Stephen Fry

Posted on February 14, 2013 by Persiflage

Never let it be said that this blog shies away from confronting sacred cows. Today’s target: Stephen Fry. In the video below, Fry tries to have his cake and eat it too — calling out grammar pedants for “showing off their superior knowledge” whilst … showing off his own superior knowledge. Perhaps the qualities we most despise in others are the ones which we most fear in ourselves. Plus, “Stephen Fry’s America” is absolutely terrible. Full disclosure, my own grammar is a mess; I cannot return a positive answer to Lawren Smithline’s immortal question “do you write like a Harvard man?” (Case in point, I’m not sure if I should have added a period to that last sentence or not; my guess is that a full stop would be ok but a period is not.)

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Torsion in the cohomology of co-compact arithmetic lattices

Posted on February 6, 2013 by Persiflage

Various authors (including Bergeron and Venkatesh) have shown that the cohomology of certain arithmetic groups have a lot of torsion. For example, if \(\Gamma\) is a co-compact arithmetic lattice in \(\mathrm{SL}_2(\mathbf{C})\), and \(\mathcal{L}\) is an acyclic local system, then

\(\log |H^*(\Gamma(N),\mathcal{L}) | \gg [\Gamma:\Gamma(N)].\)

The proof relies on the fact that the difference \(l_0\) in ranks of \(\mathrm{SL}_2(\mathbf{C})\) and \(\mathrm{SU}_2(\mathbf{C})\) is one. As the invariant \(l_0\) grows, one expects there to be less torsion. How much torsion should one expect in general? I’m not sure I have an answer, but the point of this post is that Poincare duality gives a non-trivial bound, at least if one restricts to covers up a \(p\)-adic tower. Let \(\mathbb{G}\) be a semi-simple group over \(\mathbf{Q}\), Let \(G = \mathbf{G}(\mathbf{R})\), let \(K\) be a maximal compact, let \(H^* = \bigoplus H^m\), let \(\Gamma\) be a co-compact lattice, and let \(\mathcal{L}\) be an acyclic local system. Suppose that \(n = \dim(G)\) and \(d = \dim(G/K)\). Then, for a fixed prime \(p\) (for which \(\mathbb{G}(\mathbf{Q}_p)\) is split) and varying \(m\), I claim that one has the inequality

\(\log |H^*(\Gamma(p^m),\mathcal{L}) | \gg [\Gamma:\Gamma(p^m)]^{1 – \frac{d}{n}}.\)

An elementary exercise shows that \(\mathcal{L}/p \mathcal{L}\) is trivial as a local system for \(\Gamma(p^m)\) and large enough \(m\). The inequality above can then be reduced to the following claim: there is an inequality:

\(\dim H_*(\Gamma(p^m),\mathbf{F}_p) \gg p^{m(n-d)}.\)

Assume otherwise. The main point is as follows: taking the inverse limit over all \(m\), we obtain modules \(\widetilde{H}_j\) over the Iwasawa algebra \(\Lambda\). This algebra, by results of Lazard and Venjakob, is essentially a regular local ring, in particular, it makes sense to talk about the dimension of modules over that ring. If the inequality above does not hold, then these modules will have small dimension, explicitly, co-dimension greater than \(d\). This is so small that Poincare duality will, Ouroboros like — swallow itself completely and collapse into nothingness. However, the only way that could happen is if there was nothing to start with, which is nonsense.

More mathematically, consider the completed homology groups

\(\widetilde{H}_* = \displaystyle{\lim_{\leftarrow} } \ H_*(\Gamma(p^m),\mathbf{F}_p)\)

The homology groups may be computed by a complex of free \(\Lambda\)-modules obtain by lifting an initial triangulation on the base. (Here one thinks of group cohomology as the cohomology of the associated arithmetic quotients, of course.) Poincare duality then explains what happens when one takes the dual of this sequence and considers the corresponding homology groups, namely, there is a spectral sequence:

\(\mathrm{Ext}^i(\widetilde{H}_j,\Lambda) \Rightarrow \widetilde{H}_{d-i-j}.\)

