Jordan Ellenberg's Blog, page 8

I’m not sure if I mentioned that I’m teaching a first-year undergrad seminar on “Writing and Data,” in some respects patterned after the Writing Scientists’ Workshop I ran last year. With 18-year-olds it’s a little different; for one thing, I find that doing two hour-long workshops in a row gets a little long for them, so I’m doing two 45-minute workshops with an in-class writing exercise in between. Last week’s worked particularly well so I wanted to record what I did. We started with this piece from Pew, “How Many Friends do Americans Have?” Because I want them to think about conveying the same information in different registers, and in particular writing more “conversationally,” I split the group into pairs and asked each pair to write a dialogue which conveyed some of the information from the Pew piece. I gave them 15-20 minutes to do that, then had each pair act out their dialogue. I had been wondering whether to have everyone start from the same source or let people pick; in the end, I was glad we were all working from the same article, because it was instructive to see how many different ways the same information could be deployed in speech, or an imitation of speech. If there’s one thing I’m trying to get across in this class, it’s that writing is much, much more than the factual information it conveys.

A lot of people are going to tell you that the Orioles, who won 101 games this year and have by all accounts at least half an above-average major league lineup still in the minors ready to play, are positioned to be a dynastic power in the American League for years to come.

Nope. Wrong. I wish it were so. But this was the year the Orioles needed to win. And they just got dismally swept out of the ALDS by the Rangers. So that’s it.

Yes, all the guys they say are good are good, and are going to stay good. Adley’s good. Gunnar’s good. Grayson’s good. The guys that haven’t even played yet are good.

But this year, both the Yankees and Red Sox were kind of bad, and content to be kind of bad, and didn’t make gigantic talent adds in a bit for the playoffs. That hasn’t been the case for years and it won’t be the case again anytime soon.

This year, almost nobody got injured — we didn’t have John Means for most of the season, and Cedric Mullins missed some time, but basically everybody was healthy and we played at full strength. Next year, we already know Felix Bautista is gone for 2024. And we won’t be as lucky as we were in 2023 with the lineup.

And this year, we had an incredible, unsustainable record in one-run games, and finished 7 wins better than our Pythagorean record.

This is a good team. I like watching them. You could even say that, with an ownership willing to add expensive free agents to fill the holes, it could be a championship team. But we have an ownership that’s ecstatic that the 2023 team lucked into 101 regular season wins, and that will be perfectly happy to enjoy 90-win seasons and trips to the Wild Card game for the next few years, until the unextended players mentioned above peel off into free agency one by one.

It’ll be better than the last five years have been, that’s for sure. But if we wanted to win a World Series, this was our chance.

Sam Bankman-Fried, mostly famous for being a high-profile effective altruist and subsequently defrauding lots of investors, had something to say about Shakespeare:

Literary critic, Sam Bankman-Fried, on Shakespeare: pic.twitter.com/AjrsWv60JF

— Aaron Gwyn (@AmericanGwyn) October 6, 2023

A couple of thoughts on what this has to do with effective altruism.

To do certain types (not all types) of EA you have to believe that the utility of an outcome is a real number, though it might be hard to measure, and that your uncertainty about an outcome is captured by a real number between zero and one, and that it makes sense to multiply those two real numbers.

If you get used to thinking this way, you might also become inclined to think that “literary greatness” is a real number. And if you thought literary greatness was a real number, you might find it reasonable to think of it as a random variable about which you could formulate probabilistic claims. And that’s where you get the SBF Shakespeare argument.

Mathematicians know real numbers very well; we know what they’re like and we know most things are not like them. We also often admire Shakespeare, but I think this is coincidental.

Went with AB to New York for three days and this is what we did/saw/ate, in rough chronological order:

Korean barbecue at Antoya, Natural History Museum (including new Gilder wing and finally re-opened Northwest Coast hall, my favorite), belly lox from Zabar’s eaten at Riverside Park, Little Shop of Horrors revival, bubble tea, slice at 2 Bros, breakfast at Katz’s (people, if you can stomach a 3/4-pound pastrami sandwich at 10am this is absolutely the way to beat the line), the Strand, shake at the original Shake Shack, The Play That Goes Wrong, observation deck at Top of the Rock, MUJI, Churrascaria Plataforma, breakfast at Junior’s Cheesecake, the Intrepid museum, the Staten Island Ferry (why is this free?), old high school friend, old grad school friend, Korean fried chicken at Turntable Chicken Jazz, one final bubble tea.

