## What is my Kasparov Number?

This has been a fun week in sport, what with England slaughtered at the Gabbatoir and Anand sliced up by Carlsen’s endgame magic. The latter games were fascinating if not necessarily exciting per se; consisting more of slow grinds rather than Kasparov style flourishes. Speaking of Kasparov, following Andrew Gelman, one defines the Kasparov number as the length of the shortest (ordered) chain of people (starting at you and ending at Kasparov) such that each person has beaten his or her successor at a game of chess. Let me also define the weaker “Draw Kasparov” number where one now allows either wins or draws. Being a little light on official tournament play myself, I have felt free to suitably relax the requirement of where the games take place.

The best upper bounds I could come up with for my Kasparov are around 6, which is probably pretty close to the right answer. However, my “draw” number against Kasparov is 2: I drew* with the British GM Tony Miles in 1991, and Miles’ best result against Kasparov was a draw (he was crushed by Kasparov 5.5-0.5 in 1986, but that 0.5 point counts!)

*OK, this game took place as part of a 40 player simultaneous exhibition, but that still counts!

Posted in Chess, Cricket, Waffle | Tagged , , , , | 1 Comment

## Local representations occurring in cohomology

Michael Harris was in town for a few days, and we chatted about the relationship between my conjectures on completed cohomology groups with Emerton and the recent work of Scholze. The brief summary is that Scholze’s results are not naively strong enough to prove our conjectures in full, even for PEL Shimura varieties. Motivated by this discussion, I want to give two quite explicit challenges concerning the mod-p cohomology of arithmetic locally symmetric spaces. The first I imagine will be very hard — it should already imply a certain vanishing conjecture of Geraghty and myself which has strong consequences. However, the formulation is somewhat different and so might be helpful.

Fix an arithmetic locally symmetric space $X$ corresponding to a reductive group $G$ over $\mathbf{Q}.$ Let $\ell$ and $p$ be distinct prime numbers. Consider the completed cohomology groups

$\widehat{H}^d(\overline{\mathbf{F}}_{\ell}) = \displaystyle{\lim_{\rightarrow}} H^d(X(K),\overline{\mathbf{F}}_{\ell}), \qquad \widehat{H}^d(\mathbf{C}) = \displaystyle{\lim_{\rightarrow}} H^d(X(K),\mathbf{C}),$

where we take the completion over all compact open subgroups. The limit has an action of $G(\mathbf{A})$ for the finite adeles $\mathbf{A}$, and so, in particular, has an action of $G(\mathbf{Q}_p)$. What irreducible $G(\mathbf{Q}_p)$ representations can occur in $\widehat{H}^d(\overline{\mathbf{F}}_{\ell})$? Here is a guess:

Conjecture: If the smooth admissible representation $\pi$ of $G(\mathbf{Q}_p)$ occurs as an irreducible sub-representation of $\widehat{H}^i(\overline{\mathbf{F}}_{\ell})$, then there exists an irreducible representation $\Pi$ of $G(\mathbf{Q}_p)$ in characteristic zero such that:

1. The Gelfand-Kirillov dimension of $\Pi$ is at least that of $\pi$. Equivalently,
$\mathrm{dim} \ \Pi^{K(p^n)} \gg \mathrm{dim} \ \pi^{K(p^n)}.$
2. Let $\mathrm{rec}(\Pi)$ and $\mathrm{rec}(\pi)$ be the Weil-Deligne representations associated to $\Pi$ and $\pi$ respectively by the classical local Langlands conjecture and the mod-$\ell$ local Langlands conjecture of Vigneras. Then

$(\overline{\mathrm{rec}(\Pi)})^{\mathrm{ss}} \simeq (\mathrm{rec}(\pi))^{\mathrm{ss}}.$

3. The representation $\Pi$ occurs in $\widehat{H}^j(\mathbf{C})$ for some $j \le i$.

Roughly speaking, this conjecture says that the irreducible representations occurring in characteristic $p$ are no more complicated than those which occur in characteristic zero. One naive way to try prove this conjecture would be to show that any torsion class lifts to characteristic zero, at least virtually. This conjecture is too strong, however, as can be seen by considering K-theoretic torsion classes in stable cohomology — the mod $3$ torsion class in $H^3(\mathrm{GL}_N(\mathbf{Z}),\mathbf{F}_3)$ can never lift to characteristic zero for sufficiently large N because the cohomology over $\mathbf{Q}$ is zero for all congruence sugroups by a theorem of Borel. The conjecture as stated seems very hard.

In a different direction, here is the following challenge to those trying to understand completed cohomology through perfectoid spaces. (I expect one can prove this by other means, but I would like to see a proof using algebraic geometry.)

