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*

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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

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?”

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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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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.

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“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.

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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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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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In the latest stable representation theory news, Andy Putman and (new Wisconsin assistant professor!) Steven Sam have just posted an exciting new preprint about the theory of representations of GL_n(F_p) as n goes to infinity; this is kind of like the linear group version of what FI-modules does for symmetric groups. (Or, if you like, our thing is their thing over the field with one element….!) This is something we had hoped to understand but got very confused about, so I’m looking forward to delving into what Andy and Steven did here — expect more blogging! In particular, they prove the Artinian conjecture of Lionel Schwartz. Like I said, more on this later.

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People made fun of that last scene where they take shrooms and go hiking in Big Bend. But I liked this last scene. It captures that feeling that, on the one hand, the past is past, but on the other hand, the past is always present, *all of it*, all layered on top of each other. As if the whole movie actually takes place over the course of about a second or two, in 18-year-old Mason’s mind, and we’re seeing the images that exist there in that span of time. I *think *all adults constantly have that feeling, right? That your entire adult life is sort of a mask, and you’re really 20-year-old you who’s traveled forward in time to see how it all turned out, and also you’re 15-year-old you, and 6-year-old you, and etc., all at once? You don’t actually even need shrooms for this!

Question: is it impossible to talk about a Richard Linklater movie without feeling like you’re executing a Linklater monologue pastiche?

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He starts with the following lovely observation, which was apparently in a 2007 paper of his but which I was unaware of. Suppose you make a maximalist conjecture about uniform growth of finitely generated linear groups. That is, you postulate the existence of a constant c(d) such that, for any finite subset S of GL_d(C), you have a lower bound for the growth rate

.

It turns out this implies Lehmer’s conjecture! Which in case you forgot what that is is a kind of “gap conjecture” for heights of algebraic numbers. There are algebraic integers of height 0, which is to say that all their conjugates lie on the unit circle; those are the roots of unity. Lehmer’s conjecture says that if x is an algebraic integer of degree n which is {\em not} a root of unity, it’s height is bounded below by some absolute constant (in fact, most people believe this constant to be about 1.176…, realized by Lehmer’s number.)

What does this question in algebraic number theory have to do with growth in groups? Here’s the trick; let w be an algebraic integer and consider the subgroup G of the group of affine linear transformations of C (which embeds in GL_2(C)) generated by the two transformations

x -> wx

and

x -> x+1.

If the group G grows very quickly, then there are a lot of different values of g*1 for g in the word ball S^n. But g*1 is going to be a complex number z expressible as a polynomial in w of bounded degree and bounded coefficients. If w were actually a root of unity, you can see that this number is sitting in a ball of size growing linearly in n, so the number of possibilities for z grows polynomially in n. Once w has some larger absolute values, though, the size of the ball containing all possible z grows exponentially with n, and Breuillard shows that the height of z is an upper bound for the number of different z in S^n * 1. Thus a Lehmer-violating sequence of algebraic numbers gives a uniformity-violating sequence of finitely generated linear groups.

These groups are all *solvable, *even metabelian; and as Breuillard explains, this is actually the hardest case! He and his collaborators can prove the uniform growth results for f.g. linear groups without a finite-index solvable subgroup. Very cool!

One more note: I am also of course pleased to see that Emmanuel found my slightly out-there speculations about “property tau hat” interesting enough to mention in his paper! His formulation is more general and nicer than mine, though; I was only thinking about profinite groups, and Emmanuel is surely right to set it up as a question about topologically finitely generated compact groups in general.

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Here’s one thing I liked about this movie. Every adult man in the movie talks to Mason about *responsibility. *Following up. Thinking about consequences of actions. It’s the verbal glue that holds all the men in the movie together.

But here’s the thing. Responsibility is a virtue, sure. But it turns out that good men and bad men believe in it just the same. You can’t tell who’s good by what they say. Mason’s abusive alcoholic stepdad tells him to live up to the commitments he makes. That’s good advice. Mason’s photography teacher, presented as someone who basically cares about him and means him well, tells him he can’t just do what he pleases if he wants to make art; he has to apply himself and learn technique. Also good advice. Not-necessarily-alcohol-abusing-but-drunk-and-checked-out stepdad #2 tells him he should call his mother if he’s going to be out all night because she worries. Also good advice! The manager at the cruddy restaurant where Mason works tells him he shouldn’t screw around chatting in the back when there are families waiting for their food. That’s good advice too! And the movie cleverly sets up the manager as a figure of fun (giving him a dorky polo shirt and a receding hairline) but then brings him back, in a sympathetic role, at Mason’s party, forcing the audience to say, yeah, the dorky guy was right, big ups for the dorky guy. Ethan Hawke’s second wife’s dad (following me here?) works similarly; the movie sets you up to see his gift of a gun to Mason as a piece of yokelism, but Mason visibly appreciates it, and what is the older man’s main piece of dialogue in the scene? A reminder that a gun is a serious thing and you have to use it with safety foremost in your mind. *Great* advice.

Responsibility talk isn’t really about being a good man or a bad man; it’s just about being a man. Mason’s biological father gives him a talk about birth control (at this point, Mason is about 13, and hasn’t had a girlfriend yet, I think) which is a fine model of the “This is serious, but I’m gonna be funny, but also, remember, I’m serious” approach. I’m sure a lot of dads of younger kids were taking notes.

But of course the context of this talk is that Mason Sr. himself didn’t use birth control, whence Mason! Who he then — contra all the responsibility talk — ran out on, before the movie even starts.

So it’s unfair, right, that I’m giving him credit for this speech? And for that matter, isn’t it unfair and kind of creepily patriarchal that I’m casting responsibility as part of being a *man*, as opposed to part of being an adult?

So the movie is working exactly to bring that into the light and then oppose it, I think. Because who *actually* lives up to commitments and acts responsible is Mason’s mother. He hears about responsibility from his dad, and every other man in the world. But he learns it from his mom.

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