The distance of the moon

Having just re-read the delightful Italo Calvino story of this name (which you can read here), I was also entranced by the following animation:

Also: my take on the pronunciation of the name of the narrator: the first f is silent, so it comes out at something like “quwolfque.”

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Music in London

London is a wonderful city, an I can certainly imagine moving there if I suddenly happen to become a Russian oligarch. One thing that is absolutely fantastic is the classical music scene, which is surely the best in the world. On my recent visit, I happened to be visiting the Courtauld gallery when I saw a flier indicating a concert in the museum at 4:30. The attendant didn’t appear to know anything about it, despite the fact that the sign was less than three feet from his desk. (During a brief conversation, I casually suggested the performance might consist of music inspired by the work of Cezanne; I later heard him definitively telling other patrons the same story.)

Now a “free classical music concert” with no other details could encompass a wide range of possibilities. Original compositions, perhaps? First year students playing Vivaldi in order to get in some performance practice? A rumour that the music involved a guitar and a flute was not inspiring. But inspiring it turned out to be. The flautist turned out to be Adam Walker, who, amongst other achievements, has been (since 2009) a principal flute of the LSO. Walker gave a wonderful rendition of Debussy’s “Syrinx,” which for your benefit I link to here:

As if that music was too mainstream for a random Sunday afternoon collection of patrons at an art gallery, the next performer dialed it up a notch with a performance of Henri Dutilleux’s “trois strophs sur le nom de Sacher.” This piece comes from a collection of 12 Cello works commissioned by Rostopovich for the 70th birthday of Paul Sacher. This performance (irritatingly, I can no longer recall the name of the performer) was absolutely riveting, although I have to confess this is music outside my usual comfort zone (to quote from this essay: “The only piece in this set to make use of a scordatura, a deliberate mis-tuning of strings, is Henri Dutilleux’s TROIS STROPHES SURLE NOM DE SACHER. In this piece the ‘cello’s G string is tuned down a halfstep to F sharp, and the C string is tuned down a whole step to B flat.”)
Still, this was exactly live music at it’s best — a chance to discover new music.

It turned out that the actual concerts that I attended the rest of the week (Mendelssohn’s Octet on one occasion and some Bach Violin concertos on the other) didn’t quite live up to the standard of this unexpected but wonderful recital.

Other random (non-math) observations from London: the only drinkable coffee in London is served by Australians working in Australian cafes; Soho turns out to be slightly tricky to navigate if there happens to be 50,000 people there for a pride rally; if you’re going to ride a rental bike from Soho to Kensington, there are possibly better routes than Oxford street; One day I need to schedule my trips to London to coincide with a test match at Lord’s; the HOTTEST JULY DAY IN LONDON EVER was not actually that hot.

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Course Announcement

Although this may be of limited interest, I wanted to announce a topics class that I intend to give this winter. The title of the course is, simply, “thesis problems.” The structure of the course is to devote each week to a different problem which I think would make a good thesis, or, at least, a good “first project.” The topics are sufficiently diverse that each week will be independent of the next. The problems range in difficulty, but they have several features in common, including the fact that I have ideas on how to start all of them, and I have complete solutions to none of them. I have no shortage of possible problems, but I will try my best to select those with the following properties:

  1. The background material you will be required to learn in order to attempt any of these problems will be useful even if you do not ultimately end up solving the problem.
  2. For the harder problems, there will be some low hanging fruit if you are not able to solve the problem.
  3. The problems will tie into the research of other faculty members in the department, so you will have at least two people (and frequently many more) to speak with. Pro-tip for graduate students: talk to faculty other than your advisor! For example, at least one (and possibly two) of the problems would be suitable for students of Benson, and obviously many will be suitable for students of Matt as well.

Part of the reason I am bringing this up now is to ask: has anyone taught a class like this before? Are there any pitfalls (obvious or otherwise) that I should be aware of? This course is a spiritual cousin of the numerous problem sessions which are held at conferences. However, I think that these sessions usually have limited success, in part due to the fact that the problems that come up (while usually interesting) are just too hard. I will be especially conscious therefore to choose problems for which I think real progress can be made, and for which there is somewhere to start. You can think of this class a little like what might happen if you came to my office and asked what it might be like to be my student, except spread over three hours a week for nine weeks or so.

