Monthly Archives: October 2008

Three comments on Halloween

  • Yesterday a ten-year-old girl and her mom walked by me, and the girl was saying “One Indiana Jones and two football players.” I couldn’t parse this at all until I figured out she was talking about the costumes her classmates wore to school that day.
  • A lot of little kids are wearing Spiderman costumes with sewn-in muscle chests. This is wrong. The whole point of Spiderman is that he has skinny arms and legs, like a spider, yet is super-strong. If you are a little kid and you want to wear a muscle chest, please consider being the Hulk.
  • CJ was a clown. At the last moment he decided he wanted to be an alligator instead. I suggested that he could wear the clown suit and make alligator jaw-snapping motions with his arms, thereby being an “alligator clown.” CJ argued, with some merit, that “alligator clown” was an incoherent costume concept and he wanted to be just an alligator. Finally we got him in the suit, but he wouldn’t wear the rainbow wig. So I wore that. We knocked on a few doors, but mostly he was excited to be outside past his bedtime, and just wanted to run around in the dark in his clown suit.
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Early voting in Madison

I voted yesterday after work, and if you live in the city of Madison, so can you.

The wait was about an hour; I was told it was longer at lunchtime, shorter during standard work hours. The atmosphere was pleasant and chatty. Lots of college students. A poster outside the city clerk’s office said that almost 10,000 people have already voted in person, and 10,000 more absentee ballots have been mailed.

Everyone’s expecting high turnout this year, and lawyers are already circling the polling places, making ready for the inevitable legal challenges arising from long lines, spoiled ballots, and tired voters and election workers. So if you want to do something civic-minded and non-partisan this election, go down to the city clerk and vote early. Every voter not at the polls on Election Day makes it that much more likely that everything goes smoothly.

The city clerk is at 210 MLK boulevard, and is open for voting 8am-7pm Mon-Thurs, 8am-4:30pm Friday, 8am-3pm Saturday. Last early voting is 5pm on November 3. If you’re not registered at your current address, bring proof of residence and they’ll register you on the spot. Bring a book. I had White Teeth, which is great so far.

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We’re in a new era of mathematical publishing indeed; a paper posted yesterday on the arXiv cites a post from this blog as a reference.

The paper, by Strashimir Popvassiliev, constructs for every positive integer n a simple closed plane curve with exactly n inscribed squares. (It’s an old conjecture of Toeplitz that every simple closed plane curve contains at least one inscribed square.) This seems to speak against philosophy, mentioned by Denne in her guest post here, that “the reason” every curve has at least one inscribed square is because every curve has an odd number of inscribed squares.

I’m not sure Popvassiliev’s example really contradicts this philosophy. Surely the squares should be counted with multiplicity, in the appropriate sense. With a more naive notion of “counting” you can’t expect parity conditions to hold. For instance, you certainly want to say that a straight line intersects a smooth closed curve an even number of times. Naively, you might complain that a tangent line to a circle intersects the circle only once! But of course, it really crosses the curve twice; it’s just that the two crossings are at the same point.

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Is there a Philly title drought?

If the Phillies win tonight, they’ll bring the city of Philadelphia its first championship in a major sport since the Sixers won the NBA finals in 1983.

Is 25 years really a long time to wait? Philadelphia is the eighth largest Combined Statistical Area in the U.S. No larger CSA has waited nearly so long; the closest is San Francisco / Oakland / San Jose, which hasn’t had a champion since the 1990 49ers. But go down the list a little and you find some sorrier stories. Seattle is the 12th largest CSA, about 2/3 as big as Philadelphia. Their last — and only! — champion was the 1979 Seattle SuperSonics. Go down to #15 and you’re at Cleveland, a pretty big city with a long sports history and a devoted fan base, which hasn’t seen a championship of any kind since the 1948 Indians won the World Series.

So stop crying, Philadelphia. When Rocky Colavito curses you, you stay cursed.

Update: See comments for some corrections to my hurriedly compiled statistics.

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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?

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Avenue Q at the Overture

Overture Center is facing a $3 million deficit; if you want to spend some money and help the downtown arts district get off its financial knees, why not go see Avenue Q (now through October 26?) It is that rare thing — a big musical which is not either a retread of songs everybody knows (Movin’ Out, Mamma Mia) or a thin coat of “edgy” on a boring Broadway chassis (yes, you, Urinetown.) Avenue Q has jokes. Funny jokes! Also puppets that have sex.

