Mathieu Magic

I previously mentioned that I once made (in a footnote) the false claim that for a 11-dimensional representation V of the Mathieu group M_12, the 120 dimensional representation Ad^0(V) was irreducible. I had wanted to write down representations W of large dimension n such that Ad^0(W) of dimension n^2 – 1 was irreducible. In the comments, Emmanuel Kowalski pointed to a paper of Katz where he discusses actual examples (including the 1333 dimensional representation of the Janko group J_4). On the other hand, I recently learned from Liubomir Chirac’s thesis:

that it’s an open problem to determine whether there exists such a representation for all n (although he does write down infinitely many examples in prime power dimension). Chirac’s thesis also lead me to the paper of Magaard, Malle, and Tiep, who do classify all such examples for (central extensions of) simple groups. Turns out that I could have used M_12 after all, or rather the 10-dimensional representation of the double cover 2.M_12, which does have the required property (the 99-dimensional representation factors through M_12, naturally).

One reason (amongst many) that (either of the) 11-dimensional representations V of M_12 do not have Ad^0(V) irreducible is that they are self-dual (oops). On the other hand, if you eyeball the character table, you will find that there is an irreducible representation W of dimension 120. Moreover, let me write down the characters of [V \otimes V^*] – [1] and [W]:

\begin{aligned}   & [V \otimes V^*] - [1]: & \   120, 0, \ \  8, 3, 0, 0, 8, 0, 0, -1, 0, 0, 0, -1, -1; \\  & [W]: & \   120, 0, -8, 3, 0, 0, 0, 0, 0, \ \ 1, 0, 0, 0, -1, -1. \end{aligned}

These seem surprisingly close to me! So now the question is, as one ranges over (some class perhaps all) finite groups G, what is the minimum number of conjugacy classes for which

\chi = [V \otimes V^*] – [1] – [W]

can be non-zero for irreducible V and W, assuming that it is non-zero? Since V is irreducible, by Schur’s Lemma, this virtual representation is orthogonal to [1] (unless [W] = [1] which would be silly). So \langle \chi,1 \rangle = 0, which certainly implies that there must be at least two non-zero entries of opposite signs. I don’t see any immediate soft argument which pushes that bound to 3. I admit, this is a slightly silly question. But still, a beer to anyone who proves the example above is either optimal or comes up with an example with only two non-zero terms. (To avoid silliness, say that the dimension of V has to be at least 5.) The characters above look strikingly similar to me, and it does make we wonder if there is any reason for why they are so close. Perhaps if I knew more about groups, I could feel more confident in just chalking up the resemblance above to a law of small numbers.

Probably a more sensible question is to ask for how small the number of non-zero entries of of [V]-[W] can be for two distinct irreducibles. That question has surely been studied!

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I found out a good way to describe how long my commute is: about three minutes more than the length of the second movement of Beethoven’s 9th (the greatest movement!)

On the other hand, that measure proved inaccurate the very next day, when I also found out the answer to “is the drawbridge on Lake Shore Drive ever used?”

(channelling my inner Stanley Kubrick with a little well-timed help from 98.7WFMT). The whole opening/closing of the bridge did cause quite some delay, but the process did, in in the end, finish.

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Swimming Leaderboard

The first rule of number theory swim club is…talk about number theory swim club! Anyone can join, as long as you are a number theorist at the University of Chicago with a current gym membership. The standard meeting time is at the Myers-McLoraine Pool on MWF at 11:30, but we work on an honour system here. The following will be updated on a semi-regular basis. Leave a comment if you want to join in.

  • George “the shark” Boxer: 2 3 4 5 6 visits.
  • Frank “the minnow” Calegari: 3 4 5 6 7 8 9 visits.
  • Matt “iron man metallic hydrogen man” Emerton: 0 1 visits.
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The True Heirs To Ramanujan

If you are in need of some light relief, you could do worse than peruse the opinions of Doron Zeilberger, who, if viewed strictly through the lens of these ramblings, appears to have a relationship with theoretical mathematics something along the lines of the Unabomber’s relationship with technology. (Let me add that my feelings in real life about Zeilberger is that he has proved some amazing theorems, an opinion which I am absolutely sure is not reciprocated.)

