I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash by 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.

And nobody could generate sequences of elliptic curves with unbounded ranks, either! People have constructed lots of cool examples over the years. But Noam Elkies’s elliptic curve with rank 28 has stood as champion for almost a decade now. We may be approaching a Noamsymptote.

Now here’s what Bjorn has to say. (This is based on my chats with others about his talk, since I didn’t see it. Please correct/refine in comments.) The BKLPR heuristics propose a very rich conjectural description of the distribution of Selmer, Shafarevich, and Mordell-Weil groups of a random elliptic curve over Q. In particular: the p-adic Selmer group of E should be modeled by the intersection between two randomly chosen maximal isotropic subspaces in a large orthogonal space over Z_p.

So it seems natural to model the actual Mordell-Weil group as the intersection between two random maximal isotropic lattices in a large orthogonal space over Z!

But now there’s a problem with “random.” The space of maximal p-adic isotropics is a nice compact p-adic manifold with a natural probability distribution on it. And in this probability distribution, there is zero probability that the intersection of two isotropics — the “Mordell-Weil rank” — is greater than 1. Which is the answer we’re supposed to get! But for the present problem, the claim that “0% of elliptic curves have rank greater than 1″ isn’t good enough. We don’t want 0%. We want 0.

What’s more, what can it mean to talk about a random isotropic lattice now? There are a countably infinite set of such things, with no natural distribution on them to call “uniform.”

So here’s what to do. You can count all pairs of isotropic subspaces in a “box” — say, just count those generated by vectors with entries at most B, or better, count all subspaces of height at most B. There are finitely many of them. And let p(r,B) be the probability that two of these subspaces, chosen uniformly at random from the finitely many choices, intersect in a lattice of rank r. As B goes to infinity, we ought to expect p(r,B) to go to 0.

Similarly, let P(r,X) be the probability that an elliptic curve of conductor at most X has rank r.

We would like a heuristic to say that p = P! But this is meaningless without a way of “matching” B and X. Bjorn finesses this in a very clever way. We already have random matrix predictions that tell us that P(2,X) is supposed to be on order X^{-1/24}. So you “tune” B to match X by letting B be whatever power of X makes p(2,B) ~ B^{-1/24}!

(Note: I originally had an exponent of -1/4 above, but that was for quadratic twist families. Michael Rubinstein pointed out that I should have been quoting this paper of Mark Watkins instead. Interesting — I’m very used to casually saying “quadratic twist families are the same as the general family,” and I think that’s true for questions about measures, as in BKLPR, but there’s no reason they have to behave the same way w/r/t more refined questions like this one! This big difference in exponents makes me wonder — should we expect that 100% of elliptic curves should have no quadratic twist with rank greater than, I dunno, 5?)

And what you find is that, having done so, you get that p(r,B) varies as a negative power of B, and for r big enough, p(r,B) is smaller than B^{5/6}, which is supposed to be the number of elliptic curves of conductor at most B, and so for r this big, you find yourself predicting no elliptic curves of that rank at all. Or finitely many, better to say. After all, we *know* by work of Ulmer that there are elliptic curves over F_q(t) of arbitrarily large Mordell-Weil rank, and the BKLPR heuristics work just as well in this case — but the Ulmer examples will be very sparse. Maybe a better way to read what Bjorn’s new heuristic says is that, for sufficiently large r, the number of E/Q with rank at least r and conductor at most X grows more slowly than any power of X.

All aspects of what I’ve said here are oversimplified and no doubt some are wrong! But it’s very exciting to see conventional wisdom on such a fundamental question begin to shift, so I wanted to record the moment here.

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I was skeptical of the format. 18 minutes! How can you do anything? You can really just say one thing. No opportunity to digress. Since digression is my usual organizational strategy, this was a challenge.

And there’s a format. The organizers explained it to me. Not to be hewed to exactly but taken very seriously. A personal vignette, to show you’re a human. A one-sentence takeaway. General positivity. A visual prop is good. The organizers were lovely and gave me lots of good advice when I practiced the talk for them. I was very motivated to deliver it the way they wanted it.

