Category Archives: math

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.

 

 

 

 

 

 

Tagged , , , , ,

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.
Tagged , , , ,

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.”

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!

Tagged , , , ,

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?

Rank 2 versus rank 3

One interesting feature of the heuristics of Garton, Park, Poonen, Wood, Voight, discussed here previously: they predict there are fewer elliptic curves of rank 3 than there are of rank 2.  Is this what we believe?  On one hand, you might believe that having three independent points should be “harder” than having only two.  But there’s the parity issue.  All right-thinking people believe that there are equally many rank 0 and rank 1 elliptic curves, because 100% of curves with even parity have rank 0, and 100% of curves with odd parity have rank 1.  If a curve has even parity, all that has to happen to force it to have rank 2 is to have a non-torsion point.  And if a curve has odd parity, all that has to happen to force it to have rank 3 is to have one more non-torsion point you don’t know about it.  So in that sense, it seems “equally hard” to have rank 2 or rank 3, given that parity should be even half the time and odd half the time.

So my intuition about this question is very weak.  What’s yours?  Should rank 3 be less common than rank 2?  The same?  More common?

Are ranks bounded?

Important update, 23 Jul:  I missed one very important thing about Bjorn’s talk:  it was about joint work with a bunch of other people, including one of my own former Ph.D. students, whom I left out of the original post!  Serious apologies.  I have modified the post to include everyone and make it clear that Bjorn was talking about a multiperson project.  There are also some inaccuracies in my second-hand description of the mathematics, which I will probably deal with by writing a new post later rather than fixing this one.

I was only able to get to two days of the arithmetic statistics workshop in Montreal, but it was really enjoyable!  And a pleasure to see that so many strong students are interested in working on this family of problems.

I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash talking about joint work by Derek Garton, Jennifer Park, John Voight, Melanie Matchett Wood, and himself, offering some heuristic evidence that the Mordell-Weil ranks of elliptic curves over Q are bounded above.  I remember Andrew Granville suggesting eight or nine years ago that this might be the case.  At the time, it was an idea so far from conventional wisdom that it came across as a bit cheeky!  (Or maybe that’s just because Andrew often comes across as a bit cheeky…)

Why did we think there were elliptic curves of arbitrarily large rank over Q?  I suppose because we knew of no reason there shouldn’t be.  Is that a good reason?  It might be instructive to compare with the question of bounds for rational points on genus 2 curves.  We know by Faltings that |X(Q)| is finite for any genus 2 curve X, just as we know by Mordell-Weil that the rank of E(Q) is finite for any elliptic curve E.  But is there some absolute upper bound for |X(Q)|?  When I was in grad school, Lucia Caporaso, Joe Harris, and Barry Mazur proved a remarkable theorem:  that if Lang’s conjecture were true, there was some constant B such that |X(Q)| was at most B for every genus 2 curve X.  (And the same for any value of 2…)

Did this make people feel like |X(Q)| was uniformly bounded?  No!  That was considered ridiculous!  The Caporaso-Harris-Mazur theorem was thought of as evidence against Lang’s conjecture.  The three authors went around Harvard telling all the grad students about the theorem, saying — you guys are smart, go construct sequences of genus 2 curves with growing numbers of points, and boom, you’ve disproved Lang’s conjecture!

But none of us could.

Continue reading

Tagged , , , , , ,

Mathematical progress, artistic progress, local-to-global

I like this post by Peli Grietzer, which asks (and I oversimplify:)  when we say art is good, are we talking about the way it reflects or illuminates some aspect of our being, or are we talking about the way it wins the culture game?  And Peli finds help navigating this problem from an unexpected source:  Terry Tao’s description of the simultaneously local and global nature of mathematical progress.  Two friends of Quomodocumque coming together!  Unexcerptable, really, so click through if you like this kind of stuff.

Tagged , , ,

Should Andrew Gelman have stayed a math major?

Andrew writes:

As I’ve written before, I was a math and physics major in college but I switched to statistics because math seemed pointless if you weren’t the best (and I knew there were people better than me), and I just didn’t feel like I had a good physical understanding.

But every single mathematician, except one, is not the best (and even that person probably has to concede that there are still greater mathematicians who happen to be dead.)  Surely that doesn’t make our work pointless.

This myth — that the only people who matter in math are people at the very top of a fixed mental pyramid, people who are identified near birth and increase their lead over time, that math is for them and not for us — is what I write about in today’s Wall Street Journal, in a piece that’s mostly drawn from How Not To Be Wrong.  I quote both Mark Twain and Terry Tao — how’s that for appeal to authority?  The corresponding book section also has stuff about Hilbert and Minkowski (guess which one was the prodigy!) Ramanujan, and an extended football metaphor which I really like but which was too much of a digression for a newspaper piece.

There’s also a short video interview on WSJ Live where I talk a bit about the idea of the genius.

In other launch-related publicity, I was on Slate’s podcast, The Gist, talking to Mike Pesca about the Laffer curve and the dangers of mindless linear regression.

More book-related stuff coming next week; stay tuned!

Update:  Seems like I misread Andrew’s post; I thought when he said “switched” he meant “switched majors,” but actually he meant he kept studying math and then moved into a (slightly!) different career, statistics, where he used the math he learned: exactly what I say in the WSJ piece I want more people to do!

Tagged , ,

In which I talk burritos with Nate Silver

I interviewed Nate Silver last month at the Commonwealth Club in San Francisco for an MSRI event.  Video here.

Tagged , ,
Follow

Get every new post delivered to your Inbox.

Join 553 other followers

%d bloggers like this: