Monthly Archives: June 2011

How bad is Nick Markakis? b/w Mallards misc.

  • How bad is Nick Markakis right now?  Seems a funny question considering he’s riding an 18-game hitting streak and has brought his average up to .282.  In his last 14 games he’s hit .423.  But drill down deeper and you’ll see Markakis’s problems at the plate haven’t gone away.  In those 14 games, covering 53 plate appearances, Markakis has walked once.  Home runs?  Also just one — his power shows no sign of returning.  So what’s happened in the last two weeks is that Markakis has taken his terrible season and added a lot of singles.  And how did he start hitting so many singles?  Unfortunately, it looks like it’s just luck — Markakis’ BABIP in the last 14 games is an insane .447.  Those bloop singles won’t keep falling in all year, and you can expect Markakis’s BA to drop back towards its former dispiriting level over the next few weeks.
  • Bernoulli variables like to cancel:  with all the strange business going on this season (bizarrely terrible Markakis, bizarrely good J.J. Hardy, the regression of Brian Matusz, etc. etc.) the Orioles are en route to winning 75 games or so, pretty much in line with pre-season projections. 
  • CJ and I went to the Mallards afternoon game on Father’s Day.  For the first time CJ was actually involved with the game; he spent innings 4-8 getting food and playing in the bounce house, to be sure, but after we went back to our seats he demanded to stay to the end of the game, a thriller that the Mallards won 5-4 in the bottom of the 10th after Andrew Barna walked, stole second, advanced to third on a passed ball, and scored on a wild pitch.  Barna, who plays for Davidson College during the school year, is blogging the Mallards’ season game by game:  highly recommended for anyone interested in a candid view of amateur ball, or what summer in Madison looks and feels like to a college kid away from home.
  • The bad news is that the “pigsicle” — a thick slab of bacon dipped in maple syrup and served on stick — is no longer served at the Maynard’s Slide-In stand at the Duck Pond. The good news is that it’s been replaced with the “chicken-fried pigsicle,” the same slab, battered and fried and served — on a stick, natch — with a cup of white gravy.
  • The old Mallards logo has been also been replaced, in favor of this peevish dude.  I know, everybody wants a fierce mascot.  But frankly, the maximal level of fierceness a mallard can attain is well below “I am a dangerous predator.”   I think it hovers somewhere around “You took my parking space.”
  • It was very cute watching CJ dutifully and somewhat accurately stomp-stomp-clap along to “We Will Rock You.”  But I think the Mallards are missing an opportunity by not encouraging fans to sing “We Will Duck You.”
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Poonen-Rains and lines on a quadric surface

One feature of the Poonen-Rains heuristics that might seem strange at first is that the dimension of the Selmer group isn’t 0 almost all  the time.  This is by contrast with the Cohen-Lenstra heuristic, where the p-torsion in the class group is indeed trivial about 1-1/p of the time.  Instead, the Poonen-Rains heuristics predict that the p-Selmer rank is 0 about half the time and 1 about half the time, with about 1/p’s worth of measure devoted to ranks 2 or higher.  Of course, given that we expect a random elliptic curve to have Mordell-Weil rank 1 half the time, it would be bad news for their heuristic if it predicted a lower frequency of positive Selmer rank!

But why is the intersection of two maximal isotropic subspaces 1 half the time and 0 half the time?  You can get a nice picture of what’s going on by thinking about the case of  a quadratic form Q in 4 variables.  The vanishing of the quadratic form cuts out a quadric surface in P^3.  A maximal isotropic subspace is a 2-dimensional space on which Q vanishes — in other words, a line on the quadric.  The intersection of two maximal isotropics is o-dimensional if the corresponding lines are disjoint, 1-dimensional if the lines intersect at a point, and 2-dimensional when the lines coincide.  So what’s the probability that two random lines on the quadric intersect?  The key point is that there are two families of lines.  If L1 and L2 come from different families, they intersect; if they come from the same family, they’re disjoint (except in the unlikely event they coincide.)  So there you go — the intersection of the maximal isotropics is split 50-50 between 0-dimensional and 1-dimensional.  More generally, the variety of maximal isotropic subspaces in an even-dimensional orthogonal space has two components, and this explains the leading term of Poonen-Rains.

It would be interesting to understand how to describe the “two types of maximal isotropics” in the infinite-dimensional F_p-vector space considered by Poonen-Rains, and to understand why the two maximal isotropics supplied by a given elliptic curve lie in the same family if and only if the L-function of E has even functional equation, which should lead one to expect that Sel_p(E) has even rank  (or even, thanks to recent progress on the parity conjecture by Nekovar, Kim, los Dokchitsers, etc., implies that Sel_p(E) has even rank, subject to finiteness of Sha.)





