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Charles Franklin on “Data Visualization in Political Science,” Wednesday at Union South

Good talk:  my colleague Charles Franklin, currently on leave at Marquette running serious polling on Wisconsin’s weird political microclimate, is back in town this Wednesday to give a talk about data visualization in political science:

Polling Your Resources: Using Data Visualization in Political Science

  • DateWednesday, April 11, 2012
  • Time3 p.m.
  • LocationTITU, Union South
  • DescriptionUW-Madison faculty member Charles Franklin will share examples of data visualization and discuss helping students and the public make sense of political data. If you plan to use data visualization in your teaching, come and learn how Franklin has honed this topic for his course Understanding Political Numbers. Franklin’s academic research focuses on advanced statistical and graphical analysis of public opinion and election outcomes. Light refreshments will be served. Presented by Engage.

Bovine fraternal skin graft

Another thing I learned from the August 1951 issue of The Times Review of the Progress of Science is that cows can accept skin grafts from their fraternal twins, but humans can’t.  That’s because cow fetuses actually share some blood and tissue in the womb, and automatically get desensitized to those particular foreign entities when they’re young enough not to reject them.  This was totally new to me but apparently if I knew anything about immunology I would already be familiar with this, because Peter Medawar’s work on the phenomenon earned him the 1960 Nobel Prize and more or less launched the field of acquired transplantation tolerance.

There was also an anecdote about a baby switched at birth, who doctor proved to be the identical twin of another child in his birth family by grafting a patch of his skin onto the other kid!

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Contest: worst math-related election metaphor

Just kidding: there’s not going to be a contest, because nobody’s beating Andrew Sullivan:

In one simple image, America’s soft power has been ratcheted up not a notch, but a logarithm.

What th-?

Googling reveals that I am not the first mathematician to read this sentence and say “What th-?”

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Turkelli on Hurwitz spaces and Malle’s conjecture

My Ph.D. student Seyfi Turkelli recently posted a really nice paper, “Connected components of Hurwitz schemes and Malle’s conjecture,” to the arXiv. It’s a beautiful example of the “hidden geometry” behind questions about arithmetic distributions, so I thought I’d say a little about it here.

The story begins with the old conjecture, sometimes attributed to Linnik, that the number of degree-n extensions of Q of discriminant at most X grows linearly with X, as X grows with n held constant. When n=2, this is easy; when n = 3, it is a theorem of Davenport and Heilbronn; when n=4 or 5, it is recent work of Bhargava; when n is at least 6, we have no idea.

Having no idea is, of course, no barrier to generalization. Here’s a more refined version of the conjecture, due to Gunter Malle. Let K be a number field, let G be a finite subgroup of S_n, and let N_{K,G}(X) be the number of extensions L/K of degree n whose discriminant has norm at most K, and whose Galois closure has Galois group G. Then there exists a constant c_{K,G} such that

Conjecture: N_{K,G}(X) ~ c_{K,G} X^a(G) (log X)^(b(K,G))

where a and b are constants explicitly described by Malle. (Malle doesn’t make a guess as to the value of c_{K,G} — that’s a more refined statement still, which I hope to blog about later…)

Akshay Venkatesh and I wrote a paper (“Counting extensions of function fields…”) in which we gave a heuristic argument for Malle’s conjecture over K = F_q(t). In that case, N_{K,G}(X) is the number of points on a certain Hurwitz space, a moduli space of finite covers of the projective line. We were able to control the dimensions and the number of irreducible components of these spaces, using in a crucial way an old theorem of Conway, Parker, Fried, and Volklein. The heuristic part arrives when you throw in the 100% shruggy guess that an irreducible variety of dimension d over F_q has about q^d points. When you apply this heuristic to the Hurwitz spaces, you get Malle’s conjecture on the nose.

So we were a little taken aback a couple of years later when Jurgen Kluners produced counterexamples to Malle’s theorem! We quickly figured out what was going on. There wasn’t anything wrong with our theorem; just our analogy. Our Hurwitz spaces were counting geometrically connected covers of the projective line. But a cover Y -> P^1/F_q which is connected, but not geometrically connected, provides a perfectly good field extension of F_q(t). If we’re trying to imitate the number field question, we’d better count those too. It had never occurred to us that they might outnumber the geometrically connected covers — but that’s just what happens in Kluners’ examples.

What Turkelli does in his new paper is to work out the dimensions and components for certain twisted Hurwitz spaces which parametrize the connected but not geometrically connected covers of P^1. This is a really subtle thing to get right — you can’t rely on your geometric intuition, because the phenomenon you’re trying to keep track of doesn’t exist over an algebraically closed field! But Turkelli nails it down — and as a consequence, he gets a new version of Malle’s conjecture, which is compatible with Kluners’ examples, and which I think is really the right statement. Which is not to say I know whether it’s true! But if it’s not, it’s at least the correct false guess given our present state of knowledge.

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Is there life after Potty Power?

We watched Potty Power a lot this summer. I mean, a lot. Like, John-Hinckley-watching-Taxi Driver a lot. After a while, I started wondering: who are the actors in Potty Power? Is this the kind of gig you take on your way to stardom? Or are there actors whose whole career is in toilet-training videos?

