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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?

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But no one says “oh yeah, Fido Dido.” So here I am doing it.

It is inherently hard to try to list things you’ve forgotten about. My list right now consists of

- Fido Dido
- Saying “bite me”
- Smartfood
- Devil sticks (from Jason Starr)

That’s it. What have you got?

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So I guess what I’m saying is, I’m probably going to the right coffee shop. Also, this song is cool. I’m sort of fascinated by the long instrumental break that starts around 2:50. It doesn’t seem like very much is happening; why is it so captivating? I think my confusion on this point has something to do with my lack of understanding of drums.

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But here’s something interesting; it’s only the 14th most viewed article, the 6th most tweeted, and the 6th most shared on Facebook. On the other hand, this article about child refugees from Honduras is

#14 most emailed

#1 most viewed

#1 most shared on Facebook

#1 most tweeted

while Paul Krugman’s column about California is

#4 most emailed

#3 most viewed

#4 most shared on Facebook

#7 most tweeted.

Why are some articles, like mine, much more emailed than tweeted, while others, like the one about refugees, much more tweeted than emailed, and others still, like Krugman’s, come out about even? Is it always the case that views track tweets, not emails? Not necessarily; an article about the commercial success and legal woes of conservative poo-stirrer Dinesh D’Souza is #3 most viewed, but only #13 in tweets (and #9 in emails.) Today’s Gaza story has lots of tweets and views but not so many emails, like the Honduras piece, so maybe this is a pattern for international news? Presumably people inside newspapers actually study stuff like this; is any of that research public? Now I’m curious.

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Everything else I can say is a spoiler so I’ll put it below a tab.

1. I think the right way to read the book is that, by the end, nothing has changed for the two protagonists. Their relationship at the end of the book — in which the man is a hateful worm, and the woman a murderer, and they are bound together by hatred, fear, and common lies — is meant to be the *same *relationship they had in their courtship. Just with everything a little more out in the open. Indeed I think this is what Flynn suggests marriage just, naturally, *is. *That people, in general, are sick brutes who need to hurt each other in order to gain satisfaction and who can only be kept superficially in line by the threat of being hurt or killed themselves. I don’t actually think this is true and so I don’t like novels which, by virtue of being well-made, make a compelling case that it’s true.

2. Money is important here. The structure of the story is that the couple starts rich. Then for most of the book they’re not rich. Then at the end they’re rich again, which is what enables them to go back to their normal life. *Gone Girl* suggests that what being rich means is that people pay attention to you, people believe what you say, and also that you might need to leave some broken or dead people behind in order to maintain your position. So Desi Collings is cognate to the Blue Book Boys.

3. The book is lazy in placing a lot of weight on “the psycho woman who claims to have been raped but is making it up.” The problem with misogynistic stereotypes in novels is not just that misogynistic sterotypes are bad — and they are, they are really bad — but that they’re a fundamentally cheap way of constructing characters. They are easy to believe in because we are weak people, driven by heuristics, who believe stereotypes without thinking too hard about them.

The book would have been better if it had let Nick beat up his girlfriend. In other words, if the world of the novel contains women who lie about getting beaten up by men, it ought to contain men who beat women up. And this would be truer to the moral world of the novel, where a woman falsely accusing a man of abuse is both lying and not lying, because all men abuse somebody, whether or not the accuser and the victim happen to be the same person.

And I think it would have helped prohibit the reading — which I can see from online sources is not rare — that Nick is the hero of the story, who readers are supposed to root for. No! Gross! Nick is a sick brute, Amy is a sick brute, all four of their parents are sick brutes, with the possible exception of Nick’s mother, who’s kind of a cipher.

4. It was a bad idea to name a character “Go.” Confusing in dialogue.

5. In connection with the upcoming movie, you can buy T-shirts labeled “Team Nick” or “Team Amy.” That is messed up and wrong.

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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.

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 the new stuff. (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. They finesse 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? **Update: **I am told the answer is yes; more later on this, I hope.)

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 the 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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