You might ask if I’m worried that hungry postdocs will hoover up the problems like Lunt Hall cockroaches eating crumbs after a wine and cheese. The answer is: not really! I think this class would be a success if I end up with (say) two or three students working on problems related to what I talk about. This will leave many other problems which I would still love people to work on. (Though, of course, if you are a postdoc who does want to think about one of these problems, you should probably tell me.)

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A Coq and Bull Story

Author: Michael Harris.
Title: Mathematics Without Apologies.
Source: I eventually gave up waiting for a complimentary signed copy to be sent to me in the mail, and so borrowed a copy from the Northwestern University library. I’ve read up to the beginning of chapter 5 so far.

Thoughts before reading the book: Rational thought and continental philosophy have always been non-overlapping magisteria in my mind. Who better, then, to bridge the gap than Michael Harris, the left-bank mathematician who had returned stateside (well, Manhattan-side at least).

In Mathematics Without Apologies, Michael Harris addresses what it is to be a mathematician. There are several questions raised in this book which I hope to consider on this blog. However, before we begin any analysis of the actual content, there is an issue that first has to be discussed. In a book so thick with opinions, it is quite extraordinary that the author believes that he can not only play the simultaneous roles of case study and provocateur, but also, at the same time, present himself as completely impartial observer, one who is merely “offering up” a goulash of narratives and cultural insights for our consideration. Michael Harris is not the naive innocent that he pretends to be; to me, the author is about as impartial as Socrates is when he asks Thrasymachus what makes a “just man.” On the other hand, how can one win an argument with the author on how strongly he holds certain opinions? Rather than take a detour to discuss the relationship between author and text, and argue about the extent to which an author can, by writing supplementary blog posts or otherwise, clarify the intended implications (or lack of implications) of his own words, let me offer the following solution: from this point on, we shall denote the author of “Mathematics Without Apologies” by michael harris (lower case). To be clear, michael harris is a chimera born out of Michael Harris’ words and my interpretation of them. Michael Harris is allowed to disagree with my interpretation of what michael harris says, but he is not the ultimate arbiter of deciding what michael harris thinks; that is up to the reader(s). The only further words I will say on this topic are as follows: for a book claiming to offer no apologies, there are an inordinate number of strongly worded disclaimers, for example: “[one] … should not mistake this book for a work of scholarship.” [OK, duly noted. Of course, these are the words of Michael Harris, not michael harris, the savage cultural critic.] Voloch complains here of his frustration of the author’s apparent inability to commit to any position, to which Michael Harris is sympathetic, but I suggest that both of them read closer between the lines to see that michael harris does indeed hold some strong opinions. With that out of the way, let us begin considering actual issues raised by the text. For now, let me restrict myself to explaining and largely agreeing with a single opinion of michael harris. We begin by recalling a distinction made in the book between the philosophy of Mathematics and the philosophy of mathematics. The philosophy of Mathematics is concerned purely with epistemological questions, and should be thought of as the subject whose intellectual lineage goes back to the crisis of foundations, and which tries to explain the meaning of mathematical truth and its relationship to knowledge. The philosophy of mathematics, on the other hand, is concerned with how actual mathematicians think (about mathematics), and is a topic of primary interest to the author. What becomes clear when reading the book is that michael harris believes that the philosophy of Mathematics — and indeed the subject of foundations more generally — has nothing at all useful to say or contribute to the professional lives of mathematicians, by whom I mean people like myself, or people like Michael Harris. For example, harris quotes Jeremy Gray (from Gray’s account of the Foundations Crisis) as follows:

The logicist enterprise, even if it had succeeded, would only have been an account of part of mathematics — its deductive skeleton, one might say…. mathematics, as it is actually done, would remain to be discussed.

Or, to quote michael harris directly (p.67), “Capital-F Foundations may be needed to protect mathematics from the abyss of structureless reasoning, but they are not the source of mathematical legitimacy.” Nothing controversial so far, I think. But harris goes further. In a book thick with quotations, the key to any reading is to identify the villain. And the villain in chapter 3 is definitely Voevodsky, quoted here as follows:

If one really thinks deeply about … [the possibility that the foundations of mathematics are inconsistent] … this is extremely unsettling for any rational mind.