This spectral sequence might be more familiar to some readers if one imagines \(\Lambda\) to be a field, in which case the zeroth Ext group is a Hom and the higher Exts vanish, and one obtains the duality isomorphisms between homology and cohomology over a field. Or, if \(\Lambda\) was the integers, then then zeroth Ext group is a Hom, the first Ext group is torsion, the higher Ext groups vanish, and one obtains the usual short exact sequence comparing the dual of homology to cohomology up to a torsion error term.) The dimension assumption we made implies that the limits are small as \(\Lambda\)-modules, in particular that \(\mathrm{Ext}^i(\widetilde{H}_j,\Lambda) = 0\) for all \(i \le d\). The key here is a Theorem of Ardakov and Brown relating the size of the cohomology growth under towers to the codimension of the module. Yet putting this assumption into the spectral sequence shows that all terms with \(i + j \le d\) vanish, and hence that \(\widetilde{H}_{0} = \widetilde{H}_{d-d} = 0\). Yet it is easy to see that
\(\widetilde{H}_{0} = \mathbf{F}_p\), and thus we have a contradiction.

In fact, this is the same argument that ME and I used to give lower bounds on torsion for \(p\)-adic analytic covers of \(3\)-manifolds. There is some slack where the argument can be improved – since one only needs vanishing for a triangular portion of the spectral sequence, you are in good shape if you have extra information about the lower rows. Of course, the real answer to the amount of mod p torsion in these towers (which is a different question to the original one of torsion over the integers) should be:

\(\dim H_*(\Gamma(p^m),\mathbf{F}_p) \sim p ^{m(n-l_0)},\)

where \(l_0\) was defined above.

Edit: In a previous version of this post, I confused the roles of \(\mathrm{dim}(K)\) and \(d = \mathrm{dim}(G/K)\). For complex groups one has \(n = 2d\), and this is asymptotically the correct estimate for simple real groups. In general, one has \(n \ge (3/2)d\), with the worse case, ironically, corresponding to (any number of copies of) \(\mathrm{SL}_2(\mathbf{R})\). So you get a bound of the form:

\(\log |H^*(\Gamma(N),\mathcal{L}) | \gg [\Gamma:\Gamma(N)]^{1/3}.\)

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A scandal in Romania

Posted on January 28, 2013 by Persiflage

I was invited to review some research proposals for the CNCS. They offered a modest remuneration for my time (something like €168, I believe). For privacy reasons I won’t comment on the proposals I read, suffice to say that they did actually exist (and I was impressed with the quality). However, in order to process my payment, they requested a surprisingly large amount of information, including a copy of my passport and bank account numbers. The process is long over (almost two months), but I have still have not been paid, and several emails to various people have gone unanswered. Perhaps I should check my credit report…

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Small Cyclotomic Integers

Posted on January 26, 2013 by Persiflage

Julia Robinson is a famous mathematician responsible for fundamental work in logic and in particular on Hilbert’s Tenth problem. Less well known nowadays is that her husband, Raphael Robinson, was a number theorist at Berkeley. One question R.Robinson asked concerned small cyclotomic integers. Namely, let \(\alpha\) be a cyclotomic integer, and suppose that every conjugate of \(\alpha\) has absolute value at most \(R\). Then what can one say about \(\alpha\)? If \(R \le 1\), then Kronecker’s theorem says that \(\alpha\) is a root of unity (this statement only requires that \(\alpha\) is an algebraic integer). Robinson studied the problem of what happens when \(R \le 2\) and also \(R \le \sqrt{5}\). He made five conjectures concerning these questions, four of which were solved in the 60’s by Jones, Cassels, and Schinzel. Five decades later, Frederick Robinson (no relation!) and Michael Wurtz proved the last of these conjectures (while working with me as summer students), and their paper has just been accepted by Acta Arithmetica. In particular, they answer the following problem: if \(\alpha\) is an algebraic integer the largest of whose absolute values is \(R \le \sqrt{5}\), then what are the possible values of \(R\)? Two such families of such numbers are those of the form

\(\zeta + \zeta^{-1}, \qquad i + \zeta + \zeta^{-1}\)

for a root of unity \(\zeta\). These give all \(R\) of the form

\(2 \cos(\pi/N), \qquad \sqrt{1 + 4 \cos^2(\pi/N)}.\)

Note that these sets have limit points at \(\sqrt{4}\) and \(\sqrt{5}\) respectively. It turns out that there exactly two further exceptions, as follows:

\(\displaystyle{\frac{\sqrt{3} + \sqrt{7}}{2}, \qquad \sqrt{\frac{5 + \sqrt{13}}{2}}}\)

The first element is totally real and cyclotomic, and so manifestly occurs as such an \(R\). The second turns out to be the absolute value of \(1 + \zeta_{13} + \zeta^4_{13}\). The proof by Robinson and Wurtz actually applies to slightly larger values of \(R\), and after the limit point \(\sqrt{5}\) there is another gap, and the next smallest possible \(R\) is

\(|1 + \zeta_{70} + \zeta^{10}_{70} + \zeta^{29}_{70}| \sim \sqrt{5.017655 \ldots}\)

The first two exceptional numbers turn up in relation to subfactors. How about the last example?

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Random p-adic Matrices

Posted on January 23, 2013 by Persiflage

Does anyone know if the problem of random matrices over (say) \(\mathbf{Z}_p\) have been studied?
Here I mean something quite specific. One could do the following, namely, since \(\mathbf{Z}_p\) is compact with a natural measure, look at random elements in \(M_N(\mathbf{Z}_p)\) and then ask about the distribution of several obvious quantities as \(N\) goes to \(\infty\). For example, one can consider the rank of \(M \mod p\), which translates into an elementary counting problem over \(\mathbf{F}_p\). However, I don’t mean this, that would just be rubbish for my purposes. What I am looking for is something that models a random compact operator, and then I want to understand the behavior of the normalized eigenvectors as the eigenvalue
\(\lambda \rightarrow 0\). To be concrete, let \(B = \mathbf{Q}_p \langle T \rangle\) be the Tate algebra corresponding to the open unit ball. Then consider a “random” compact operator \(U\) acting on \(B\). What does random mean? This is a good question, to which I do not know the answer. But let me give several properties that it should satisfy. Because the ball \(B\) is a disk, it is “dimension 2 as a real manifold”, and so — imagining that our compact operator is a \(p\)-adic avatar of \(e^{-\nabla}\) for the Laplacian \(\nabla\) — the eigenvalues of \(U\) should satisfy Weyl’s Law:

\(N(T):=\{ \# \lambda \ \| \ -v(\lambda) < T \} \sim \displaystyle{ \frac{\mathrm{Vol}(B)}{4 \pi}} \cdot T.\)

Here \(v(\lambda)\) denotes the valuation of \(\lambda \in \overline{\mathbf{Q}}_p\). Ignoring the volume factor, this just means that the Fredholm determinant \(\det(1 – U T)\) has a Newton Polygon with certain quadratic growth. I’m not sure exactly what ensembles one can come up with to define such operators, which is one of my questions. Let us also assume, although this may not be necessary, that \(U\) is semi-simple and admits nice convergent spectral expansions. We can’t quite insist that \(U\) is a self-adjoint operator, because one doesn’t have p-adic Hilbert spaces. For such an operator, what behavior should one expect of the normalized eigenvalues \(\phi_j\) of \(U\)? For example, suppose one knows that the number of zeros of \(\phi_j\) goes to infinity. What limit distribution should the zeros of \(\phi_j\) satisfy when \(\lambda \rightarrow 0\)? (Somewhat troubling here is that the eigenvalues will lie in \(\overline{\mathbf{Q}}_p\) in general and \(\overline{\mathbf{C}}_p\) has compactness issues…)

As you might guess, this is related to p-adic arithmetic quantum CHAOS, a group of subjects which gets sexier every time an extra adjective is added, and will form part of my student project at the Arizona Winter School.

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NT Seminar: A haruspicy

Posted on January 18, 2013 by Persiflage

Following JSE’s advice, I will blog on something that I know absolutely nothing about. Apologies in advance for mathematical errors!

