on the monster Lie algebra, but it grew out of control, so I’m hacking off a chunk. Here, I’ll describe the basics of integer lattices.

Lattices show up in many mathematical contexts, some of which may be unexpected to the uninitiated. These contexts include the study of optimal periodic sphere-packings, the topology of 4-manifolds (where lattices give a full classification in the simply connected case), algebraic number theory, finite group theory, and theoretical high-energy physics. I will say almost nothing about these applications, though.

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Wilczek: What is Space?

Frank Wilczek gave an impressive physics colloquium at MIT last month, called “What is Space?” and it really gave me a new view of the physical world. I had read about spontaneous symmetry-breaking in quantum field theory texts, but I had not appreciated that the space around us can be viewed as a condensate. This was also the first time I’ve heard the word “superconductor” used in the sense he meant, but it seems like a somewhat natural generalization after some pondering.

He put up slides of his talk here (warning: this file did not agree with the Firefox on Linux in my office, but I was able to view it in Safari at home), but there were a couple differences between the slides and the lecture he gave. First, he only presented the first 2/3 of the slides, so it was interesting to read the slides that he rejected and tossed to the end. Second, in the talk, he really emphasized the point that truly empty space is a fundamentally explosive medium, because quark-antiquark pairs have negative energy. In particular, the space we see is quite full of such pairs, which mutually repel, so there is an equilibrium concentration. Also, these pairs are made of real particles, and are not virtual (although I don’t understand the significance of this statement). I had never seen this idea before, and I wonder if the negative energy claim is a result of some reasonably recent lattice QCD computations, or if I’ve simply failed to pay attention in the past.

At the end, there were a lot of questions from the audience about extra dimensions and strings, and he ducked all of them. I had been meaning to ask if he expected the nature of the condensates to change in or near the event horizon of a black hole, but unfortunately, I was unable to think of a good way to say it in English words at the time.

Question about convergence norms

I have a question for the mathematical community about the language mathematicians use for a series of functions, and it has been bugging me for a while. I was thinking of posting it on MathOverflow, but it seems to fit the “subjective and argumentative” criterion for closing rather well.

Suppose someone introduces a countably infinite set of holomorphic functions \{ f_n \}_{n \geq 0} on some open subset U of the complex numbers, and wants to argue that the sequence sum converges to a holomorphic function. One valid way to prove it would be to show that the sequence sum converges locally uniformly absolutely (or uniformly absolutely on compact subsets of U), and then point toward the complex analysis text of choice.

Question: If someone only argues that the sequence sum converges absolutely, should I complain? (If so, how loudly?)

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Remarks on career advice

There have been a few questions about the job application process on MathOverflow, and I’d like to make a few remarks in an open forum.

First of all, I think there have been some really good questions, and really good answers. I found it especially illuminating when mathematicians who have been on hiring committees weighed in on what they thought was important in an application. Depending on your social circle and who your advisor is, it can be difficult to get accurate information when you are a graduate student (or a postdoc – I recently learned that my research statement was too long by a factor of 2 or 3). So, hats off to the people who give well-informed advice. Please keep it up.

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Classified problem

Today at tea, some grad students were discussing the following enumeration problem:

How many elements of GL_n(\mathbb{F}_q) have zeroes in all diagonal entries?

I think they [Redacted]. The answer is apparently known but classified. It’s a sort of q-analog of derangements (i.e., permutations without fixed points), but if you take the derangement formula and add q-numbers in the naive way, the formula (q-1)^n \sum_{k=0}^n (-1)^k \frac{[n]!}{[k]!} doesn’t seem to work for n > 2.

Conference Networking

Early in my graduate student career, I was told by several people that I should go to conferences and talk to professors. If you work in mathematics, you’ve probably heard this piece of advice before, and it’s hard to see how you could damage your career by following it (given reasonable assumptions on your behavior). I encountered two problems:

  1. What sort of talking am I supposed to do with a professor if I don’t know anything?
  2. How do I make my way into one of those small circles of people that inevitably form between talks?

I’ve heard that some advisors actually go to conferences with their students and introduce them to colleagues, and this pretty much solves both problems, but I’d like to focus on the case that this doesn’t happen, since I imagine it will be the norm for a while. This isn’t meant to be a definitive guide, and I’d really appreciate further suggestions and anecdotes.
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