Breuillard’s ICM talk: uniform expansion, Lehmer’s conjecture, tauhat

Emmanuel Breuillard is in Korea talking at the ICM; here’s his paper, a very beautiful survey of uniformity results for growth in groups, by himself and others, and of the many open questions that remain.

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

\lim |S^n|^{1/n} > c(d).

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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Notes on Boyhood

Richard Linklater’s Boyhood is certainly the best movie I’ve seen this year, likely the best movie I’ll see this year.  But I don’t see a lot of movies.  After the spoiler bar, some notes on this one.  I meant to write this right after I saw it, but got busy, so no doubt I’ve forgotten some of what I meant to say and gotten other things wrong. Continue reading

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August linkdump

  • The company that makes OldReader, the RSS reader I fled to after the sad demise of Google Reader, is from Madison!  OK, Middleton.  Still part of Silicon Isthmus.
  • I never new that Mark Alan Stamaty, one of my favorite cartoonists, did the cover of the first They Might Be Giants album.
  • Hey I keep saying this and now Allison Schrager has written an article about it for Bloomberg.  Tenure is a form of compensation.  If you think tenure is a bad way to pay teachers, and that compensation is best in the form of dollars, that’s fine; but if California pretends that the elimination of tenure isn’t a massive pay cut for teachers, they’re making a basic economic mistake.
  • New “hot hand” paper by Brett Green and Jeffrey Zweibel, about the hot hand for batters in baseball.  They say it’s there!  And they echo a point I make in the book (which I learned from Bob Wardrop) — some of the “no such thing as the hot hand” studies are way too low-power to detect a hot hand of any realistic size.
  • Matt Baker goes outside the circle of number theory and blogs about real numbers, axioms, and games.  Daring!  Matt also has a very cool new paper with Yao Wang about spanning trees as torsors for the sandpile group; but I want that to have its own blog entry once I’ve actually read it!
  • Lyndon Hardy wrote a fantasy series I adored as a kid, Master of the Five Magics.  I didn’t know that, as an undergrad, he was the mastermind of the Great Caltech Rose Bowl Hoax.  Now that is a life well spent.
  • Do you know how many players with at least 20 hits in a season have had more than half their hits be home runs?  Just two:  Mark McGwire in 2001 and Frank Thomas in 2005.
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Plagiarism, patchwriting, Perlstein

Some people are complaining about Rick Perlstein’s new book, claiming that some passages are plagiarized.  Most of my friends think this is nonsense.

Here’s a passage from Craig Shirley’s Reagan’s Revolution:

Even its ‘red light’ district was festooned with red, white, and blue bunting, as dancing elephants were placed in the windows of several smut peddlers.

And from Perlstein:

The city’s anemic red-light district was festooned with red, white and blue bunting; several of the smut peddlers featured dancers in elephant costume in their windows.

Shirley:

Whenever he flew, Reagan would sit in the first row so he could talk to people as they boarded the plane.  On one occasion, a woman spotted him, embranced him, and said, “Oh Governor, you’ve just got to run for President!”  As they settled into their seats, Reagan turned to Deaver and said, “Well, I guess I’d better do it.”

Perlstein:

When Ronald Reagan flew on commercial flights he always sat in the front row.  That way, he could greet passengers as they boarded.  One day he was flying between Los Angeles and San Francisco.  A woman threw her arms around him and said “Oh, Governor, you’ve got to run for president!” “Well,” he said, turning to Michael Deaver, dead serious, “I guess I’d better do it.”

The second passage is cited to Shirley, the first isn’t.  But I don’t think it matters!  You shouldn’t paraphrase someone else’s book sentence by sentence, even if you cite them.  If you’re going to say exactly what they said, you should quote them.

Is this plagiarism?  It is, at the very least, patchwriting:  “restating a phrase, clause, or one or more sentences while staying close to the language or syntax of the source.”  Mark Liberman at LanguageLog has a long, magisterial post about patchwriting in Perlstein’s book, pointing out some places where Shirley himself patchwrites from the New York Times.

I once came across a magazine article whose lede was patchwritten from an article of my own.  I talked to a few trusted friends about how to handle it.  Uniformly, they said:  it’s not nice, but it’s not plagiarism, and you shouldn’t accuse the other author of stealing your stuff.  In the end, I alerted the other author to the issue without accusing her, and she apologized, saying she’d done it in a hurry and didn’t realize it was so close.  Which is probably true.

So I guess it’s not plagiarism and Shirley is not going to win his $25 million lawsuit against Perlstein.  But I don’t really like it and I think when we do journalism we should strive to write our own stuff.

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Conservative commentators on education are mad about the new AP US History standards.

The group’s president, Peter Wood, called the framework politically biased. One of his many complaints is about immigration: “Where APUSH sees ‘new migrants’ supplying ‘the economy with an important labor force,’ others with equal justification see the rapid growth of a population that displaces native-born workers from low-wage jobs and who are also heavily dependent on public services and transfer payments.”

Here’s the full text of the relevant bullet point in the standards.