SLM gave a number theory seminar this week about the first Betti number of \(\Gamma(n)\) — as \(n\) varies — for certain lattices in \(\mathrm{SU}(2,1)\). In particular, he proved an upper bound of the form:

\(\mathrm{dim} \ H^1(\Gamma(n),\mathbf{Q}) \ll [\Gamma:\Gamma(n)]^{3/8 + \epsilon},\)

which turns out (in certain cases) to be essentially the best possible estimate. As was known to Rogawski, the forms contributing to \(H^1\) all arise via endoscopy. In particular, if \(\Gamma\) is simple in the sense of Kottwitz, then the first cohomology vanishes (this also is due to Rogawski). So assume we are not in that case. The argument proceeds mostly as one would expect: Rogawski classifies the endoscopic forms which contribute to cohomology — they come from certain representations \(\xi \times \mu\) for \(U(2) \times U(1)\). Here I think the choice of Grossencharacter \(\mu\) is almost determined by \(\xi\), so I will drop it from the notation below. The possible packets can be described as follows:

1. Singletons for the split primes.
2. A set \(\{J^{+},D^{-}\}\) for the interesting infinite prime, where \(J^{+}\) contributes (via \((\mathfrak{g},K)\) cohomology) to \(H^1\) and another representation \(D^{-}\) which doesn’t (although it contributes to \(H^2\), I think).
3. A set \(\{\pi_s, \pi_p\}\) consisting of a supercuspidal representation and another representation at the inert primes.
4. Something similar to 3. for the ramified primes.

Using Matsushima’s formula, in order to count the contribution to cohomology one has to deal with the following:

1. The global multiplicity: this is either \(1\) or \(0\) depending on certain signs related to epsilon factors. As one varies \(n\) this should vanish half the time, but one can ignore it as far as an upper bound goes.

2. Suppose that \(p\) divides \(n\), and let \(K\) be a hyperspecial maximal compact at \(p\). Then one has to bound the trace of the characteristic function of \(K(p^k)\) on the representations \(\pi_s\) and \(\pi_p\).

Let \(f\) be such a characteristic function. One would like to write down a corresponding transfer function \(f^H\) on the endoscopic group such that:

\(\mathrm{Tr}(\pi_s,f) + \mathrm{Tr}(\pi_t,f) = \mathrm{Tr}(\xi,f^H)\)

By the Fundamental Lemma, if \(f\) is the characteristic function of the hyperspecial \(K\) itself, then \(f^H\) turns out to be the characteristic function on the maximal compact of \(U(2)\). SML shows that (using some of the same computations required for the fundamental lemma for \(U(3)\)) the same identity holds for the corresponding characteristic function for \(K(p^n)\), that is, the transfer \(f^H\) is the characteristic function of \(U(2)(p^n)\). Is this true for any deeper reason? More generally, to what extent do characteristic functions transfer to characteristic functions?

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Number theory and 3-manifolds

Posted on January 13, 2013 by Persiflage

It used to be the case that the Langlands programme could be used to say something interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now, after the work of Agol, Wise, and others has blown the subject to smithereens, this gravy train appears to be over. It seems to me, however, that the great advance in our knowledge of hyperbolic 3-manifolds has precious little to say about arithmetic 3-manifolds qua lattices in semi-simple groups. As a basic example, suppose that \(X\) is a maximal compact arithmetic three orbifold associated to a quaternion algebra \(Q/F\) for some field \(F\) (with the appropriate behavior at the infinite primes). Then one may ask whether \(X\) has positive Betti number after some finite congruence cover \(\widetilde{X} \rightarrow X\). Let’s call this the virtual congruence positive Betti number conjecture. (This conjecture should be true – it is a consequence of Langland’s conjectural base change for \(\mathrm{SL}(2)\), which everyone believes but is probably very difficult.) AFAIK, there’s not really much one can say about this problem from the geometric group theory/RAAG/LERF/etc perspective, where the arithmetic structure of the tautological \(\mathrm{SL}(2)\)-representation does not seem to play so much of a role. A related question is the extent to which arithmetic 3-manifolds are intrinsically different from their non-arithmetic hyperbolic brethren. Is the virtual congruence Betti number conjecture (for arithmetic manifolds) something that could plausibly answered using geometric group theory?

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Where even the flat tax is a liberal dream

Posted on January 3, 2013 by Persiflage

As Democrats and Republicans squabble over the expiration of the Bush tax cuts, it’s worth noting that at least in one corner of America, even the flat tax is considered too progressive. It used to be the case that if you landed between Whitechapel Road and King’s Cross Station (or their local equivalents) you were forced to pay $200 or 10% of your assets, which ever was lower. Now, bearing in mind that even this amounts to a marginal tax rate which decreases as one’s wealth increases (and is thus less progressive than a flat tax), apparently it wasn’t enough for Rich Uncle Pennybags. Now everyone must pay $200, regardless of income. Oh the humanity!