Transportation note: we didn’t take a taxi or Lyft the entire time. I understand why almost no US city can have a subway and bus network this thick and this good, but boy is it nice. (Maybe relevant is that we didn’t leave Manhattan the entire time except for the two-block radius around the ferry terminal on the Staten Island side.)

CJ and I took a father-son trip to Spain (or, depending on how you partition nations, to Spain and Catalonia.) A very enjoyable, nimble, seat-of-the-pants-planned trip. Just back yesterday.

Some things I found surprising about Spain:

Pollworkers for elections are chosen by lot from the population, the way juries are in the United States. If the pollworker assigned to a polling station doesn’t show up, the police can select an unexpecting voter and assign them to work the polls on the spot.Crosswalks aren’t really at the intersection, but set back quite a ways from the intersection, maybe 10% of the way to the middle of the block. This seems like a good system!I expected to enjoy seeing Harry Styles play the Estadi Olimpic in Barcelona but I was surprised by how much I enjoyed it. I didn’t understand how much the Catholic Church is wound into the government there. To some extent this is lingering Francoism, to some extent just how Spain has been forever. There are religion classes in public schools, and when you do your taxes there’s a box you can mark that allocates 0.7% of your taxes to the Church. In the Granada Cathedral there was a free newsletter which turned out to be entirely devoted to convincing readers to mark the box.Burrata, which I think of as Italian food, seems to be a standard menu item in Spain. I think it’s been fully incorporated and is now also Spanish food.We went to a sports bar in Barcelona to watch Carlos Alcaraz play in the Wimbledon quarterfinal. We expected to be among a crowd of cheering fans but in fact the bar wasn’t even showing the match until we asked the bartender to put it on, and it took him quite a while to find the channel. Alcaraz is one of the biggest Spanish athletes in the world, so why is this? Some potential explanations: 1) Watching sports in bars isn’t popular in Spain (evidence; there weren’t very many sports bars listed!); 2) In Barcelona, Alcaraz is seen as Spanish-as-opposed-to-Catalonian; 3) Tennis just isn’t a popular spectator sport in Spain. I don’t know which it was!Also, the sports bar was founded in 2008, is called “Obama Gastropub,” and has a… colonial British Africa theme? Like, khaki jungle gear and 1920 maps of Africa everywhere? Very weird.During the brief period of anarchist rule in Barcelona before 1939, the radically anti-clerical Republicans dug up the bodies of priests and nuns from under the church and displayed the decayed corpses in the town square, as a way of falsifying the popular belief that the clergy lay undecomposed beneath the earth in preparation for their eventual bodily ascension.CJ and I went to a bullfight in Madrid. I’m not sure what I expected — but I did not expect it to be as thoroughly sad as it was. I had taken the name to mean that a bullfight was a fight. But the actual name for this event in Spanish, corrida, makes no such promise, saying only that the bull will run. At the beginning, it does run, even jabs with its horns a little bit. But then the picadors wound the bull and the banderilleros drive barbs into its neck. At that point, the bull looks tired and confused. It is clearly not mad anymore. It would be happy to walk away and call the whole thing off. And then the matador, who at this point conveys no sense of being in danger at all, whose spangly uniform is not even mussed, drives a sword into the bull and the crowd sits and waits while the bull vomits a dribble of blood, starts to wobble, and then finally goes down to its knees and dies. And then everybody cheers. And then the horses drag the bull’s body across the ring and a couple of janitors sweep dirt over the bloody trail. I don’t know what I saw, but it wasn’t a fight. There were six bulls slated to be stabbed that night but we left after two.

