The conformal modulus of a mapping class

So I learned about this interesting invariant from a colloquium by Burglind Jöricke.

(Warning — this post concerns math I don’t know well and is all questions, no answers.)

Suppose you have a holomorphic map from C^* to M_g,n, the moduli space of curves.  Then you get a map on fundamental groups from $\pi_1(\mathbf{C}^*)$ (otherwise known as Z) to $\pi_1(\mathcal{M}_{g,n})$ (otherwise known as the mapping class group) — in other words, you get a mapping class.

But not just any mapping class;  this one, which we’ll call u, is the monodromy of a holomorphic family of marked curves around a degenerate point.  So, for example, the image of u on homology has to be potentially unipotent.  I’m not sure (but I presume others know) which mapping classes u can arise in this way; does some power of u have to be a product of commuting Dehn twists, or is that too much to ask?

In any event, there are lots of mapping classes which you are not going to see.  Let m be your favorite one.  Now you can still represent m by a smooth loop in M_g,n.  And you can deform this loop to be a real-analytic function

$f: \{z: |z| = 1\} \rightarrow \mathcal{M}_{g,n}$

Finally — while you can’t extend f to all of C^*, you can extend it to some annulus with outer radius R and inner radius r.

Definition:  The conformal modulus of a mapping class x is the supremum, over all such f and all annuli, of (1/2 pi) log(R/r).

So you can think of this as some kind of measurement of “how complicated of a path do you have to draw on M_{g,n} in order to represent x?”  The modulus is infinite exactly when the mapping class is represented by a holomorphic degeneration.  In particular, I imagine that a pseudo-Anosov mapping class must have finite conformal modulus.  That is:  positive entropy (aka dilatation) implies finite conformal modulus.   Which leads Jöricke to ask:  what is the relation more generally between conformal modulus and (log of) dilatation?  When (g,n) = (0,3) she has shown that the two are inverse to each other.  In this case, the group is more or less PSL_2(Z) so it’s not so surprising that any two measures of complexity are tightly bound together.

Actually, I should be honest and say that Jöricke raised this only for g = 0, so maybe there’s some reason it’s a bad idea to go beyond braids; but the question still seems to me to make sense.  For that matter, one could even ask the same question with M_g replaced by A_g, right?  What is the conformal modulus of a symplectic matrix which is not potentially unipotent?  Is it always tightly related to the size of the largest eigenvalue?

Southern California Number Theory Day, the airport Chili’s, Evan Longoria counterfactuals

I came back this morning from a very brief trip to California to speak at Southern California Number Theory Day, hosted this year at UC Irvine. The other speakers were terrific, well worth undergoing the pain of a red-eye flight back Midwest. (Non-math material follows below the SCNTD sum-up, for those readers who don’t cotton to the number theory.)

• Brian Conrad talked about his work (some of it with Gabber and G. Prasad) on finite class numbers for algebraic groups, and an alternative to the notion of “reductive group” over global function fields of characteristic p, where the usual notion doesn’t behave quite as well as you expect. Very clear, and very much in Brian’s style in its admirable refusal to concede any “simplifying assumptions.” Well, except the occasional avoidance of characteristic 2.
• Jeff Achter talked about a circle of results, many joint with Pries, about the geography of the moduli space of curves in characteristic p. Here you have lots of interesting subvarieties that don’t have any characteristic 0 analogue, such as the “p-rank r stratum” of curves whose Jacobians have exactly p^r physical p-torsion points. Typical interesting theorem: the monodromy representation of the non-ordinary locus (a divisor in M_g) surjects onto Sp_2g, just as the monodromy representation of M_g itself does. I asked Jeff whether we know what the fundamental group of the non-ordinary locus is — he didn’t know, which means probably nobody does.
• Christian Popescu closed it out with a beautiful talk arguing that we should replace the slogan “Iwasawa theory over function fields is about the action of Frobenius on the Tate module of a Jacobian” with “Iwasawa theory over function fields is about the action of Frobenius on the l-adic realization of a 1-motive related to the Jacobian.” This point of view — joint work of Popescu and Greither — cleans up a lot of things that are customarily messy, and shows that different-looking popular conjectures at the bottom of the Iwasawa tower are in fact all consequences of a suitably formulated Main Conjecture at the top.

On the way over I’d eaten a dispiriting lunch at the St. Louis airport Chili’s, where I waited twenty minutes for a hamburger I can only describe as withered. Last night, I got to LAX with an hour and a half to spare, and the Rays and Phillies in the 7th inning of a close game 3. And the only place to watch it was Chili’s. This time I was smart enough just to order a Diet Coke and grab a seat with a view of the plasma screen.

The airport Chili’s, late on a Word Series night, turns out to be a pretty pleasant place. People talk to you, and they talk about baseball. On one side of me was a pair of fifty-something women on their way to Australia to hang out with tigers in a nature preserve. One was a lapsed Orioles fan from Prince George’s County, the other had no team. On the other was a guy from Chicago in a tweed jacket who writes for the Daily Racing Form. He liked the Mets. We all cheered for Philadelphia, and pounded the table and cussed when Jayson Werth got picked off second in the 8th in what seemed at the time the Phils’ best chance to score. (Werth, you might remember, used to be the Orioles’ “catcher of the future”; in the end, he never played a major-league game for the Orioles, or behind the plate.)

The game went into the bottom of the 9th tied 4-4, about a half hour before I was supposed to board. I figured I’d miss the end. But a hit batsman, a wild pitch, and an off-line throw to second put Eric Bruntlett on third with nobody out. Tampa Bay intentionally walked the next two hitters to get to Carlos Ruiz.

Question 1: Was this wise? I understand you set up the force, and I understand you want to put the worst Phillies hitters in the critical spot. But even a pretty bad hitter suddenly turns pretty good if you can’t walk him. And the extra two baserunners mean that Tampa Bay is still in big trouble even if Bruntlett is out at the plate after a tag-up. Mitchel Lichtman of The Hardball Times says Joe Maddon blew this decision.

And then: well, you probably saw this on TV, but Ruiz hits a slow, goofy chopper up the third-base line. Evan Longoria charges it, but by the time he gets there Bruntlett is almost home; Longoria heaves a desperate moonball in the general direction of home plate, only much, much higher; Phillies win.

Question 2a: Would Longoria have had a play if he’d stopped, set, and thrown, instead of trying to fling the ball to the catcher mid-dive?

Question 2b: Should Longoria have tried to make the play at all? Suppose he’d just stood at third, recognizing he had no play. Maybe Bruntlett scores and the Phillies win; but maybe the ball rolls foul, sending everyone back to their base with the game still tied. My Racing Form neighbor was convinced the ball was headed foul, and that Longoria had blown the game by picking it up. Subquestion: Would any human being alive have the self-control not to charge the ball in this situation?

Question 2c: A commenter on Baseball Think Factory proposed a counterfactual ending for this game even more outlandish than what actually occurred. Say Longoria runs towards the ball, sees he has no play, decides not to pick it up and hope it rolls foul. The ball rolls past Longoria, headed towards third base, as Bruntlett crosses the plate. If the ball stays fair, Phillies win; if not, Ruiz bats again. So the ball’s rolling along the line, and meanwhile, Shane Victorino, who started on second, is rounding third — and as he passes the ball he kicks it fair. Now Victorino is clearly out for interfering with the ball in play. But in this scenario, has Philadelphia won the game?