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Thoughts on Dan Sharfstein’s “The Invisible Line”

I blogged about Dan’s book before I read it and said “it’s surely terrific.”  Now I’ve read it, and it is!

The book follows three families, each occupying its own complicated position on the boundary between black and white, from Revolutionary times to the early 20th century.  Sharfstein pieces together a miraculously detailed picture of his subjects from newspaper accounts, archives, and especially legal records.  It’s a great work of American history, but it’s also pretty straight-up exciting —  daring slave rescues, courtroom dramas, Kentucky blood feuds, and steamship explosions make good seasoning for Dan’s contemplations of the American racial conundrum.

Disorganized thoughts:

  • Dan has been researching and writing this book for 15 years or so.  While reading it I kept thinking “this is why we have books and not just blogs; this is why we have historians and not just editorialists.”
  • Every American should, once a year, read a book about US history between the 1865 and 1910.  In high school, you get the Civil War, then Lincoln is assassinated, and the next thing you know there are biplanes flying around shooting at each other.  And maybe they’ll say “oh and by the way Theodore Roosevelt opened a bunch of national parks.”  I certainly made it through school (in what’s officially the South!) without ever hearing about Reconstruction.  The complicated, multifront forward-and-backwards struggle toward racial equality gets flattened into something like this:   the slaves get freed — then a hundred years later their descendants suddenly realize they should be allowed to eat at Woolworth’s.
  • A running theme of Dan’s book is the extreme attention paid to and importance placed on “racial purity”.  It really mattered to people (though not to all people, and not in the same way to all the people to whom it mattered) whether you had 16 white great-great-grandparents or only 15.  “Racial purity” is one of those words, like “honor,” that now seems to us a strange abstraction, not referring to anything in the actual world, but was experienced by our ancestors as a real thing.   I wonder whether “privacy” will go down the same path.  If so, I hope the future contains book-writing historians like Dan to explain to my great-great-grandchildren what I meant by it.
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To say nothing of Gary Carter The Unstoppable Sex Machine

Great baseball names sound just like bad band names!  Quiz:  which of these are baseball players and which are musical acts?

  • Jumpy Garcia
  • Death to Flying Things
  • Luscious Jackson
  • Todd Van Poppel
  • Three-Finger Brown
  • T-Bone Shelby
  • Vida Blue

Es apulkat (avocado coffee shake) at Calasan Diner

  Calasan Diner is an Indonesian fast-food place that opened last month at Old Sauk and High Point, just inside the Beltline (same shopping center as Alicia Ashman library.)  Es alpukat is an Indonesian sweet drink made with avocado, instant coffee, and condensed milk.  You can get the latter at the former and I was glad I did.  At lunch they just serve fried chicken, which comes sweet or salty — I preferred the salty.  I didn’t know Indonesian fried chicken was a thing, but it is!  At dinner there’s a full Indonesian menu.  A welcome addition to this relatively restaurant-free neighborhood.

(Ultra-brief rundown of other restaurants around there:  in the same shopping center is the pretty good Swagat, notable for often having some Indian-Chinese dishes on the buffet.   Oliva is a Turkish pizzeria which doesn’t serve iskender kebab so I don’t go there.  But they recently expanded into a bigger space (formerly yet another depressing Atlanta Bread Company) so I’m guessing their pizza’s good.  Down Gammon most of the way to Mineral Point is the likeable Cilantro, “interior Mexican” at a high price point and a lethally untrafficked location.)

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Poonen-Rains, Selmer groups, random maximal isotropics, random orthogonal matrices

At the AIM workshop on Cohen-Lenstra heuristics last week I got to hear Bjorn Poonen give a terrific talk about his recent work with Eric Rains about the distribution of mod p Selmer groups in a quadratic twist family of elliptic curves.

Executive summary:  if E is an elliptic curve, say in Weierstrass form y^2 = f(x), and d is a squarefree integer, then we can study the mod p Selmer group Sel_d(E) of the quadratic twist dy^2 = f(x), which sits inside the Galois cohomology H^1(G_Q, E_d[p]).  This is a finite-dimensional vector space over F_p.  And by analogy with the Cohen-Lenstra heuristics for class groups, we can ask whether these groups obey a probability distribution as d varies — that is, does

Pr(dim Sel_d(E) = r | d in [-B, … B])

approach a limit P_r as B goes to infinity, and if so, what is it?