Jessica Cannon, the peppy MC and vocalist who manages to deliver lyrics like “Wash your hands / wash ‘em real good / wash your hands like you know you should” with a winning supper-club flair, appears in the New York Daily News in August 2006 as a struggling actor, working kids’ birthday parties and cocktail-waitressing between auditions to keep afloat. She’s now giving piano lessons in New York City. Matt Dyer, who plays the King to Cannon’s Queen in the play-within-a-play, “The Princess and the Potty,” got good reviews this year in a Norwich, CT production of “The Last Session,” a musical about AIDS and the music industry. Also appearing in “The Princess and the Potty” is the biggest success story of all, the scene-stealing jester Todd Alan Crain. He’s continued to appear in kids’ videos, but also seems to get consistent Off-Off-Broadway work, has some appearances on Comedy Central and the Onion News Network, and, best of all, toured the U.S. as Slim Goodbody. It sounds like what pays the bills is a steady series of jobs in the pharmaceutical industry, being the guy in the suit behind the desk who looks a little like a news anchor and introduces the in-house promotional film. I never wondered about who did jobs like that, but now I know — graduates of Potty Power.

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My friend the delegate

Friend-of-the-blog Monica Youn was a delegate at the Democratic convention. In Slate, she reports on her four days in Denver:

“The yellow-vests have been overwhelmed by sheer numbers, and the convention floor has devolved to a state of nature in which only brute force prevails. Transverse movement has slowed to the point that it’s only by referring to external landmarks that you can tell, for instance, that the guy in the Uncle Sam hat has progressed eight feet in the last 20 minutes. From my vantage point, a few rows back from the floor, it’s like watching a slow-motion feed of the LaBrea, Calif., tar pits, observing once mighty creatures—Secret Service agents, EMTs, cameramen, Chuck Schumer—struggling with increasingly feeble gestures, then succumbing, brought down by the sheer weight of inertia.”

Based on the photo, Monica is in no position to be making fun of other people’s hats.

Here’s a poem of Monica’s, recently published in Guernica:

Ignatz Oasis

When you have left me
the sky drains of color

like the skin of a tightening fist.

The sun begins
its gold prowl

swatting at tinsel streamers
on the electric fan.

Crouching I hide
in the coolness I had stolen

from the brass rods of your bed.

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Zebrafish are very interesting

I was working in Memorial Union today, trying to figure out what I think the phrase “random pro-p group” should mean, when I noticed that the guy in the booth next to mine was reading an 800-page conference proceedings about zebrafish. Well, I just had to ask. What’s so interesting about zebrafish?

It turns out that developmental biologists are BFF with zebrafish, whose growth to maturity is both very visible — their eggs are transparent — and very, very fast — from a single cell to a creature with a functioning nervous system in 24 hours, and to something resembling a fish in 4 days. So you can follow many hundreds of generations of these guys from fertilization on, watching closely on a microsopic scale, making different kinds of cells light up so you can see what they’re up to, flicking different genomic switches on and off … SCIENCE!

All material above paraphrased from my conversation with unnamed zebrafish expert, and not checked against an authoritative source — please do not use in your term paper, zebrafish Googlers! Perhaps a better resource would be Zebrafish — the peer-reviewed journal. Or the University of Oregon zebrafish FAQ, where you can find the answer to “How can we obtain mutant stocks of zebrafish for our high school lab?” Gotta go, I think I just had a great idea for a low-budget horror movie.

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Harvard reunion — the chin scratched yet again

Matthew Yglesias reports that the Harvard class of 2003 has gotten more liberal over the last five years. I have the 15th reunion survey of the class of 1993 in hand, and it’s the same with us: of the 467 respondents, 42% describe themselves as “liberal” or “extremely liberal,” up from 34% in 2003. I’d like to see the figures for 1998 — it’s not clear whether our class is actually drifting steadily to the left, or just dislikes the current President in sync with the rest of the country.

The poll was taken early in election season, when Clinton was down in the primaries but not out of the running. A big plurality of the class, 59%, back Barack Obama, with 16% liking Clinton and 15% for McCain; pretty much identical with the figures Yglesias gives for our 5th reunion counterparts.

Most surprising results: 7.3% of male respondents say they’ve paid for sex. 39% of men with children have spouses staying home full-time. 25% of all alums are married to another Harvard grad. Lawyers are much less happy than other people. OK, maybe that last one isn’t so surprising.

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Diversity Road

I spent last Wednesday morning working in the profoundly pleasant Prairie Cafe in Middleton Hills. This is the kind of unassuming place that you’d assume would make really first-rate breakfast and soups and maybe a heavily besprouted chicken-salad sandwich, but where you might hesitate to order a hot lunch. In fact, the corned beef hash, while homemade, was just so-so, while the reuben was really first-rate. The cold black-bean and corn salad that came alongside in lieu of coleslaw was even better, a crisp contrast to the thoroughly correct hot goopiness of the reuben.