Zeilberger’s opinion 151 is somewhat of a doozy, calling out Ken Ono as a member of the “fancy math gang” who “stole” Ramanujan. First, the idea that one could imagine what Ramanujan thought of the modern theory of mock theta functions (or any other part of mathematics influenced by his legacy) is pure BS. Second, once you create something in mathematics, it transcends its creator; what Ramanujan actually would have thought of his legacy is mostly irrelevant. Maybe Robert Langlands thinks we are all fools for not taking up the double-bitted axe and the cross-cut saw and devoting our lives to the trace formula, and maybe he’s right, but in reality the Langlands program will go in a very different direction to what Langlands anticipated and most likely be better for it. Third, it’s hard to maintain any legitimacy making criticisms of the way modern mathematics is done while insisting that you don’t actually know any “fancy” mathematics, whether that is true or not. But finally, and this is what is most amusing about this opinion, is that I am usually inclined to criticise Ken on precisely the same point, except in the exact opposite direction. That is, my frustration with the theory of mock modular forms is not that it’s too fancy, but rather that it’s nowhere near fancy enough! The subject is crying out for a treatment which incorporates representation theory, where “shadows” are related to the (reducible) principal series which has a discrete series quotient, and where the amazing special structures which have been discovered are given a more algebraic framework rather than the subject as it currently stands: a wild west of crazy q-series, Lerch sums, and indefinite theta series. Whether such an approach would actually be useful is hard to guarantee in advance, but the relevance for the classical theory of modular forms cannot be overestimated.

The process of mathematics is as follows. You discover or observe some phenomenon, and you try to explain it. While trying to explain it, you may come up with a more general theory which explains not only the original example, but also an entire family of examples. And usually at this point, the general theory is more interesting than the original example, because it has more explanatory power to explain why things are true. That doesn’t mean the original example is no longer interesting, but it has to be viewed in the more general context. There is room in mathematics for crazy unique examples that don’t fit a pattern and very general theory. But there are no “heirs” to Ramanujan, because mathematics doesn’t work that way. The fact that Ramanujan’s name will always be linked to Deligne (a mathematician of a quite different sort, to say the least) is testament to that.

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Public Displays of Mathematics

Let me start by saying that I’m in favor of making the effort to both educate the public about mathematics (as well as science more generally) and to convey to them a sense of the excitement of our discipline. But the science always has to come first, and should never be twisted for the purposes of sensationalism. I understand that I have more antipathy than most towards this sort of thing, but I wanted to discuss a few examples in this post of things which I have seen recently that have particularly annoyed me.

Examples of “math in the news” which didn’t quite live up to the hype have been around for a while. There was the whole E8 thing. I’m not sure how these things start, but it has to involve some unholy trinity of sensationalist journalism, self-promoting universities (or institutes), and complicit [or merely naive] authors. It’s not so easy to untangle the web of blame in any particular situation, but, at the very least, let me recommend to anyone to avoid AIM when it comes to matters of publicity. Not even wrong had an discussion on this particular case, but there’s also an interesting follow up on the E8 article here, where, to give credit where it is due, Oliver Roeder somewhat redeems himself for his previous crimes. (To be clear, the math behind the E8 story — removed from any breathless claims about how it will change the word — is pretty interesting.)

Second, let me also admit that I could be completely wrong about all of this. Obviously I’m not the target audience for popular articles on mathematics, and maybe, even when the truth is stretched beyond recognition or even just a little in that direction, such publicity is good for mathematics. I really don’t know. I acknowledge that I can be out of touch on some issues. I am, after all, someone who gets annoyed by the idea of NPR discussing non-classical music. But (at least in that case) I am sufficiently self-aware to appreciate that my opinion on the matter can safely be disregarded. Maybe that’s true in this case as well. Still, I feel that one can write with excitement about mathematics to a general audience and still be faithful to the mathematics. I grew up with a collection of articles reprinted from Scientific American in the ’60s and ’70s, and they were never afraid to challenge their audience with difficult technical concepts in order to elucidate some often difficult but important idea. One can also find such writings nowadays, although almost always in blog form rather than in print. (I don’t really read popular math blogs that much, but this blog is an example of how one can demonstrate the delights of our subject without being super-technical and yet still being honest about the underlying mathematics.)