And in the end, I found the restrictiveness of the format to be really useful. It’s like a sonnet. Sonnets are, in certain ways, all the same, by force; and yet there’s a wild diversity of sonnets. So too for TED talks. No two of the talks at TEDxMadison were really the same. And none of them was really like Steve’s TED talk (though I did read a poem like Steve) or Amanda Palmer’s TED talk or (thank goodness) like the moleeds TED talk.

No room in the talk to play the Housemartins song “Sitting on a Fence,” which plays a key role in the longer version of the argument in *How Not To Be Wrong. *So here it is now.

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If the feel of sand between your toes gets you thinking about Zeno’s Paradox or Pascal’s Wager, Ellenberg’s book is ideal beach reading. But even if your interests lie elsewhere, you may find it a challenging but welcome companion.

at NewCity Lit:

To the mathematician, math is a curious process of assumption and provocation. “How Not To Be Wrong” is part exposé—concepts most of us are never privy to are explained along with obvious surprises we just need to hear over again.

at Nature:

Ellenberg, an academic and

Slate‘s ‘Do the Math’ columnist, explains key principles with erudite gusto

and at Canada’s The Globe and Mail.

For audio fans, here’s an interview at the New Books podcast.

But actually, most of the publicity this week came from the WSJ “Hawking Index” article, which got covered all over the place. I like this Washington Post followup, which applies the methodology (such as it is!) to political memoirs. More good coverage from the National Post, featuring obligatory CanLit content. And here’s how it looks in Indonesian.

Christopher D. Long decided to see what happened if you tried to model “quotability” using a more serious dataset, scraped from Goodreads, instead of just screwing around like I did. His top 10 included some expected entries and some surprises. Any ranking where Eleanor Roosevelt and Groucho Marx place first and second is obviously doing something right.

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How can we find today’s greatest non-reads? Amazon’s “Popular Highlights” feature provides one quick and dirty measure. Every book’s Kindle page lists the five passages most highlighted by readers. If every reader is getting to the end, those highlights could be scattered throughout the length of the book. If nobody has made it past the introduction, the popular highlights will be clustered at the beginning.

Thus, the Hawking Index (HI): Take the page numbers of a book’s five top highlights, average them, and divide by the number of pages in the whole book. The higher the number, the more of the book we’re guessing most people are likely to have read. (Disclaimer: This is not remotely scientific and is for entertainment purposes only!)

At the end I suggest we call this number the Piketty Index instead, because Piketty’s unlikely megahit *Capital in the Twenty-First Century* comes in with an index of 2.4%, the lowest in my sample.

But it’s not the winner anymore! Hillary Clinton’s *Hard Choices* scores an amazing 1.9%. But somehow I feel like HRC’s book is in a different category entirely; unlike Piketty, I’m not sure I believe it’s a book people even *pretend to intend* to read.

The piece got lots of press, including a nice writeup at Gizmodo today. So I thought I’d add a few more comments here, to go past what I could do in an 800-word story.

- Lots of people asked: what about
*Infinite Jest?*In fact, that book was in the original piece but got cut for length. Here’s the paragraph:There was a time, children, when you couldn’t ride the 1/9 without seeing a dozen recent graduates straining under the weight of Wallace’s big shambling masterpiece. Apparently it was too heavy for most. Yes, I included the endnotes in the page count. This is another one whose most famous line – “I am in here” – doesn’t crack the Kindle top five.*Infinite Jest*, by David Foster Wallace. HI 6.4%. - Other books I computed that didn’t make it into the WSJ: Stephen King’s new novel
*Mr. Mercedes*scores 22.5%.*How To Win Friends and Influence People*gets 8.8%. And*How Not To Be Wrong*comes in at 7.7%. In fact, the original idea for the piece came from my dismay that all the popular highlights in my book were from the first three chapters. But actually that puts*How Not To Be Wrong*in the middle of the nonfiction pack! - Important: I highly doubt the Piketty Index of the book is actually a good estimate for the median proportion completed. And I think different categories of books can’t be directly compared. All nonfiction books scored lower than all novels (except
*Infinite Jest*!) I don’t think that’s because nobody finishes nonfiction; I think it’s because nonfiction books usually have introductions, which contain lots of direct assertions and thesis statements, exactly the kind of thing Kindle readers seem to like highlighting. - Challenges: can you find a book other than
*The Goldfinch*whose index is greater than 50%? Can you find a nonfiction book which beats 20%? Can you find a book of any kind that scores lower than Hillary Clinton’s*Hard Choices*?