The Poonen-Rains heuristic is based on the following charming observation.  The product of the local cohomology groups H_1(G_v, E[p]) is an infinite-dimensional F_p-vector space endowed with a bilinear form coming from cup product.  In here you have two subspaces:  the image of global cohomology, and the image of local Mordell-Weil.  Each one of these, it turns out, is maximal isotropic — and their intersection is exactly the Selmer group.  So the Selmer group can be seen as the intersection of two maximal isotropic subspaces in a very large quadratic space.

Heuristically, one might think of these two subspaces as being randomly selected among maximal isotropic subspaces.  This suggests a question:  if P_{r,N} is the probability that the intersection of two random maximal isotropics in F_p^{2N} has dimension r, does P_{r,N} approach a limit as N goes to infinity?  It does — and the Poonen-Rains heuristic then asks that the probability that dim Sel_d(E)  = r approaches the same limit.  This conjecture agrees with theorems of Heath-Brown, Swinnerton-Dyer, and Kane in the case p=2, and with results of Bhargava and Shankar when p <= 5 (Bhargava and Shankar work with a family of elliptic curves of bounded height, not a quadratic twist family, but it is not crazy to expect the behavior of Selmer to be the same.)  And in combination with Delaunay’s heuristics for variation of Sha, it recovers Goldfeld’s conjecture that elliptic curves are half rank 0 and half rank 1.

Johan de Jong wrote about a similar question, concentrating on the function field case, in his paper “Counting elliptic surfaces over finite fields.”  (This is the first place I know where the conjecture “Sel_p should have size 1+p on average” is formulated.)  He, too, models the Selmer group by a “random linear algebra” construction.  Let g be a random orthogonal matrix over F_p; then de Jong’s model for the Selmer group is coker(g-1).  This is a natural guess in the function field case:  if E is an elliptic curve over a curve C / F_q, then the Selmer group of E is a subquotient of the etale H^2 of an elliptic surface S over F_q; thus it is closely related to the coinvariants of Frobenius acting on the H^2 of S/F_qbar.  This H^2 carries a symmetric intersection pairing, so Frobenius (after scaling by q) is an orthogonal matrix, which we want to think of as “random.”  (As first observed by Friedman and Washington, the Cohen-Lenstra heurstics can be obtained in similar fashion, but the relevant cohomology is H^1 of a curve instead of H^2 of a surface; so the relevant pairing is alternating and the relevant statistics are those of symplectic rather than orthogonal matrices.)

But this presents a question:  why do these apparently different linear algebra constructions yield the same prediction for the distribution of Selmer ranks?

Here’s one answer, though I suspect there’s a slicker one.

A nice way to describe the distributions that arise in problems of Cohen-Lenstra type is by computing their moments.  But the usual moments (e.g. “expected kth power of dimension of Selmer” or “kth power of order of Selmer” tend not to behave so well.  Better is to compute “expected number of injections from F_p^k into Selmer,” which has a cleaner answer in every case I know.  If the size of the Selmer group is X, this number is


Evidently, if you know these “moments” for all k, you can compute the usual moments E(X^k) (which are indeed computed explicitly in Poonen-Rains) and vice versa.

Now:  let A be the random variable (valued in abelian groups!)  “intersection of two random maximal isotropics in a 2N-dimensional quadratic space V” and B be “coker(g-1) where g is a random orthogonal N x N matrix.”

The expected number of injections from F_p^k to B is just the number of injections from F_p^k to F_p^N which are fixed by g.  By Burnside’s lemma, this is the number of orbits of the orthogonal group on Inj(F_p^k, F_p^N).  But by Witt’s Theorem, the orbit of an injection f: F_p^k -> F_p^N is precisely determined by the restriction of the orthogonal form to F_p^k; the number of symmetric bilinear forms on F_p^k is p^((1/2)k(k+1)) and so this is the expected value to be computed.

What about the expected number of injections from F_p^k to A?  We can compute this as follows.  There are about p^{Nk} injections from F_p^k to V.  Of these, about p^{2Nk – (1/2)k(k+1)} have isotropic image.  Call the image W;  we need to know how often W lies in the intersection of the two maximal isotropics V_1 and V_2.  The probability that W lies in V_1 is easily seen to be about p^{-Nk + (1/2)k(k+1)}, and the probability that W lies in V_2 is the same; these two events are independent, so the probability that W lies in A is about p^{-2NK + (1/2)k(k+1)}.  Summing over all isotropic injections gives an expected number of p^{(1/2)k(k+1)} injections from F_p^k to A.  Same answer!