The new migrants affected U.S. culture in many ways and supplied the economy with an important labor force, but they also became the focus of intense political, economic, and cultural debates.

You can decide for yourself whether the standard sweeps under the rug the fact that many people wish there were fewer immigrants.  But shouldn’t Newsweek print the whole sentence, instead of letting its readers rely on selective quotes?  Why do I have to look this stuff up myself?

Grothendieck’s parents

From “Who is Alexander Grothendieck?  Anarchy, Mathematics, Spirituality, Solitude,” by Winfried Scharlau (trans. Melissa Schneps)

If one is to believe the account given in Eine Frau, Sascha saw Hanka’s photograph by chance, probably one of the photographs that still exist today, and immediately informed the dismayed husband: “I will take your wife away!”  A few days later Hanka appeared, still rather weak from her abortion — and it was love at first sight.”

Where are people buying How Not To Be Wrong?

Amazon Author Central shows you Bookscan sales for your books broken down by metropolitan statistical area.  (BookScan tracks most hardcover sales, but not e-book sales.)  This allows me to see which MSAs are buying the most and fewest copies, per capita, of How Not To Be Wrong.  Unsurprisingly, Madison has by far the highest number of copies of HNTBW per person.  But Burlington, VT is not far behind!  Then there’s a big drop, until you get down to DC, SF, Boston, and Seattle, each of which still bought more than twice as many copies per person as the median MSA.

Where do people not want the book?  Lowest sales per capita are in Miami.  They also have little use for me in Los Angeles, Atlanta, and Houston.  Note that for reasons of time I only looked at the 30 MSAs that sold the most copies of the book; going farther down that list, there are more pretty big cities where the book is unpopular, like Tampa, Charlotte, San Antonio, and Orlando.

It would be interesting to compare the sales figures, not to population, but to overall hardcover book sales.  But I couldn’t find this information broken down by city.

 

Bounded rank was probable in 1950

Somehow I wrote that last post about bounded ranks without knowing about this paper by Mark Watkins and many other authors, which studies in great detail the variation in ranks in quadratic twists of the congruent number curve.  I’ll no doubt have more to say about this later, but I just wanted to remark on a footnote; they say they learned from Fernando Rodriguez-Villegas that Neron wrote in 1950:

On ignore s’il existe pour toutes les cubiques rationnelles, appartenant a un corps donné une borne absolute du rang. L’existence de cette borne est cependant considérée comme probable.

So when I said the conventional wisdom is shifting from “unbounded rank” towards “bounded rank,” I didn’t tell the whole story — maybe the conventional wisdom started at “bounded rank” and is now shifting back!

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Show your work

Here’s another comment on that New York Times piece:

“mystery number game …. ‘I’m thinking of a mystery number, and when I multiply it by 2 and add 7, I get 29; what’s the mystery number?’ “

See, that’s what I mean, the ubiquitous Common Core approach to math teaching these days wouldn’t allow for either “games” or “mystery”: they would insist that your son provide a descriptive narrative of his thought process that explains how he got his answer, they would insist on him drawing some matrix or diagram to show who that process is represented pictorially.

And your son would be graded on his ability to provide this narrative and draw this diagram of his thought process, not on his ability to get the right answer (which in child prodigies and genius, by definition, is out of the ordinary, probably indescribable).

Actually, I do often ask CJ to talk out his process after we do a mystery number.  I share with the commenter the worry of slipping into a classroom regime where students are graded on their ability to recite the “correct” process.  But in general, I think asking about process is great.  For one thing, I learn a lot about how arithmetic facility develops in the mind.  I asked CJ the other night how many candies he could buy if each one cost 7 cents and he had a dollar.  He got the right answer, 14, not instantly but after a little thought.  I asked him how he got 14 and he said, “Three 7s is 21, and five 21s is a dollar and five cents, so 15 candies is a little too much, so it must be 14.”

How would you have done it?

“Like a girl”

I wrote a New York Times op/ed last week about the relationship between teaching math and coaching Little League.  Several people wrote me to say that I shouldn’t have written the following passage:

My level of skill at baseball — actually, with every kind of ball — is pretty much the opposite of my mastery of math. I’ve reached 40 and I still throw in the way that we used to call, before they started showing college softball on TV, “like a girl.”

So obviously my goal here is to undercut the stereotype and present it as obsolete.  But the people who wrote me argued that to use the force of a sexist phrase to give my sentence a little oomph is a problem, even if (as I once heard J. P. Serre say about a piece of notation) “I mention it only in order to object to it.”

So I asked about this on Facebook, and maybe 60% of people thought it was fine, and 40% said that they winced when they read it.

Which means it’s not fine.  Because why write something that makes 40% of readers wince in annoyance?  Especially when a) it’s in no way intrinsic to the piece, which is otherwise not about gender roles, and b) the piece itself ties math to baseball, a boy-coded activity, and has much more material about my son than it does about my daughter.

I think “like a girl” can be an OK place to go if you need to.  But I didn’t need to.  So I think I shouldn’t have.

One of my friends suggested I should have said instead that I throw “like a mathematician.”  Better!

 

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