Middleton Hills, it turns out, is a Duany Plater-Zyberk development in the “New Urbanist” style. Which means mixed retail and housing, walkability, density, stores fronting directly on sidewalks, cheap houses and expensive ones on the same block, and so on. Basically, if you take every feature of America’s soul-killing suburbs that people like to complain about, invert them, and build housing developments based on the result, you get something like New Urbanism.

As for me, I grew up in one of America’s soul-killing suburbs, and I like them! One of the nicest features of the Near West Side of Madison is that you can get on your bike and be in an authentically urban landscape in 15 minutes; or, after a 15-minute drive in the other direction, you can pull up in the oversized parking lot outside the even more oversized grocery store and load your station wagon until it groans.

Anyway, Middleton Hills. My first impression is that it’s charming; the houses all share a mild kind of Prairie style, but no two on the block look exactly alike. The main drag, Frank Lloyd Wright Boulevard, winds around a big and agreeably wild pond; lots of cattails, lots of birds, grass not too kempt. The street names do a good job of congratulating you for your participation in sustainable development — John Muir Drive, Aldo Leopold Way, and, best of all, Diversity Road.

My second impression is that it’s completely empty. You can see that the streets are laid out to encourage pedestrianism and unplanned human interaction, as in Princeton, a favorite town of Duany Plater-Zyberk’s, and mine. But at three in the afternoon, the only people I saw were a trickle of kids coming home from school, and a birdwatcher. The birdwatcher and I watched a sandhill crane for a few minutes. Then I sat down to continue revising a long-overdue paper with Michel and Venkatesh about sums of three squares. (Among other things, the paper features a careful explanation of the group structure — more properly, torsor structure — on the set of representations of a squarefree integer n as the sum of three squares. More on this when the paper’s finished.)

What makes Princeton’s streets lively and new-urban, of course, is that it has a big and interesting downtown, whose shops and restaurants serve not just Princetonians but residents of the surrounding towns. Middleton Hills has a grocery store, the Prairie Cafe, a pizza place, and a Starbucks — not enough to draw foot traffic away from Madison, or, for that matter, downtown Middleton. If this post pulls in a throng of reuben-lovers, I guess I’ll have done my bit for the New Urbanism.

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Non-simple abelian varieties in a family

Here’s a funny question. Let f in C[x] be a squarefree polynomial of degree at least 6. Let S be the set of complex numbers t such that the Jacobian of the hyperelliptic curve

y^2 = f(x)(x-t)

is not simple. Is S always finite? Even more, is there a bound on |S| which doesn’t depend on f, or depends only on the degree of f?

This question comes from the introduction to “Non-simple abelian varieties in a family: geometric and analytic approaches” , a new paper by me, Christian Elsholtz, Chris Hall, and Emmanuel Kowalski. In its original form this was a four-author, six-page paper — fortunately we’ve now added enough material to make the ratio a bit more respectable!

The paper isn’t about complex algebraic geometry at all — it explains how to get bounds on S when f has rational coefficients and t ranges over rational numbers, which is quite a different story. The point of the paper is partly to prove some theorems and partly to make a metamathematical point — that problems of this kind can be approached via either arithmetic geometry or analytic number theory, and that the two approaches have complementary strengths and weaknesses. Bounds from arithmetic geometry are stronger but less uniform; bounds from analytic number theory are weaker but have better uniformity.

Here’s my favorite example of this phenomenon. Let X be a smooth plane curve over Q of degree d at least 4. Then by Faltings’ Theorem we know that X has only finitely many rational points.

On the other hand, a beautiful theorem of Heath-Brown tells us that the number of rational points on X with coordinates of height at most B is at most C B^(2/d), for some constant C depending only on d. At first, this seems to give much less than Faltings. After all, as B gets larger and larger, the upper bound given by Heath-Brown gets arbitrarily large — whereas we know by Faltings that there are only finitely many points on the whole curve, no matter how large we allow the coordinates to be.

But note that the constant in Heath-Brown’s result doesn’t depend on the curve X. It is what’s called a uniform bound. Faltings’ theorem, by contrast, gives an upper bound on the number of points which depends very badly on the choice of X. Depending on what you’re trying to accomplish, you might be willing to sacrifice uniformity to get finiteness — or the reverse. But it’s best to have both options at hand.

Is it possible to have uniformity and finiteness simultaneously? Conjecturally, yes. Caporaso, Harris, and Mazur showed that, conditional on Lang’s conjecture, there is a constant B(g) such that every genus-g curve X/Q has at most B(g) rational points. The Caporaso-Harris-Mazur paper came out when I was in graduate school, and the idea of such a uniform bound was considered so wacky that CHM was thought of as evidence against Lang’s conjecture. Joe Harris used to wander around the department, buttonholing graduate students and encouraging us to cook up examples of genus-g curves with arbitrarily many points, thus disproving Lang. We all tried, and we all failed — as did many more experienced people. And nowadays, the idea that there might be a uniform bound for the number of rational points on a genus-g curve is considered fairly reputable, even among people who have their doubts about Lang’s conjecture. As far as I know, the world record for the number of rational points on a genus-2 curve is 588, due to Kulesz. Can you beat it?

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