Silas Johnson on weighted discriminants with mass formulas

My Ph.D. student Silas Johnson just posted his thesis paper to the arXiv, and it’s cool, and I’m going to blog about it!

How should you count number fields?  The most natural way is by discriminant; you count all degree-n number fields K with a given Galois group G in S_n and discriminant bounded in absolute value by B.  This gives you a value N_G(B) whose asymptotic behavior in B you might want to study.  The classical results and exciting new ones you’ve heard about — Davenport-Heilbron, Bhargava, and all that — generally concern counts of this kind.

But there are reasons to consider other kinds of counts.  I once had a bunch of undergrads investigate the behavior of N_3(X,Y), the number of cubic fields whose discriminant had squarefree part at most X and maximal square divisor at most Y.  This gives a more refined picture of the ramification behavior of the fields.  Asymptotics for this are still unknown!  (I would expect the main term to be on order $X Y^{1/2}$, but I don’t know what the secondary terms should be.)

One nice thing about the discriminant, though, is that it has a mass formula.  In brief:  a map f from Gal(Q_p) to S_n corresponds to a degree-n extension of Q_p, which has a discriminant (a power of p) which we call Disc(f).  Averaging Disc(f)^{-1} over all homomorphisms f gives you a polynomial in p^{-1}, which we call the local mass.  Now here’s the remarkable fact (shown by Bhargava, deriving from a formula of Serre) — there is a universal polynomial P(x) such that the local mass at p is equal to P(p^{-1}) for every P.  This is not hard to show for the tame primes p (you can see this point discussed in Silas’s paper or in the paper by Kedlaya I linked above) but that it holds for the wild primes is rather mysterious and strange.  And it certainly seems to ratify the idea that there’s something especially nice about the discriminant.  What’s more, this polynomial P is not just some random thing; it’s the product over p of P(p^{-1}) that gives the constant in Bhargava’s conjectural asymptotic for the number of number fields for degree n.

But here’s the thing.  If we replace G by a subgroup of S_n, there need not be a universal mass formula anymore.  Kedlaya (and Daniel Gulotta, in the appendix) show lots of examples.  The simplest example is the dihedral group of order 8.

All is not lost, though!  Wood showed in 2008 that you could fix this problem by replacing the discriminant of a D_4-extension with a different invariant.  Namely:  a D_4 quartic field M has a quadratic subextension L.  If you replace Disc(L/Q) with Disc(L/Q) times the norm to Q of Disc(L/M), you get a different invariant of M — an example of what Silas calls a “weighted discriminant” — and when you compute the local mass according to {\em this} invariant, you get a polynomial in p^{-1} again!

So maybe Wood’s modified discriminant, not the usual discriminant, is the “right” way to count dihedral quartics?  Does the product of her local masses give the right asymptotic for the number of D_4 extensions with Woodscriminant at most B?

It’s not at all clear to me how, if at all, you can cook up a modified discriminant for an arbitrary group G that has a universal mass formula.  What Silas shows is that having a mass formula is indeed special; when G is a p-group, there are only finitely many weighted discriminants that have one.  Silas thinks, and so do I, that this is actually true for every finite group G, and that some version of his approach will eventually prove this.  And he classifies all such weighted discriminants for groups of size up to 12; for any individual G, it’s a computation which can be made nicely algorithmic.  Very cool!

Is the two-Burke ballot the new butterfly ballot?

BURKE:  One other area outside of that that people really should take a look at is the Wisconsin Economic Development Corporation, which was a nonprofit, public-private corporation created in 2011 which Governor Walker used to make himself the chair of. What’s most interesting is that Governor Walker’s experience in private business is in selling warranties for IBM and doing blood drives and fund-raising for the American Red Cross. While these are both worthy positions and individuals who do them obviously are working to build a life, that doesn’t give someone the experience necessary to make themselves a chair of a venture capital firm. Because that’s what it is. They’re giving away private taxpayer dollars to public businesses. We would end that practice.

Except that’s not Mary Burke; it’s Robert Burke, a lifelong Republican from Hudson who switched to the Libertarian party to run for governor.  Burke talks in the interview about how he hopes the “name recognition” — misrecognition? — he draws from the Mary Burke campaign will help him get votes.  The question is:  will he get votes from people who like libertarianism, or miscast votes that are actually meant for her?

Are you wondering whether Burke the Libertarian is running precisely in order to siphon votes from Burke the Democrat in this way?  I was, too, but I have to admit that the linked interview really does make him sound like a sincere libertarian dude who just found out Republicans dig market distortions as much as Democrats do.