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F_1 and the braid group — a note of skepticism

Since I wrote this post, I’ve become less sure about this assertion that the braid group can be thought of as GL_n(F_1[t]). Here are three reasons to be doubtful:

  • As Jim points out in commments, GL_n(F_q) embeds in GL_n(F_q[t]), but S_n doesn’t embed in the braid group. This has to be counted against the braid group, I think. Jim also says that in his version of F_1 geometry, which comes out of lambda-rings, GL_n(F_1[t]) is just S_n.
  • Terry Tao observed that one might expect GL_n(F_1[t]) to embed into GL_n(F_q[t]), just as GL_n(F_1) embeds into GL_n(F_q). But this doesn’t appear to be the case, at least not for any obvious reason. Keep in mind, it was quite hard to prove that the braid group had any faithful linear representations at all! The n-dimensional linear representations developed by Lawrence and Krammer, and proven faithful by Bigelow (and then again by Krammer) have coefficients in Z[t,1/t,u,1/u]. So the idea that one might find the braid group inside GL(F_1[t,1/t,u,1/u]) remains, from this point of view, alive! But I wonder whether Jim thinks this latter group is also S_n…?
  • Finally, the argument given by Kapranov and Smirnov looks like it’s making a case that the braid group admits a map to GL_n(F_1[[t]]), not so much that it should be thought of as GL_n(F_1[t]).

GL_n(F_1[[t]]), by the way, seems a little easier to get our hands on. Note that the order of the finite group GL_n(F_q[t]/t^k) is just a power of q times |GL_n(F_q)|. So, setting q = 1, one might expect

|GL_n(F_1[t]/t^k)| = |GL_n(F_1)| = n!

and in particular

GL_n(F_1[[t]]) = GL_n(F_1[t]/t^k) = S_n.

(Short version of this argument: “Pro-1 groups are trivial.”) In this case, the braid group certainly does map to GL_n(F_1[[t]])!

By the argument in the previous post, one would then want to say GL_n(F_1((t))) is the affine Weyl group Z^{n-1} semidirect S_n. Which means that the Hecke algebra

GL_n(F_1[[t]]) \ GL_n(F_1((t))) / GL_n(F_1[[t]])

is a pretty standard object — the double cosets above are in bijection with S_n-orbits on Z^{n-1}, which can be identified with the symmetrized monomials in n variables whose degree is a multiple of n, up to multiplication by x_1, … x_n. So the Hecke algebra should just be some version of the algebra of symmetric functions on n variables.

Believers that the braid group is GL_n(F_1[t]) are strongly encouraged to revivify my faith in comments.

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Reader survey: which journals are fast?

I’m often asked by students and junior colleagues where I think they should submit a paper they’ve just finished. It’s not so hard to figure out whether a given journal is a good mathematical fit for the paper. What’s harder is when the author has a job application coming up, and really wants the paper not to sit in referee hell for a year and a half.

My go-to journals for a fast response are Mathematical Research Letters and International Math Research Notices. But it would be great to have some others. So: which journals, in your experience, are reliably fast? Bonus points if they’re cheap, or even free.

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In which the Red Sox keep playing

Down 7-0 in the 7th inning of a must-win game, the Scarlet Footwear come back for an improbable 8-7 win to keep their season alive. Obama fans should now begin to worry about my “John McCain is the Red Sox” theory. Note that, days after I made that post, McCain announced that the Sox are his “sentimental favorite” to win the World Series.

Politics aside: if the Red Sox can win yet another seven-game series after being down 3-1, they’re the comeback team of the modern era. They’ll have to retire “Sweet Caroline” and replace it with this:

Actually, maybe McCain should adopt the song too. It might help him with the all-important “grunge nostalgia” demographic.

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Shana Wittenwyler is a really good photographer

If you’re in Memorial Union, take a few minutes to see her exhibition, Portrait of My Father’s Dairyland, which is hanging across from the Lakefront on Langdon food court through November 11. Beautiful and effective portraits of Wittenwyler’s hometown of Monticello, WI, where the dairy way of life is vanishing and there’s nothing in sight to replace it. If you can think of nothing but the upcoming election, she also has a great gallery of campaign shots here. Below:

The television at the Quality Inn broadcasts Hillary Clinton being interviewed by the local media the morning of the South Carolina Democratic Primary, Newberry, South Carolina, January 26, 2008. (Image © Shana Wittenwyler.)

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