Monthly Archives: November 2011

Jay Michaelson on God vs. Gay, Dec 1

My friend Jay Michaelson, my go-to guy for all matters of Jewish learning, is speaking in Madison this Thursday evening about his new book God vs. Gay?:  The Religious Case for Equality. Recommended for all who care what feist left-wing observant Jews have to say about religion and sex.  Which is everyone, right?

Book trailer:

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Why do people think government workers are stupid?

From Michael Lewis’s The Big Short:

“You know how when you walk into a post office you realize there is such a difference between a government employee and other people,” said Vinny. “The ratings agency people were all like government employees.” Collectively they had more power than anyone in the bond markets, but individually they were nobodies. “They’re underpaid,” said Eisman. “The smartest ones leave for Wall Street firms so they can help manipulate the companies they used to work for.

Where does it come from, this idea that people whose employer is the city, state, or nation are made of inferior stuff?  What is the “difference” Vinny perceives between the person helping him at the post office and the teller at his bank?  Does he really get worse service at the DMV than he gets from United Airlines?    Does he not have cable?

 

The Arrow-Debreu model wishes you a happy Thanksgiving

I keep going to talks that raise the question:  what is an equilibrium, in the sense of economics?  Not “what is the mathematical definition,” but “what is it, really?”  (The Big Short is relevant here too.)   I don’t have any thoughts of my own articulate enough for the blog, but in the spirit of the holiday I should certainly link to Cosma Shalizi’s explanation of why conceptual art is the most economically efficient use of a dead turkey.  Gobble gobble.

 

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Math Girls

The holiday season approaches and surely you are looking for a new translation of a bestselling young adult novel from Japan which is half adolescent love story and half elementary number theory text.  You’re in luck.  Bento Books sent me a review copy of the book, Math Girls by Hiroshi Yuki (tr. Tony Gonzalez) and by the second chapter the narrative has already addressed not only the fact that 1 is not prime, but the fact that it’s entirely in our hands whether to define 1 to be a prime or not, and why we made the choice we did.  How romantic!

Sample chapters here.

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Miscellaneous linkdump Nov 11

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Are math departments better at recruitment than elite financial firms?

Via Bryan Caplan, Lauren Rivera at Northwestern studied hiring practices at top financial, law, and consulting firms and found some surprises:

[E]valuators drew strong distinctions between top four universities, schools that I term the super-elite, and other types of selective colleges and universities. So-called “public Ivies” such as University of Michigan and Berkeley were not considered elite or even prestigious… In addition to being an indicator of potential intellectual deficits, the decision to go to a lesser known school (because it was typically perceived by evaluators as a “choice”) was often perceived to be evidence of moral failings, such as faulty judgment or a lack of foresight on the part of a student.

I’m not sure what those four schools are, but they exclude some pretty good undergraduates:

You will find it when you go to like career fairs or something and you know someone will show up and say, you know, “Hey, I didn’t go to HBS [Harvard Business School] but, you know, I am an engineer at M.I.T. and I heard about this fair and I wanted to come meet you in New York.” God bless him for the effort but, you know, it’s just not going to work.

And don’t neglect those extracurriculars:

[E]valuators believed that the most attractive and enjoyable coworkers and candidates would be those who had strong extracurricular “passions.” They also believed that involvement in activities outside of the classroom was evidence of superior social skill; they assumed a lack of involvement was a signal of social deficiencies… By contrast, those without significant extracurricular experiences or those who participated in activities that were primarily academically or pre-professionally oriented were perceived to be “boring,” “tools,” “bookworms,” or “nerds” who might turn out to be “corporate drones” if hired.

All this stuff sounds bizarre to people outside the world of corporate recruitment.  And it is natural for academics like me to read this and silently congratulate myself on our superior methods of judgment.  But surely there are things about our process which would seem just as irrational and counterproductive to people outside of academic mathematics.  What are they?

It might make more sense to concentrate on graduate recruitment as against tenure-track hiring, since then both we and the financiers are talking about recent BAs with little track record in the workplace.

(Linguistic note:  “Counterproductive” is surely a word that people would deride as horrible managementese if it weren’t already in common use.  But it’s a great word!)

(Upcoming blog note: At some point soon I’ll blog about Michael Lewis’s The Big Short, which I just finished, and which is the reason the credentials of financial professionals are on my mind.)

 

 

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Scott Walker: not toast

Much was made of the WPR/St. Norbert poll released last week, in which 58% of respondents said they’d vote for Scott Walker’s opponent if a recall comes to pass, with only 38% saying they’d vote to keep the Governor in office.  Worth noting the numbers below the top line, though:  in the sample of 482 voters, 34% reported voting for JoAnne Kloppenburg in April’s Supreme Court election, against 27% who said they voted for Prosser.  In fact, those votes were evenly split.  So it’s way, way, way too soon to say that Walker’s behind in a potential recall election, especially with Wisconsin D’s still in search of a candidate.

(Another interesting result from that poll:  people in Wisconsin apparently really like electing their Supreme Court, and in fact would prefer that the prospective justice’s party affiliation be listed on the ballot!)

 

 

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Gonality, the Bogomolov property, and Habegger’s theorem on Q(E^tors)

I promised to say a little more about why I think the result of Habegger’s recent paper, ” Small Height and Infinite Non-Abelian Extensions,” is so cool.

First of all:  we say an algebraic extension K of Q has the Bogomolov property if there is no infinite sequence of non-torsion elements x in K^* whose absolute logarithmic height tends to 0.  Equivalently, 0 is isolated in the set of absolute heights in K^*.  Finite extensions of Q evidently have the Bogomolov property (henceforth:  (B)) but for infinite extensions the question is much subtler.  Certainly \bar{\mathbf{Q}} itself doesn’t have (B):  consider the sequence 2^{1/2}, 2^{1/3}, 2^{1/4}, \ldots  On the other hand, the maximal abelian extension of Q is known to have (B) (Amoroso-Dvornicich) , as is any extension which is totally split at some fixed place p (Schinzel for the real prime, Bombieri-Zannier for the other primes.)

Habegger has proved that, when E is an elliptic curve over Q, the field Q(E^tors) obtained by adjoining all torsion points of E has the Bogomolov property.

What does this have to do with gonality, and with my paper with Chris Hall and Emmanuel Kowalski from last year?

Suppose we ask about the Bogomolov property for extensions of a more general field F?  Well, F had better admit a notion of absolute Weil height.  This is certainly OK when F is a global field, like the function field of a curve over a finite field k; but in fact it’s fine for the function field of a complex curve as well.  So let’s take that view; in fact, for simplicity, let’s take F to be C(t).

What does it mean for an algebraic extension F’ of F to have the Bogomolov property?  It means that there is a constant c such that, for every finite subextension L of F and every non-constant function x in L^*, the absolute logarithmic height of x is at least c.

Now L is the function field of some complex algebraic curve C, a finite cover of P^1.  And a non-constant function x in L^* can be thought of as a nonzero principal divisor.  The logarithmic height, in this context, is just the number of zeroes of x — or, if you like, the number of poles of x — or, if you like, the degree of x, thought of as a morphism from C to the projective line.  (Not necessarily the projective line of which C is a cover — a new projective line!)  In the number field context, it was pretty easy to see that the log height of non-torsion elements of L^* was bounded away from 0.  That’s true here, too, even more easily — a non-constant map from C to P^1 has degree at least 1!

There’s one convenient difference between the geometric case and the number field case.  The lowest log height of a non-torsion element of L^* — that is, the least degree of a non-constant map from C to P^1 — already has a name.  It’s called the gonality of C.  For the Bogomolov property, the relevant number isn’t the log height, but the absolute log height, which is to say the gonality divided by [L:F].

So the Bogomolov property for F’ — what we might call the geometric Bogomolov property — says the following.  We think of F’ as a family of finite covers C / P^1.  Then

(GB)  There is a constant c such that the gonality of C is at least c deg(C/P^1), for every cover C in the family.

What kinds of families of covers are geometrically Bogomolov?  As in the number field case, you can certainly find some families that fail the test — for instance, gonality is bounded above in terms of genus, so any family of curves C with growing degree over P^1 but bounded genus will do the trick.

On the other hand, the family of modular curves over X(1) is geometrically Bogomolov; this was proved (independently) by Abramovich and Zograf.  This is a gigantic and elegant generalization of Ogg’s old theorem that only finitely many modular curves are hyperelliptic (i.e. only finitely many have gonality 2.)

At this point we have actually more or less proved the geometric version of Habegger’s theorem!  Here’s the idea.  Take F = C(t) and let E/F be an elliptic curve; then to prove that F(E(torsion)) has (GB), we need to give a lower bound for the curve C_N obtained by adjoining an N-torsion point to F.  (I am slightly punting on the issue of being careful about other fields contained in F(E(torsion)), but I don’t think this matters.)  But C_N admits a dominant map to X_1(N); gonality goes down in dominant maps, so the Abramovich-Zograf bound on the gonality of X_1(N) provides a lower bound for the gonality of C_N, and it turns out that this gives exactly the bound required.

What Chris, Emmanuel and I proved is that (GB) is true in much greater generality — in fact (using recent results of Golsefidy and Varju that slightly postdate our paper) it holds for any extension of C(t) whose Galois group is a perfect Lie group with Z_p or Zhat coefficients and which is ramified at finitely many places; not just the extension obtained by adjoining torsion of an elliptic curve, for instance, but the one you get from the torsion of an abelian variety of arbitrary dimension, or for that matter any other motive with sufficiently interesting Mumford-Tate group.

Question:   Is Habegger’s theorem true in this generality?  For instance, if A/Q is an abelian variety, does Q(A(tors)) have the Bogomolov property?

Question:  Is there any invariant of a number field which plays the role in the arithmetic setting that “spectral gap of the Laplacian” plays for a complex algebraic curve?

A word about Habegger’s proof.  We know that number fields are a lot more like F_q(t) than they are like C(t).  And the analogue of the Abramovich-Zograf bound for modular curves over F_q is known as well, by a theorem of Poonen.  The argument is not at all like that of Abramovich and Zograf, which rests on analysis in the end.  Rather, Poonen observes that modular curves in characteristic p have lots of supersingular points, because the square of Frobenius acts as a scalar on the l-torsion in the supersingular case.  But having a lot of points gives you a lower bound on gonality!  A curve with a degree d map to P^1 has at most d(q+1) points, just because the preimage of each of the q+1 points of P^1(q) has size at most d.  (You just never get too old or too sophisticated to whip out the Pigeonhole Principle at an opportune moment….)

Now I haven’t studied Habegger’s argument in detail yet, but look what you find right in the introduction:

The non-Archimedean estimate is done at places above an auxiliary prime number p where E has good supersingular reduction and where some other technical conditions are met…. In this case we will obtain an explicit height lower bound swiftly using the product formula, cf. Lemma 5.1. The crucial point is that supersingularity forces the square of the Frobenius to act as a scalar on the reduction of E modulo p.

Yup!  There’s no mention of Poonen in the paper, so I think Habegger came to this idea independently.  Very satisfying!  The hard case — for Habegger as for Poonen — has to do with the fields obtained by adjoining p-torsion, where p is the characteristic of the supersingular elliptic curve driving the argument.  It would be very interesting to hear from Poonen and/or Habegger whether the arguments are similar in that case too!

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Help me be a great Nim teacher

I’ll be at Marvelous Math Morning at CJ’s school this Saturday, playing Nim with kids ranging from K-5.  One simple goal is to teach them the winning strategy for the version of the game where there’s one pile and each player can draw 1 or 2 chips.  I’ve done that with CJ and he really liked it — and I think the idea of a perfect strategy is one of those truly deep mathematical concepts that even little kids can grasp.

But what else should I do?  What other Nims and Nimlikes should I teach these kids and what lessons should I try to impart thereby?

Update:  First two commenters both mentioned Tic-Tac-Toe.  At what age do kids typically learn how to play Tic-Tac-Toe and at what age have they learned a perfect strategy?  CJ is in kindergarten and has not seen this, or at least he hasn’t seen it from me.  I’ll ask him tonight.

Update:  Nim a success!  I played mostly one-pile, and the kids were definitely able to grasp pretty quickly the idea of winning and losing positions, and the goal of chasing the former and avoiding the latter.  I didn’t encounter anyone who’d played nim before.  I felt some math was transmitted.  Mission accomplished.

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R.E.M. and Cal Ripken

Dave Daley delivers a great, frank interview with Michael Stipe on R.E.M.’s breakup.

I would rather throw myself off a cliff or be boiled in lead than listen to “Life’s Rich Pageant” demos – [and here Stipe groan sings unintelligible syllables as if he is in pain] — my doing this horrible moaning over a song that then became a beautiful song. Peter and Mike love that stuff.

Somehow I have managed not to write anything in this space about the end of my very favorite rock band.  The post I was going to write was about R.E.M. and Cal Ripken — both of whom were, from start to finish, so recognizably themselves as to be a pleasure to watch, even with half their power gone.  Both of them announced an new way to play their position, and neither would ever be mistaken for the imitators they made possible.  (I like to think that Derek Jeter is Live in this scenario, but really only because I hate both so very much.)  I guess the way that Stipe doesn’t sound like a great singer, but is one, matches the way that Ripken didn’t look like a great fielder, but was one.  Where were the dives, where were the behind-the-back flips?  No need — Ripken was just always standing where the ball was going to be.  And Stipe was always singing what the song wanted him to sing, whether or not you could make well-defined words, or even well-defined notes, out of it.

Murmur and Ripken’s first MVP season were surely the two best things about 1983  — though Pete Thorn, decades ago, was saying that Ripken’s astounding defense in 1984 made that an even better season — a suggestion that was ignored at the time, but WAR agrees.  Maybe Pete Thorn is the analogue of the diehards who think Fables is better than Murmur.  (Except Thorn was probably right.)   People who love Automatic for the People would no doubt call that record the analogue of Ripken’s out-of-nowhere MVP year in 1991 — not me, though.  And R.E.M. never really did anything that matches the lonely beauty of a half-season that Ripken turned in in 1999, at the age of 38, for a team going nowhere — in fact, a team that, once Ripken left, would spend most of a decade going nowhere, and the rest of the time actually being  nowhere.

Most days I think “Pilgrimage” is the greatest song they ever made:

though at the time I might have preferred “It’s The End of the World As We Know It (and I feel fine),” whose lyrics I resolutely memorized, and on behalf of which I launched a doomed campaign for my high school’s 1989 prom theme.

It would have been a really great prom theme.

 

 

 

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