Francis Galton could be kind of a jerk

As here (from Hereditary Genius, p. 21)

Every tutor knows how difficult it is to drive abstract conceptions, even of the simplest kind, into the brains of most people—how feeble and hesitating is their mental grasp—how easily their brains are mazed—how incapable they are of precision and soundness of knowledge. It often occurs to persons familiar with some scientific subject to hear men and women of mediocre gifts relate to one another what they have picked up about it from some lecture—say at the Royal Institution, where they have sat for an hour listening with delighted attention to an admirably lucid account, illustrated by experiments of the most perfect and beautiful character, in all of which they expressed themselves intensely gratified and highly instructed. It is positively painful to hear what they say. Their recollections seem to be a mere chaos of mist and misapprehension, to which some sort of shape and organization has been given by the action of their own pure fancy, altogether alien to what the lecturer intended to convey. The average mental grasp even of what is called a well-educated audience, will be found to be ludicrously small when rigorously tested.

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Poem for the ALDS

These are the names that are freaking me out,
Verlander, Scherzer, and Price,
Plaguing my Oriole fandom with doubt,
Verlander, Scherzer and Price.
A trio of felines, bringing the heat,
Verlander, Scherzer, and Price,
Are these guys that a team writing “Ryan Flaherty” and “Jonathan Schoop” on the lineup card every day actually has a chance to beat??
Verlander, Scherzer, and Price.

Update:  I should make clear that this is meant to be apres “Tinkers to Evers to Chance,” by Franklin Pierce Adams.

Squares and Motzkins

Greg Smith gave an awesome colloquium here last week about his paper with Blekherman and Velasco on sums of squares.

Here’s how it goes.  You can ask:  if a homogeneous degree-d polynomial in n variables over R takes only non-negative values, is it necessarily a sum of squares?  Hilbert showed in 1888 that the answer is yes only when d=2 (the case of quadratic forms), n=2 (the case of binary forms) or (n,d) = (3,4) (the case of ternary quartics.)  Beyond that, there are polynomials that take non-negative values but are not sums of squares, like the Motzkin polynomial

$X^4 Y^2 + X^2 Y^4 - 3X^2 Y^2 Z^2 + Z^6$.

So Greg points out that you can formulate this question for an arbitrary real projective variety X/R.  We say a global section f of O(2) on X is nonnegative if it takes nonnegative values on X(R); this is well-defined because 2 is even, so dilating a vector x leaves the sign of f(x) alone.

So we can ask:  is every nonnegative f a sum of squares of global sections of O(1)?  And Blekherman, Smith, and Velasco find there’s an unexpectedly clean criterion:  the answer is yes if and only if X is a variety of minimal degree, i.e. its degree is one more than its codimension.  So e.g. X could be P^n, which is the (n+1,2) case of Hilbert.  Or it could be a rational normal scroll, which is the (2,d) case.  But there’s one other nice case:  P^2 in its Veronese embedding in P^5, where it’s degree 4 and codimension 3.  The sections of O(2) are then just the plane quartics, and you get back Hilbert’s third case.  But now it doesn’t look like a weird outlier; it’s an inevitable consequence of a theorem both simpler and more general.  Not every day you get to out-Hilbert Hilbert.

Idle question follows:

One easy way to get nonnegative homogenous forms is by adding up squares, which all arise as pullback by polynomial maps of the ur-nonnegative form x^2.

But we know, by Hilbert, that this isn’t enough to capture all nonnegative forms; for instance, it misses the Motzkin polynomial.

So what if you throw that in?  That is, we say a Motzkin is a degree-6d form

expressible as

$P^4 Q^2 + P^2 Q^4 - 3P^2 Q^2 R^2 + R^6$

for degree-d forms P,Q,R.  A Motzkin is obviously nonnegative.

It is possible that every nonnegative form of degree 6d is a sum of squares and Motzkins?  What if instead of just Motzkins we allow ourselves every nonnegative sextic?  Or every nonnegative homogeneous degree-d form in n variables for n and d less than 1,000,000?  Is it possible that the condition of nonnegativity is in this respect “finitely generated?”

What correlation means

In a meta-analysis of sixty-six studies tracking interests over time (the average study followed subjects for seven years), psychologists from the University of Illinois at Urbana–Champaign found that our interests in adolescence had only a point-five correlation with our interests later in life. This means that if a subject filled out a questionnaire about her interests at the age of, say, thirteen, and again at the age of twenty-one, only half of her answers remained consistent on both.

I think it’s totally OK to not say precisely what correlation means.  It’s sort of subtle!  It would be fine to say the correlation was “moderate,” or something like that.