Certainly, by this measure, I am either not in a possession of a rational mind, or not a deep thinker about these questions. If someone told me today that Voevodsky had discovered an inconsistency in ZFC, I would care slightly less than if someone told me the Collatz problem had been solved, and care much less than if someone (trustworthy) told me that a serious error had been found in the proof of cyclic base change. In the first case, I would presume that what ever fix (large or small) to foundations needed to be made, it wouldn’t make any difference to the mathematics that I think about. Whereas in the third case, it would make quite a lot of difference. Underlying all of this, of course, is some assumption (by me) that there is some formulation of mathematics which is consistent. There seems to be some evidence for this, including a several thousand year history of mathematics which has required barely any modification at all in light of whatever logical issues arose in the 19th and 20th centuries. This is not to say that michael harris completely dismisses any efforts to study foundations. However, to his mind, and to mine, the ultimate judgment about such an enterprise should be on its effect on mathematics qua mathematics (“If and when univalent foundations is adopted as a replacement for today’s … foundations, it will probably be … triggered by a demonstration of the new method’s superiority in addressing old problems”, p.65). This ties together with the attitude that both michael harris and I share towards computer assisted proofs, which is somewhere between “who cares” and a general skepticism that it will have any relevance to mathematics as we practice it (caveat: I believe michael harris discusses these issues in later chapters which I have not yet read). Again, to understand what michael harris thinks, it suffices to judiciously select those whom he quotes. Here is another quote by harris on p.66:

When [Benecerraf] limits the articulation of mathematical truth to logic and then complains that the ability of mathematicians to refer has been lost, it is no wonder; it is also no wonder that number theorists and geometers have not borrowed the language of logic to do their work.

(Confusingly, I can’t quite work out where this quote is from: the footnote refers to a 2009 reprint of writings of Herman Weyl (who died in 1955), but the quote above implicitly (in the context of the book) seems to refer to an article of Benecerraf from 1973.) Here michael harris is drawing the following distinction. The view, coming from the philosophy of Mathematics, is that mathematicians are trying to understand some truth, and then to try to decipher what form of truth mathematics actually consists of; whether it be a chain of modus ponens all the way down, or what have you. After all, if the goal of mathematicians is to seek the “truth” (whatever that is), then surely it follows that it is important to put the notion of truth on some firm philosophical footing. Yet the philosophy of mathematics view is really quite different. If you asked me why I do mathematics, I wouldn’t, if I was honest, say that I was seeking “the truth.” My attraction to mathematics (and even analytic philosophy) is that I find it a lot of fun, and moreover completely addictive. Analytic philosophy (of the flavour I enjoy) is fun, because it is a (logical or linguistic) puzzle. For example, the observation:

“Hesperus is Phosphorus” is not a tautology

can be used to negate certain claims about the meaning of proper names and the relationship of the necessary to the a priori. (Or even better, to go Quine rather than Frege, “9 = the number of planets” is not a tautology, not least of which because it is currently false.) But do these exercises actually tell us anything about epistemology, or are they really just some form of enjoyable intellectual exercise? Those questions are beyond my pay grade. But honestly, they don’t even interest me that much. I simply enjoy those puzzles for what they are, rather than lay any claim that they are revealing deep truths about human thought. (To be honest, I think if I really felt that I wanted to understand what words mean, I would become a computational linguist, not an analytic philosopher.) In mathematics, too, I enjoy playing with objects (Galois representations, automorphic forms) that are my stock in trade; the fun and beauty aspects are more compelling than the “search for absolute truth.” No doubt michael harris will have more to say on these topics in later chapters. These issues also remind of a conflict between mathematicians and historians of Mathematics. A colleague once reported to me a conversation (or argument) with a historian of Mathematics who claimed that Galois did not know that A_5 was simple. On the other hand, Serre claims here that Galois not only knew that A_n for n \ge 5 was simple, but also that L_2(p):=\mathrm{PSL}_2(\mathbf{F}_p) was simple (for p \ge 5), and that (moreover) Galois really understood these groups. For example, Serre claims that Galois knew that L_2(p) for p > 3 has a transitive action on p points only for p = 5,7,11. (Exercise for the reader!) Of course, it’s a bit of a running joke in mathematics that everything is really due to Gauss, but I side with Serre here (a bold stance, I know), and moreover think that one of the issues is (or could be) the tendency for a historian of Mathematics to view mathematics through the perspective of a Philosopher of Mathematics rather than of mathematics; to fail to distinguish between a formal proof and a genuine understanding.

Other remarks on the first few chapters: Harris quotes Neil Chriss (p.72 “who chose to forgo a promising future in the Langlands program to work for a … hedge fund”) as saying that “The Glass Bead Game is a favorite novel among my mathematician friends.” (the implication being here that Mathematicians would love to have no responsibility to the outside world). Really? Either those mathematicians completely misunderstood the point of that novel, or I did. It seemed to me that one of the key themes of Hesse’s book was that a complete disconnect between intellectual pursuits and the ultimate responsibilities of humanity towards society was a bad thing; possibly not a surprising message for a book which was published in 1943. On the other hand, the implication of the quote above is that Mathematicians dream of Castalian paradise where they can pursue mathematics unencumbered by the realities of society. Curiously enough, Both Michael Harris and michael harris are entities to whom the theme of Hesse’s book (in my reading) seems to hold some appeal. One certainly gets the sense even from the first few chapters that michael harris is deeply concerned with the interaction between intellectuals and broader society. (I am with harris completely when it comes to his opinions on foundations; I expect to differ on matters related to our role in society.)

(p.123) I think that Tom Stoppard did a better job of demonstrating that the symmetry group of Rosencrantz and Guildenstern is \mathbf{Z}/2 \mathbf{Z}.

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Harassed by Springer

Those of you who have ever submitted a paper to any mathematical journal may have noticed that it’s not a particularly speedy process. Nowadays, even a one year turnaround is nothing out of the ordinary. Thus, I always find it slightly amusing (once the paper is accepted) to receive breathlessly worded emails from the publisher demanding that you “review the proofs within 48 hours” with the (implied) risk that acceptance of your paper might be at risk if you don’t rush to meet their deadline.

What happens if you ignore these emails? Well, it turns out you get follow-up emails: “the message below was sent to you several days ago but we have not yet received your corrections. Please return your proof as soon as possible so as not to delay the publication of your article.” Of course, these emails are subtitled First Reminder, so it’s probably safe to ignore those as well. At this point, I’m kind of curious as to how long this process continues. Maybe they will write a short abstract for me and then publish the paper anyway. Presumably my co-author doesn’t mind (though I don’t want to get my editor in trouble).

This all leads me to the following suggestion for disrupting journal publications: submit your paper to journals, but then when they are accepted, put them on your website with an annotation along the lines of accepted by Journal X (you can even include the original acceptance email from the editor for authenticity purposes). All the imprimatur of the journal system, none of the cost!

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Counting solutions to a_p = λ

We know that the eigenvalue of T_2 on \Delta is 24. Are there any other level one cusp forms with the same Hecke eigenvalue? Maeda’s conjecture in its strongest form certainly implies that there does not. But what can one prove along these lines? Conjecturally, one would certainly predict the following:

Conjecture: Fix a tame level N prime to p. If \lambda \ne 0, there are finitely many eigenforms of level N an arbitrary weight such that a_p = \lambda. If \lambda = 0, there are finitely many eigenforms with the additional condition that they do not have CM by a quadratic field in which p is inert.

I have no idea how to prove this conjecture. If one counts the number of such forms of weight \le X, then the trivial bound for eigenforms with a_p = \lambda is O(X^2). When I visited Princeton a few weeks ago, Naser Sardari, a student of Sarnak, showed me a short preprint he is writing which improves this bound by a power saving (additionally, it gives a power saving for each individual weight as well). The most interesting case of this result is when \lambda = 0, but today I want to talk about the much easier case when \lambda \ne 0, where, via some p-adic tricks, one can obtain a substantial improvement on the trivial bound. Let’s start from the following:

Proposition I: Let S_{\lambda}(X) denote the number of cuspforms of level N and weights \le X such that a_p = \lambda. Assume that \lambda \ne 0. Then

S_{\lambda}(X) = O(X).