I have little faith that university publicists or (even worse) journalists have much interest in being accurate. But I save the most opprobrium for mathematicians who make unreasonable claims about their own work. To make it clear, I was previously a little critical and cheeky about the publicity surrounding the LMFDB here, but that wouldn’t even rise to the level of a small misdemeanor compared to the crimes outlined below. I’m taking here about levels of intellectual dishonesty which would be more appropriate at a business school.

Plimpton 322: I was first alerted to this by a recent post on facebook linking to this article. A few sample sentences are as follows:

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.”

At first, I interpreted this as merely the type of quality rubbish one expects from the Guardian. But then, to my horror, I clicked on the accompanying video to discover that this nonsense was coming directly from the authors (mathematicians, not historians) themselves. Due to ignorance, I will leave aside here the historical claims of the authors (which to be fair are the main point of the article, although they also have been met with skepticism by actual historians) and merely comment on a few of their mathematical claims, specifically, that “Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today.” This is so patently absurd that it’s not worth spending much time discussing it, but here goes. There is nothing (and indeed far less) in the “exact” nature of the tablet that isn’t immediately a consequence of the usual rational parametrization of the circle. (Yes, you can use slopes instead of angles!) The “sexagesimal” aspect of the tablet is also a red herring. If you take any base B which is not a prime power, it admits a similar tablet comprised of Pythagorean triples with a side length one and the two other sides given by terminating decimals whose ratios (for large enough triples) are dense in any interval. Here’s a fragment of the corresponding decimal version:

Decimal Plimpton

(The column consists of triples (\delta,a,c) with (a,b,c) a Pythagorean triple, b an S-unit with S=10, and \delta = (c/b)^2 given as an exact decimal. The triples are ordered by \delta and limited by some height restriction.) Since 60 has more distinct prime factors than 10, the size of the entries in this table is little larger, but that’s about it. Of course, even if you wanted to base a primitive trigonometric system on exact ratios, you would much prefer to use rational points on the unit circle of small height, rather than insisting that the ratios involved had finite expansions, which is very restrictive. (I’m not denying that this may have been convenient historically for numerical computation, I’m only addressing the absurd claim that we have something mathematically to learn from this tablet.) I would stop short of saying that the claims of the authors are fraudulent, but I would go way further than to say they are simply overreaching. Let’s stick with saying that they are vastly overstated purely in order to drum up public interest for their own professional enhancement. And this type of irresponsible behavior leads, inevitably, to this:

OK, enough of that nonsense, let us move on.

Numberphile: OK, perhaps this will be a little bit more controversial. Perhaps the correct thing to bear in mind with this rant is to recall my comment about NPR and classical music above: just because it really annoys me doesn’t mean that I can’t simultaneously accept that it’s probably a good idea for NPR to be somewhat inclusive (I guess). Numberphile is funded by MSRI and that’s probably a good thing, but it still (sometimes) annoys the hell out of me. Should anyone care? I’m not sure. It’s also important to note that there are better and worse numberphile videos — if they restricted themselves to the good ones I would only have very positive things to say. Readers may be aware of the infamous 1+2+3+4+5…=-1/12 video (see here for a takedown). But it gets worse. And not necessary worse in the “this is just rubbish” kind of way, but in the “this gives absolutely the wrong impression about what mathematics is and dresses it up as some ridiculously stupid parlour game instead of something with deep and profound connections” kind of way. There’s a lot of dross to draw from, but here is one typical example:

OK, so what’s the problem? A tiny bit of mathematical knowledge reveals that the (concept of) the Mills’ constant is an interesting observation about what we know concerning (upper bounds for) gaps between primes. But that’s not what one gets out of this video at all. At first, it seems as though there is some mysterious prime generating constant — perhaps you as a youtube viewer can discover a closed form and reveal the mystery of the primes! But this is just rubbish, the actual number (or smallest such number) is of little interest. It’s true that they are slightly more honest towards the end of the video, but the actual mathematics behind this story is always completely obscured. Honestly, if they had just spent a little time (maybe even a minute) saying something at least tangentially related to the real point behind Mills’ constant I would have been much happier. Is enthusiasm better than accuracy? (genuine question). To me, Numberphile can sometimes seem to be the video series that will launch a thousand cranks rather than a thousand mathematicians. It doesn’t help that there’s a bit too much emphasis on recreational mathematics in the worst sense (2 million views between them), which are to real mathematics roughly what eating play-doh is to molecular gastronomy. (Hmmm, maybe a bad analogy, I can totally imagine a play-doh course at Alinea.)

It’s not easy to get it right when it comes to publicizing mathematics (and mathematicians), but it can be done (here, here, and here to name three recent examples). But it helps to start with something serious and try to explain how interesting it is.

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Continuing on the theme of the last post (Buzzard related viral videos), you can now view Buzzard’s MSRI course (in progress at the time of this post) online here. Having previously excoriated MSRI for restricting how many people can attend such workshops, I must now congratulate them on doing an excellent job in the audio-visual department and making the lectures available to everyone. Students at many levels could learn a lot by watching these and making an honest effort to think about the (implicit) exercises. Even if you know the material, it is still fun to watch; a little like your cool uncle telling you a familiar bedside story but with his own subversive twist. For various psychological reasons, I suspect that those watching the videos now as they come out will have a lower dropout rate than those who watch them later. So go watch them now! (Unless you are a student at Brown, of course.)

Kevin is always refreshingly honest about things he was confused by as a student (or is still confused by now), although sometimes it is reminiscent of Volodya “reminding” almost every speaker at the start of his seminar that his is a beginner and so the speaker will have to go very slowly. Along those lines, here are some (very) tangential remarks on the lectures so far.

When I was a student, I always got very confused when someone talked about the “closure” of the commutator subgroup [G,G]. The basic problem was that I couldn’t conceive of taking the quotient of \widehat{\mathbf{Z}} by 1 and getting anything other than the trivial group. Of course that is what you should get unless you are doing it wrong, because anyone who thinks about profinite groups as abstract groups are probably crazy.

That said, here’s an idle question: is the commutator subgroup [W,W] of the the Weil group of a local field K actually already closed? I believe that the corresponding result for the (local) Galois group G itself is positive (essentially as a consequence of the fact that G is a finitely generated pro-finite group), but W has a distinctly non-compact quotient \mathbf{Z}, so I’m not sure. Maybe this is an easy question, I don’t know.

Another random fact: I was a graduate student at Berkeley in 2000 when Richard gave a colloquium on the local Langlands conjectures for GL(n). One aspect of the talk I remember was Richard defining the p-adic numbers, to which Mariusz Wodzicki cried out: “excuse me, this is Berkeley, do you really think you need to define the p-adic numbers?.” At this point, someone else cried out “Yes!” and the talk continued as planned. But the part of this story that is relevant here is that I somehow internalized (either at this talk or before) the fact that, long before Harris-Taylor, the local Langlands conjectures had been proved for GL(n) when p > n (which mirrors the story for n = 2). But I was surpised to find out recently (i.e. this week) that this result was not something from the distant past, but rather was a theorem of 1998 from (friend of the blog) Michael Harris in Inventiones.

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Redraw the Balance

This video would be inspiring even if it didn’t star someone that I know:

I suspect that Tamzin’s other youtube video will probably not end up with 20+ million views, being somewhat more…technical, I guess you could say.

Of course, having just watched this, your task is now to imagine what a new incarnation of Doctor who might look like.

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