But I don’t think it’s OK to say “This means that…” and then say something which isn’t what it means.  If the questionnaire was a series of yes-or-no questions, and if exactly half the answers stayed the same between age 13 and 21, the correlation would be zero.  As it should be — 50% agreement is what you’d expect if the two questionnaires had nothing to do with each other.  If the questionnaire was of a different kind, say, “rate your interest in the following subjects on a scale of 1 to 5,” then agreement on 50% of the answers would be more suggestive of a positive relationship; but it wouldn’t in any sense be the same thing as 0.5 correlation.  What does the number 0.5 add to the meaning of the piece?  What does the explanation add?  I think nothing, and I think both should have been taken out.

Credit, though — the piece does include a link to the original study, a practice that is sadly not universal!  But demerit — the piece is behind a paywall, leaving most readers just as unable as before to figure out what the study actually measured.  If you’re a journal, is the cost of depaywalling one article really so great that it’s worth forgoing thousands of New Yorker readers actually looking at your science?

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That Jeff Koons feeling

The sculptures and paintings of this fifty-nine-year-old artist are so meticulously, mechanically polished and groomed that they rebuff any attempt to look at them, much less feel anything about them.

But four paragraphs later:

Koons knows how to capitalize on the guilty pleasure that the museumgoing public takes in all his mixed messages. He knows how to leave people feeling simultaneously ironical, erudite, silly, sophisticated, and bemused.

Does Koons make people feel things, or does he not?  Or are irony, erudition, silliness, sophistication, and bemusement feelings that don’t count as feelings?

Jed Perl writes well but I find his judgment strange.  About Jeff Koons I have no opinion.  But I remember his name because of the piece he wrote about Francis Bacon, which seems to suggest that people like Bacon not because of anything in the paintings, but because the artist sports a biography and attitude that appeals to mushy-minded would-be avant-gardists.  “The Bacon mystique,” Perl writes, “is not grounded in his paintings so much as in a glamorous list of extenuating circumstances.”

To me this makes no sense.  I went to a small museum which was showing some of Bacon’s paintings and I was knocked over by them.  Whoa, what is that?  I had no idea who he was, or whether he was glamorous, or whether it was cool to like him.

I think it’s OK to say (as Perl also does, later in that piece) that Bacon is a stupid painter and only people who are stupid about painting like his paintings.  But it’s crazy to deny that people actually do like Bacon’s paintings, as such, not just the idea of Bacon’s paintings, or the idea of being the kind of person who likes Bacon’s paintings.

So… yeah

Lately CJ has a habit of ending every story he tells by saying

“So… yeah.”

I first noticed it this summer, so I think he picked it up from his camp counselors. What does it mean? I tend to read it as something like

“I have told my story — what conclusions can we draw from it? Who can say? It is what it is.”

Is that roughly right? Per the always useful Urban Dictionary the phrase is

“used when relating a past event and teller is unsure or too lazy to think of a good way to conclude it”

but I feel like it has more semantic content than that. Though I just asked CJ and he says it’s just his way of saying “That’s all.” Like “Over and out.”

So yeah.

Squeeze! Squeeze!

I hope the world never runs out of awesome Earl Weaver stories.

I saw Earl Weaver put on a suicide squeeze bunt, in Milwaukee. It worked. Everybody asked him, ‘Wait, we thought you told us you didn’t even have a sign for a suicide squeeze, because you hated it so much.’ Earl said, ‘I still don’t.’ I asked him, ‘How did you put it on then?’ He said, ‘I whistled at Cal Ripken, Sr., my third base coach. Then I shouted at him, ‘Squeeze! Squeeze! Then I motioned a bunt.’ I said, ‘Paul Molitor was playing third. Didn’t he hear you?’ Earl said, ‘If he did, I’m sure he thought there was no way we were putting it on, or I wouldn’t have been yelling for it.’

This is from the Fangraphs interview with the greatest announcer of our time, Jon Miller. His memoir, Confessions of a Baseball Purist, is full of great stuff like this. I didn’t know until just this second that it had been reissued by Johns Hopkins University Press.

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The 1979 Houston Astros hit only 49 home runs

49 home runs! That’s nuts. They hit more triples than home runs. Their home run leader was Jose Cruz, who hit 9. In September they went 20 straight games without hitting a home run, the longest such streak in modern baseball. And that was after they went 15 games without hitting a rome run in July!

Must have been a pretty bad team, right? But no! They won 89 games and finished second, just a game and a half behind the Reds. That 15 game homerless streak in July? They went 11-4 in those games.

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