Proof: Since \lambda \ne 0, the p-adic valuation of \lambda is finite. However, all forms with bounded slope belong to one of finitely many Coleman families, so the number of such forms in any weight is bounded. Using Wan’s explicit results, one can even give an explicit bound here that depends only on N, p, and the valuation of \lambda.

The point of this post, however, is to give an improvement on this bound.

Proposition II: Let S_{\lambda}(X) denote the number of cuspforms of level N and weight \le X such that a_p = \lambda. Assume that \lambda \ne 0. Then, as X \rightarrow \infty,

S_{\lambda}(X) \ll_{\lambda} \log \log \log \log \log \log \log X.

The argument will (obviously) allow for an arbitrary number of logs. But then the statement would become more cumbersome.

Proof: As in the proof of the previous result, we may reduce to the case where we are considering a single Coleman family \mathcal{F}. Over this family, the function U_p is continuous, and hence so is U_p(U_p - \lambda). More importantly, over a small enough disc, it is an Iwasawa function. Let \Sigma denote an infinite set of integral weight such that, for the relevant points of \mathcal{F}, we have T_p = \lambda, or

U_p(U_p - \lambda) = - p^{k-1}.

If s is a limit point of \Sigma, then certainly U_p(U_p - \lambda) will vanish at s. Since this function is a non-zero bounded function on a disc, it has only finitely many zeros, and so the set of weights \Sigma will have only finitely many limit points. Thus, we may reduce to the case where the set of weights has a single limit point. In particular, if S_{\lambda}(X) is not bounded, we may imagine that the set \Sigma consists of a sequence of integers (which we may assume to be increasing in the Archimedean norm): k_0, k_1, k_2, \ldots which converge p-adically to s, and, at the relevant point of \mathcal{F}, correspond to an eigenform which satisfies the equation

U_p(U_p - \lambda)(k_i) = - p^{k_i - 1}.

Around a zero s, any Iwasawa function has an asymptotic expansion of the form

F(s + \epsilon) \simeq A \cdot \epsilon^m + \ldots

where the LHS has the same valuation as the leading term of the RHS for sufficiently small \epsilon. If F = U_p(U_p - \lambda), we deduce that, for sufficiently large integers k_i,

v(s - k_n) = r k_n + c

for some r = 1/m > 0, which implies that v(k_{n+1} - k_n) = v(s - k_n), and hence also that

k_{n+1} - k_n > C p^{r k_n}

for some r  > 0. This iterated exponential growth proves the result. QED.

The argument also shows that if the set \Sigma is infinite, the limit roots of U_p - \lambda = 0 will be transcendental Liouville numbers, which seems unlikely. The result also applies if one replaces \lambda by a sufficiently continuous function without zeros, say a_2 = 24(1 + 2(k -12)^2). On the other hand, I don’t think these analytic methods will ever be enough to prove the conjectural bound, which is O(1).

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The space of classical modular cuspforms of level one and weight 24 has dimension two — the smallest weight for which the dimension is not zero or one. What can we say about the Hecke algebra acting on this space without computing it?

Formally, the Hecke algebra \mathbf{T} is a rank two \mathbf{Z}-algebra, which is either an order in the ring of integers of a real quadratic field, or a subring of \mathbf{Z} \oplus \mathbf{Z}. Let’s investigate the completion of this algebra at various primes p.

Let’s first consider the prime p =23. The curve X_0(23) has genus two, and the corresponding Hecke algebra in weight two is \mathbf{Z}[\phi], where \phi is the Golden Ratio. The prime p =23 does not split in this field, and hence modulo p there is a pair of conjugate eigenforms with coefficients in \mathbf{F}_{p^2}. Multiplying by the Hasse invariant, we see that this eigenform also occurs at level one and weight 24 over \mathbf{F}_{p}. It follows that:

\mathbf{T} \otimes \mathbf{Z}_{23} = W(\mathbf{F}_{23^2}).