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Classic Papers in Number Theory

Posted on December 27, 2012 by Persiflage

One of my students came to me with the idea of having a reading course on “classic papers in number theory”. The idea is for everyone to spend the week reading a particular paper, and then have one student lead a discussion on the key details of the paper. The paper should ideally be a “classic”, but yet accessible enough that students can read it by themselves and at least be able to contibute useful thoughts/questions to the discussion section (it definitely should *not* end up as a lecture by me). The hope is, that by the end of the semester, the students have a good background knowledge of some of the “big ideas” in the subject. My question is: what are a good choice of papers to use for this seminar? Ideal is something in the 20-30 page range for which the key ideas can be absorbed in one to two weeks, for example, Ken’s paper on the converse to Herbrand. Some wag suggested the collected works of Tate, and truthfully, any mathematician who read and absorbed all the ideas of Tate would probably be a pretty strong mathematician. Here are some things I jotted down, along with some extra comments I’m adding now:

Ribet converse to Herbrand
Mazur-Wiles (too much like the previous paper?)
Mazur Eisenstein ideal (perhaps too long?)
Something on Iwasawa theory for class groups of cyclotomic fields (I haven’t read the original papers here myself)
Serre’s Duke paper on Serre’s Conjecture
Deligne?? (I’m not even sure what paper I meant here)
Tate p-divisible groups (could be a semester course)
Tate thesis
Oort-Tate
Gross-Zagier
Ravi Annals (on lifting Galois representations, I think)
Oort-Tate
Fontaine (not sure what paper I meant here either, could be AB/Q or the definition of B-cris)
Tunnell Congruence #
Faltings
Grothendieck Tohoku
Serre GAGA

I’m sure there are many great papers that I am missing, and some here that won’t work very well. One problem with this list is that it is a little too narrow (as I write this, I am thinking perhaps of adding a paper by Hardy on the circle method – every algebraic number theorist should know about the circle method!). Suggestions very much appreciated!

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Olfactory Dreams

Posted on December 24, 2012 by Persiflage

I was having dinner with NB at a restaurant, when I noticed that several patrons at nearby tables were smoking. I found this a little bit peculiar, in part because smoking is illegal in Chicago restaurants, but mostly because NB is far more sensitive to cigarette smoke than I am, and she hadn’t noticed anything. As NB paid the (relatively modest) bill by filling out a second mortgage application, I decided that I must be dreaming, because otherwise NB would have complained about the smoke. By that point, I couldn’t remember whether I had detected the smokers by visual cues or by olfactory ones, which made me curious as to whether one can “smell” during a dream.

Cue to lunch the next day, when (now at a sandwich shop) I bring up the topic of whether one can have olfactory dreams. Nobody seems to have any idea, but I notice Toby Handfield and Mike Liddell having lunch at the adjoining table, so I ask them. Toby tells me that before Liv could talk, she had vivid dreams involving her sense of smell, which she explained by pointing to the corresponding eigenvalues on a piece of paper. Unfortunately, this was still part of my dream, so it didn’t really answer the question. Later on, we ended up on a tour of the workshop of a master brewer, which was filled to the brim with bottles of armagnac and port which the brewer used to make fine adjustments to the final product. (I did end up tasting the result, but it did not leave much of an impression – perhaps because I couldn’t smell anything.)

Cue to today, which, while hopefully no longer dreaming, I am still curious about the same question. It is suggested
http://www.firstnerve.com/2008/10/dreaming-of-smell.html

http://baywood.metapress.com/app/home/contribution.asp?referrer=parent&backto=issue,5,6;journal,31,120;linkingpublicationresults,1:300311,1

here that such dreams are indeed possible, but not particularly common (especially amongst men). Unfortunately, I rarely remember my dreams, so I started a dream diary to record them; this is the sixth entry in about as many months. (None of the previous entries contain any remarks concerning smell, however, there is an entry about Toby Gee joining the biology department at Berkeley.)

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