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Scarlett Johansson gainfully posed in underwear and spiked heels for Esquire’s cover last year after the magazine named her the “sexiest woman alive.” But a French novelist’s fictional depiction of a look-alike so angered the film star that she sued the best-selling author for defamation.

The inappropriate “but” is one of the sneakiest rhetorical tricks there is. It presents the second sentence as somehow contrasting with the first. It isn’t. Scarlett Johansson agreed to be photographed mostly undressed; does that make it strange or incongruous or hypocritical that she doesn’t want to be lied about in print? It does not. To be honest, I can’t think of any explanation other than weird retrograde sexism for writing the lede this way. “She got *paid* for looking all *sexy*, so who is she to complain that she was defamed?” Patricia Cohen of the New York Times, I’m awarding you an

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This is by Little Red Wolf, a Madison band, who have a great new record, *Junk Sparrow,* recorded by Brian Liston at Clutch Sound, the same guy who did my audiobook. Range!

Of course the strange piano note, the one that kind of insists despite everything that it’s the *right* note and thereby colors the whole song with its weirdness and stubbornness, is sort of the same one that Weezer uses to devastating effect in “The Sweater Song.” And yet the two songs are completely different. Though the latter is also very, very beautiful. And now that I listen to both again there’s also something in common about the way the wordless aah-ahh’s are deployed, but it might just be that everybody in the world, whether AOR-indie or alt-country, loves *Doolittle.*

Wait, are there readers of this blog so young as not to have heard “The Sweater Song?” Very likely. So OK:

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The Koji Uehara burger at Mr. Bartley’s. A new one, very good. With onion rings, of course.

Peking ravs at the Hong Kong. Traditional.

A double cheeseburger at Charlie’s Kitchen.

Big sub at the amazing Bub and Pop’s.

Green curry from Regional Thai, which 15 years ago was my favorite place to eat in Chelsea (maybe tied with Rocking Horse Cafe.) Still good.

A crottin, taken to go at Murray’s Cheese Shop and eaten while walking.

Schnitzel and bright-pink Berliner Weissbier at Lederhosen deep in the West Village.

My Ferry Terminal usual: salami cone from Boccalone and mac and cheese at the Cowgirl Sidekick. This mac and cheese possibly my national favorite apart from the one at Miss Mamie’s Spoonbread Too, which was farther uptown than I got this NYC swing. (This also explains why no belly lox this time. Though now that I think of is, this could have been my chance to try Russ and Daughters.)

I’m over Mission burritos. Sorry. So this time I had Mission pierogi at Stuffed. Dumb name, decent pierogi, but surprisingly awesome sauerkraut, more like halbsauerkraut with a jolt of I think caraway? My recommendation: just buy their sauerkraut, buy a taco somewhere else, put the sauerkraut on the taco, resell it at your popup fusion cart. Become wealthy beyond human ability to imagine.

BBQ sampler, including kalua pig, from the 808 Grinds Hawaiian cart in Portland’s city of food carts. The fried chicken, surprisingly, was the standout. But if it doesn’t move, Portland, it’s not a cart. You must accept this, Portland. You’ll feel better when you do.

Four-chowder sampler at Pike Place Chowder. Long line? Tourists? Yes and yes (though shorter lines, and fewer tourists, than at the Original Starbucks down the block.) But really, really good chowder. And eating chowders in a flight formation is, I think, the right way.

Terrific black fideus at Aragona.

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