In particular, \mathbf{T} = \mathbf{Q}(\sqrt{D}) for some square-free integer D > 0.

Now let us consider primes p < 23. Any Galois representation modulo such a prime will occur — possibly up to twist — in lower weight. Yet all the spaces in lower weight have dimension at most one, and hence it follows that the residue fields of \mathbf{T} are all of the form \mathbf{F}_p. Suppose further that 5 \le p < 23. Then, using theta operators, we may find two distinct eigenforms in weight 24, from which it follows that \mathbf{T} has two distinct residue fields of characterstic p, and so, for 5 \le p < 23, we have:

\mathbf{T} \otimes \mathbf{Z}_p = \mathbf{Z}_p \oplus \mathbf{Z}_p.

One expects at level one that a_2(f) always generates the Hecke field. This is still a conjecture, but we may deduce this unconditionally in weight 24 because the dimension of the cuspforms is two, and so this follows automatically from the Sturm bound! Hence we may write:

\mathbf{T} = \mathbf{Z}[a_2(f)], \quad a_2(f) = \displaystyle{\frac{a + b \sqrt{D}}{2} \in \mathbf{Z} \left[ \frac{1+\sqrt{D}}{2} \right]}

where b \ne 0. Even better, using Hatada’s Theorem — giving congruences for a_2 and a_3 for eigenforms of level one modulo 8 and 3 respectively — we may write

a_2(f) = 12(a + b \sqrt{D}), \quad a,b \in \mathbf{Z}

where b \ne 0. This gives an upper bound on D in light of the Deligne bound |a_2| \le 2 \cdot 2^{23/2}. More precisely, we obtain the bound b^2 D < 2^{27}/24^2, and hence that D < 233017.

Let’s now think more carefully about p = 2 and 3. For these primes, there will be a unique Coleman family of slope v(-24) = 3 for p =2 and v(252) = 2 for p = 3. I can’t quite see a pure thought way of proving this, but at least this would be a consequence of the strong form of the GM-conjecture as predicted by Buzzard. So we should expect that, in these cases

\mathbf{T} \otimes \mathbf{Z}_p \hookrightarrow \mathbf{Z}_p \oplus \mathbf{Z}_p.

In addition to congruences for small primes, there will also be congruences between the unique cusp form with an Eisenstein series modulo the numerator of B_{24}, which is

\displaystyle{B_{24} = \frac{-1}{2 \cdot 3 \cdot 5 \cdot 7 \cdot 13} \times 103 \times 2294797.}

I claim that these primes will also have to split in \mathbf{T}. For example, it is impossible for b to be divisible by 2294797, because that would violate the inequality on b^2 D above, and hence it follows that p = 2294797 must also split in \mathbf{T} \otimes \mathbf{Q}. The same argument works for p = 103 having ruled out some very small D. To summarize, we have the following:

The primes 5 \le p < 23, p = 103, 2294797 split in K = \mathbf{Q} (\sqrt{D}), but p = 23 does not split, and D < 233017. Moreover, we expect that 2 and 3 also split.

This is enough to determine D completely up to 72 possibilities, and 9 with the unproven assumption at 2 and 3. On the other hand, all of these D are quite large (the smallest are 3251 and 15791 respectively), which forces b to be very small. But we also have the congruence

12(a + b \sqrt{D}) \equiv 1 + 2^{23} \mod 2294797.

For the remaining D, we can determine, with |b| satisfying the required inequality, whether there exists such a congruence with |a| \le 2^{27/2}/24 \sim 483. A simple check shows that is a unique solution (with the assumption on two or three or not), and hence, by (something close to) pure thought, we have shown that D = 144169, and moreover (using Deligne’s bound again) that

a_2(f) = 12(45 \pm \sqrt{144169}), \qquad \mathbf{T} = \mathbf{Z}[12 \sqrt{144169}].

One can indeed check this is the case directly, if you like. Curiously enough, this Hecke eigenvalue is quite close to the Deligne bound — the probability it is (in absolute value) this big is, assuming a Sato-Tate distribution, slightly under 5%.

Extra Credit Problem: Hack Ken Ribet’s Yelp password by using the fact that 144169 is his favorite prime number.

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