Category Archives: math

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.

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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.

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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!

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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.

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Amie Wilkinson’s commencement speech

You want a good commencement speech?  This is a good commencement speech.  From Amie Wilkinson, at the Berkeley math department graduation ceremony.

The only way to begin is to start.

So let’s start with death. From today onward you will die a little death every time you bother to notice it. By this I mean a death of possibilities. Imagine a tree with many branches, directed into the future. Each is a potential future, a life path. Up until now, you’ve probably been climbing the large trunk of this tree, following what seems the natural path, ignoring some smaller branches along the way. Looking up, the tree has always been lush, dense and even impenetrable, rich with potential.

Click here to read the whole thing (pdf)

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Amit Singer on 3d reconstruction

Amit Singer just gave a very cool talk at Wisconsin’s Applied Algebra Day.  Slides from a similar talk he gave at ICERM are here.

Briefly:  the problem is to reconstruct an image (so let’s say a function f in L^2(R^3) measuring density, or potential, or whatever) from a bunch of linear 2d projections.  This is what you get when you try to do cryo-EM on molecules of biological origin; you have no control of how the molecules are oriented, so when you pass a beam through them and record the shape of the “shadow” you get, you’re looking at a projection of the molecule in an unknown direction.  But of course you may have 10,000 of the molecules, so you have 10,000 of these (noisy) projections at a bunch of different angles, and one may hope that this is enough information to reconstruct the original 3d shape!

Suppose f’ is one of these projections.  If F is the Fourier transform of f, and F’ the Fourier transform of f’, then F’ is just the restriction of F to some plane through the origin.

So we have the following problem:  there’s a mystery function F on R^3, and we have a bunch of functions F’_1, .. F’_n on R^2 which are restrictions of F to planes, but we don’t know which planes.  How to reconstruct those planes, and F?

Let G = SO(3).  We can set this problem up as an optimization problem over G^n as follows.  We want to find F and g_1, … g_n in G such that F’_i matches the restriction to the xy-plane of g_i F.

But optimizing over G^n is hard — an essentially exponential problem.  So what Singer and his collaborators do is discretize and convexize the problem in a very clever way.  You put a net of L points on S^2; then every rotation is going to induce (after some rounding) a permutation of these points, i.e. an LxL permutation matrix.  What’s more, n rotations just give you n permutations, which is an L x Ln matrix.

But optimizing over permutation matrices has a nice convex relaxation; the convex hull of the permutation matrices is the polytope of doubly stochastic matrices, which is very manageable.  It gets better:  a general permutation of the witness points on the sphere, of course, looks nothing at all like a rotation.  But the fact that a rotation preserves distances means that the corresponding permutation approximately commutes with the LxL covariance matrix of the points; this is a linear condition on the permutation matrices, so we end up optimizing over a linear slice of the doubly stochastic matrices.  And the point is that the difficulty now scales polynomially in n instead of exponentially.  Very nice!  (Of course, you also have to show that the cost function you’re actually optimizing can be made linear in this setup.)

Idle question:  how essential is the discretization?  In other words, is there a way to optimize directly over the convex hull of SO(3)^n, an object I know people like Bernd Sturmfels like to think about?

 

Boyer: curves with real multiplication over subcyclotomic fields

A long time ago, inspired by a paper of Mestre constructing genus 2 curves whose Jacobians had real multiplication by Q(sqrt(5)), I wrote a paper showing the existence of continuous families of curves X whose Jacobians had real multiplication by various abelian extensions of Q.  I constructed these curves as branched covers with prescribed ramification, which is to say I had no real way of presenting them explicitly at all.  I just saw a nice preprint by Ivan Boyer, a recent Ph.D. student of Mestre, which takes all the curves I construct and computes explicit equations for them!  I wouldn’t have thought this was doable (in particular, I never thought about whether the families in my construction were rational.) For instance, for any value of the parameter s, the genus 3 curve

2v + u^3 + (u+1)^2 + s((u^2 + v)^2 - v(u+v)(2u^2 - uv + 2v))

has real multiplication by the real subfield of \mathbf{Q}(\zeta_7).  Cool!

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Why aren’t math professors sociopaths?

Great open from Chris Hayes:

Imagine you’re a scientist in some sci-fi alternate universe, and you’ve been charged with creating a boot camp that will reliably turn normal but ambitious people into broken sociopaths more or less willing to do anything.

There are two main traits you’d want to cultivate in your recruits. The first would be terror: You’d want to ensure that the experimental subjects were kept off-­balance and insecure, always fearful that bad things would happen, that they would be humiliated or lose their position and be cast out. But at the same time, it would be crucial that you assiduously inculcate a towering sense of superiority, the belief that the project they happen to be engaged in is more important than anything and that, because of their remarkable skills and efforts, they are among the select few chosen to be a part of it. You’d want to simultaneously make them neurotically insecure and self-doubting and also filled with the conviction that they and their colleagues are smarter and better and more deserving than anyone else.

He’s writing about young investment bankers, whose lives, such as they are, are described in Kevin Roose’s new book “Young Money.”  But doesn’t this boot camp actually describe the Ph.D. experience pretty well?  And if so, why aren’t math professors sociopaths?

I can think of one reason:  in finance, the thing you are trying to do is screw over somebody else.  If you win, someone has lost.  Mathematics is different.  We’re all pushing together.  Not that there’s no competition; but it’s embedded in a fundamental consensus that we’re all on the same team.  Apparently this is enough to hold back the sociopathy, at least for most of us.

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Math blog roundup

Lots of good stuff happening in math blogging!

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