Problem: Fix an integer $d$, and let $X_g$ be the Shimura variety corresponding to the moduli space of polarized abelian varieties of genus $g$. Prove that, for $g$ sufficiently large, the completed cohomology group $\widetilde{H}^{d}(X_g,\mathbf{F}_p)$ is finite over $\mathbf{F}_p$.

An equivalent formulation of this problem is to show that the only smooth admissible $\mathrm{GSp}_{2g}(\mathbf{Q}_p)$-representations $\pi$ which occur inside $\widetilde{H}^{d}(X_g,\mathbf{F}_p)$ are one dimensional.

## Abelian Varieties

Jerry Wang gave a nice talk this week on his generalization of Manjul’s work on pointless hyperelliptic curves to hyperelliptic curves with no points over any field of odd degree (equivalently, $\mathrm{Pic}^1$ is pointless). This work (link here) is joint with Manjul and Dick, so the exposition is predictably of high quality. But I wanted to mention a result that arose during the talk which I found quite intriguing. Namely, given the intersection $X$ of two quadrics $P$ and $Q$ in projective (2n+1)-space, the variety of projective n-spaces passing through $X$ turns out (over the complex numbers) to be an abelian variety. For $n = 1$ this is pretty familiar, but, for general $n$, I hadn’t seen any construction like this before. It gives, for example, explicit constructions of equations for abelian varieties in surprisingly low degree. It brought me back to a lecture I once went to by Beauville as a graduate student when he talked about intermediate Jacobians (wait – perhaps this construction also has to be isomorphic to an intermediate Jacobian…). Is it possible (in some weak sense) to classify all varieties whose variety of maximal linear subspaces is an abelian variety of suitably high dimension? Are there varieties in which this construction gives rise to abelian varieties which are not isogenous to Jacobians? The geometric result is due (independently) to several authors, but, in a solo paper here, Jerry showed that the result is true arithmetically, and, even better, the construction can more precisely be described as giving an explicit torsor for the corresponding Jacobian. This very nicely generalizes the classical picture between pairs of quadrics and 2- and 4-descent.

Posted in Mathematics | | 2 Comments

## The problem with baseball

Jordan Ellenberg, in a lovely slate article, explains perfectly what I don’t like about baseball.

I think the fundamentals of baseball as a sport are sound. I like the pace of the game, the variation, the statistics, the quirkiness, the history. But my problem is that I only started following baseball after I came to the US. I supported the A’s and the Giants (no doubt already poor form amongst serious baseball fans), and I went to a few games at each park. The A’s were (as they perennially seem to be) thriving on young talent, in this case the trio of Hudson, Mulder, and Zito; the Giants were in the peak of the Bonds era. But seasons passed and players came and went — either to other teams or into disgrace (Giambi, Tejada, Bonds, etc.). Whilst my move to Boston (presumably) inspired the Red Sox to their ’04 world series win, my allegiance to any single team became even more fractured. I’m not sure you can truly love baseball unless you either grew up with a team or have a deep sense of loyalty to a particular city. I’m not going to lament a past lost era when players spent their entire career with one team (Persiflage supports capitalism!), but I know that my top sporting allegiance will never stray from the Australian cricket team.

Posted in Cricket, Waffle | Tagged , , | 1 Comment

## Does Harvard discriminate in favour of Jews?

In The American Conservative, Ron Unz published an essay that was ostensibly about whether the top Ivy schools discriminate against Asian students but, upon closer reading, was mainly concerned with arguing that Harvard/Yale/Princeton discriminate against white Gentiles in favour of Jews. The essay was widely discussed in various blogs, in part because the question of whether Asians are discriminated against is one that has a resonance in the academic community, and also because the essay contained voluminous appendices to back up its claims.

However, the wheels started to come off around February, when Columbia statistician Andrew Gelman reported on some significant inconsistencies in Unz’s methodology, as pointed out by Nurit Baytch (full disclosure: she is my wife), and Unz’s gross underestimates of the percentage of Jews among US recent IMO team members (as pointed out by Prof. Janet Mertz). There’s now an update on Gelman’s blog here, which links to a more in-depth rebuttal of Unz by Nurit (which you can find directly here). Nurit’s critique rips apart Unz’s argument on many levels (it mainly addresses the issues concerning [the lack of] discrimination in favour of Jews, not against Asians).

The entire episode points to a few disturbing facts. First, if you present a lot of data, people will trust your arguments even if your statistical analysis is completely flawed (the sociological “power of math”!). Second, if you couch your crazy argument (“The Jews” control Harvard — look, even President Drew Faust’s husband is Jewish [seriously, Unz uses this argument]) in something which is a genuine concern (discrimination against Asians), people will take you seriously. Finally, the essay was promoted by David Brooks; that should be a warning to anyone that it should not be taken seriously.

## Who is D.H.J. Polymath?

D.H.J. Polymath is the assumed collective pseudonym for the authors of a number of papers which have arisen as a result of the polymath project initated by Gowers. Presumably, since it is a matter of open record, one can go through and identify the participants and their various individual contributions.

But what is it that mathematicians on the street think when they answer this question? The answer is that D.H.J. Polymath is equivalent to T.Gowers, T.Tao, et. al. I don’t claim that this is an accurate reflection of the contribution of the participants, but simply the perception in the community (as far as I have gauged through a number of conversations). Credit for joint publications can be a tricky issue at the best of times. In this case, the people responsible for making the decision to choose a pseudonym are presumably those for whom receiving credit is the least relevant. I find myself sympathetic to the remarks made by T.Brown in the mathscinet review of the first polymath paper:

This reviewer would have preferred to see, rather than the pseudonym “Polymath”, a list of authors. In other fields there are papers with a hundred co-authors. Why not in mathematics a paper with twenty or thirty co-authors, with extra credit for the person(s) who wrote the exposition?

Posted in Mathematics, Politics | Tagged , , , | 1 Comment

## Why is my paper taking so long to review?

The question in the title does not refer to any of my own papers; rather, I want to *answer* the question from the perspective of an editor. Here, roughly, is how the sausage is made (this is a medium case scenario, your mileage may vary). Keep in mind that this is a journal which has relatively good standards (for number theorists, we are talking somewhere between JNT and Duke).

• Day 0: After carefully selecting a suitable journal and performing a final check on your paper for typographical errors, you submit your precious baby to the whims of fate.
• Day 20(?): The paper works its way though the editorial system and is assigned to me as an editor.
• Day 40: I have had a chance to take a look at the paper and determine whether it is obviously rubbish or not. Moreover, I have identified someone (usually at the level of professor) whom I trust to give an honest opinion of both how interesting the paper is and whether it is suitable for the journal in question. I email that person asking for a quick opinion and any suggestions they may have for possible reviewers.
• Day 60: I email the expert again because they have not yet responded to my original request. Often, at this point, the expert will say that they are not qualified to give an opinion, and I return to the previous step.
• Day 80: The expert has usually found time to respond, often to suggest another expert to consult (go back two spaces).
• Day 100: I have a response from the expert. If they are only lukewarm, I reject the paper. So far, 80% of papers have now been rejected. Measured by the “standards of the industry,” I think that rejecting papers within about 3 months is acceptable to good. If the expert is enthusiastic, they either agree to referee the paper themselves or suggest someone else (often someone younger) to to the job. I then send out a detailed review request, either to the person suggested by the expert or to someone else.
• Day 120: I email a different reviewer, because the first review declines for one of the standard excuses (busy/not qualified/lazy and so makes up something about not liking commercial publishers). I email someone else.
• Day 130: They agree to review! I give them three months.
• Day 230: I email the reviewer to follow up on my previous email. They start reviewing the paper.
• Day 250: The paper is accepted. 25% of the time, the comments consist of minor typographical remarks. 50% of the time, there are a few requests for clarification, references, and corrections of minor inaccuracies. 25% of the time, there are substantial comments and corrections. In the majority of cases, the referees do a conscientious job (some papers don’t need many corrections!)

Some General Remarks:.

• Of all the papers I have edited, a small number (at most 2 or 3) have ultimately been rejected because of a fatal mathematical error (i.e., the paper would have been accepted if it had turned out to be correct). In all of those cases, I was the one who found the error.
• I end up rejecting quite a few papers because there is a fixed number of pages I can accept per year. I would anticipate doubling the number of acceptances if there were no such constraint.
• Sometimes papers do fall through the cracks. It can be very hard to find a reviewer for a very technical paper, especially one that builds off previous technical work of the author. Can one reject a paper on the basis that you couldn’t find anyone to review it? I honestly think we may be heading in that direction.
• The main task of the editor is not summary judgement, but administration. It’s not enough to email someone (say, a reviewer) and then consider one’s job done; you have to keep track of when you emailed them, so you know when to email them again (or someone else) if (or frequently when) they don’t respond. (I admit, I’m by no means perfect as a reviewer, either.)
• Any online system set up to coordinate and facilitate communication with authors/editors is more annoying than useful; I work off the grid as much as possible.
• Posted in Mathematics, Politics | 21 Comments