(Note:  in the above paragraph, “about” always means “this is the limit as N gets large with k fixed.”)

What’s the advantage of having two different “random matrix” formulations of the heuristic?  The value of the “maximal isotropic intersection” model is clear — as Poonen and Rains show, the Selmer really is an intersection of maximal isotropic subspaces in a quadratic space.  One value of the “orthogonal cokernel” model is that it’s clear what it says about the Selmer group mod p^k.

Question: What does the orthogonal cokernel model predict about the mod-4 Selmer group of a random elliptic curve?  Does this agree with the theorem of Bhargava and Shankar, which gives the first moment of Sel_4 in a family of elliptic curves ordered by height?

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Statistics, Politics, and Policy

The Berkeley Electronic Press launches a new journal:

The increasing amount and complexity of available data is constantly creating new challenges for statistical thinking in policy problems. While many academic statisticians tend to share among themselves their latest methods and models, less attention has been paid to the usefulness of those statistical methods and models to inform public policy decisions, and what statistical approaches might be most effective in designing how policies are implemented. In the policy sphere, statistical methods are sometimes taken as a given, with less attention to all the variations, assumptions, and effects of different methods in differing contexts. But it is in the policy sphere that statistical debates can have the great value and impact, and the intersection of statistics and public policy is a fertile ground for statistical research and analysis to address important policy issues that may have widespread ramifications.

As an electronic journal, Statistics, Politics, and Policy will use a mix of voices and approaches to reach a broad audience. The journal aims to open avenues of communication between statisticians and policy makers on questions that pique the interest of the public. The journal will appeal to statisticians, policy analysts, and anyone interested in the implicit yet powerful ways that statistical thinking influences decisions that affect many aspects of public life.

The debut issue features an article by Indiana mathematician Russell Lyons attacking the statistical basis of widely-publicized research results on “social contagion” effects on obesity, addiction, and other social ills.

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Aggregating degrees of belief: a puzzle

There are two events X and Y whose probability you’d like to estimate.  So you ask a hundred trusted, reasonable people what they think.  Half of them say that the probability of X and the probability of Y are both 90%, and the probability of both X and Y occurring is 81%.  The other half say that P(X) = 10%, P(Y) = 10%, and P(X and Y) = 1%.

What is your best estimate of P(X), P(Y), and P(X and Y)?

If you said “50%, 50%, 41%,” does it bother you that you deem these events not to be independent, even though every single person you polled said the opposite?  If not, what did you say?

(The subtext of this post is:  is the “Independence of Irrelevant Alternatives” axiom in Arrow’s theorem a good idea?  Feel free to discuss that too.)


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“Three things”: Ravi Vakil on how to listen to a math talk

Good seminar hack from Ravi:

The theory is as follows. If you can get even three small things out of a talk, it is a successful talk. And if you can’t get even three small things out of a talk, it was not a successful experience. Note that the things you get out of a talk needn’t be the things that your neighbor got out of a talk, or the things the speaker expected you to get out of the talk.

Here is how it works. Take a clean sheet of paper, or an index card. Your goal is to have three things, and only three things, on this sheet at the end of the talk.

My own practice is that at the end of every seminar talk I should have a question to ask the speaker.  Not that I always, or even usually ask!  But I find “What more do I want to know?” to be a good mantra for maintaining engagement with the lecture.

Share your own best practices in comments!

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Does New Mexico’s governor have a Wisconsin veto?

She thinks so, and to make the point has vetoed the first digit of a $150,000 appropriation, cutting the funding for the program to a third of the amount approved by the legislature.  Steve B., knowing me to be  a partial veto aficionado, wrote me to ask whether Gov. Martinez can actually do this.

I don’t think so.  New Mexico governors have a partial veto, but as far as I can tell there’s no Wisconsinist tradition of vetoing individual words, much less digits.  (For that matter, the digit-vetoing power has been off-limits even in Wisconsin for many years now.)  The New Mexico Supreme Court weighed in on the scope of NM-GOV’s partial veto power in Sego vs. Kirkpatrick (1974):

“The Governor may not distort, frustrate or defeat the legislative purpose by a veto of proper legislative conditions, restrictions, limitations or contin- gencies placed upon an appropriation and permit the appropriation to stand. He would thereby create new law, and this power is vested in the Legislature and not in the Governor.”

Per Louis Fisher and Neal Devins:

The New Mexico court’s ruling against the use of the partial veto to alter legislative policy sharply conflicts with that of the Wisconsin Supreme Court, which recognized the governor’s authority to “change the policy of the law” through a similar partial veto provision.

So I predict redigitation of the bill in question.

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