I like a close, hard-fought game as much as the next baseball fan, and I’ve seen a lot of those lately, but there is a peculiar and specific pleasure to the game in which the team you’re rooting for gets absolutely, relentlessly pummeled. It was a beautiful night on Friday, though chilly enough that they closed the roof at American Family Field. The Brewers were in their City Connect “Brew Crew” uniforms. We got there just as Christian Yelich was grounding into an RBI double play with the bases loaded. That was about as good as it got for Milwaukee. Freddy Peralta, starting for the Brewers, didn’t have it. The next reliever didn’t have it either. Ethan Small, brought up that morning from triple-A Nashville, didn’t have it, and by that time the game was out of reach and Craig Counsell just left Small up there on the hill to take his lumps and save the rest of the pen. The Brewers were booting balls, botching throws, just generally Bad News Bearsing it out there, and the crowd was, well, good-natured. Like I said, it was a beautiful night. Our guys were having a bad day and we were there for them.

Mike Brosseau moved over from first base to pitch the ninth and it was a real pleasure to see the Giants’ batters stymied at lsat, unable to adjust to the 68-mph fastball and the changeup that cruised in at 62. He got them 1-2-3. By that time a lot of fans had gone home. But we stayed through to the end. And you can see us pretty clearly, sitting along the third base line above the Giants dugout, in the broadcast.

Next visit to AmFam will be when the Orioles come to town. So I’m hoping to see the Brewers lose one more time this spring.

I was just at a conference where someone asked me if I had coined any mathematical terms. Well, sort of! I was the one who decided on the name “FI-modules” for the abelian category Tom Church, Benson Farb and I wrote about in this paper. More informally, I’m pretty sure I’m the originator of using “Bhargavology” to mean “the program of counting arithmetic things by putting them in bijection with orbits of the integral points of a group acting on the integral points of a space.” At least, I can find this usage in emails I wrote in 2003, after Manjul’s thesis but before any of the papers came out. And that still seems to be something people say.

My coinages have not always been successful. Nobody ever again mentioned the “esperantist graphs” from my paper with Hall and Kowalski. (They were named so in honor of Harald Helfgott, who speaks Esperanto, and because in some sense because they are typically graphs we hope are expanders.) Nor did “superduperstrong approximation” catch on.

The “made-up language” task that GPT-3 couldn’t do is handled much better by GPT-4, or at least the version of GPT-4 built into Bing Chat. The prompt:

Henderson’s native language is Gravinic, and when he speaks English, it sounds like this:

“I to the store did go and some cabbage did buy. Now I some dinner am making and I some cake will later eat. I you am not very much liking. I for you to go away am politely asking.”

Write more dialogue where Henderson tells you about his day.

The output:

Henderson continued to speak in his native language, Gravinic, which was very difficult for me to understand. He seemed to have a lot of things to say about his day, but I could only catch some words here and there.

“I today at work did have a meeting with my boss. He me did give a new project that I by next week must finish. It very hard and complicated is. I not sure if I it can do. He me also did say that I more hours must work and less pay will get. He very mean and unfair is. I him do not like at all.”

He paused for a moment and looked at me expectantly. I nodded politely, pretending to follow his story.

“Then I at lunch did go with my friend Bob. He me did tell that he a new girlfriend has. She very beautiful and smart is. He her met online on a dating site. He me showed her picture on his phone. I it did see and very jealous was. I also a girlfriend want but no one me likes.”

He sighed and shook his head sadly.

“Then I after work did go to the gym. I some exercise did do and some weights did lift. I very strong and fit am. But no one me notices or compliments gives. They only at me stare and laugh do behind my back. They very rude and mean are. I them do not like either.”

He frowned and clenched his fists.

“Then I home did come and you here did find waiting for me. You me always questions ask and annoy do with your presence. You very annoying and boring are. I you do not like too.”

He glared at me angrily.

“I you am now telling that you must leave do right now or else I you will hit do with this frying pan.”

He pointed at the frying pan on the stove menacingly.

I decided it was time to end this conversation before he got violent.

I quickly got up from my seat and headed towards the door.

Not perfect, but it really has picked up that in this version of English, verbs go last and past tense is indicated with “did.” It hasn’t really picked up how to do present tense (i.e. it should have said “They only are at me staring and are behind my back laughing”) and it is inconsistent in what it does with adverb phrases and there are some weird uses of “do” towards the end, but altogether a big improvement.

I was doing guest lecture / Q&A in a science communication course at Cornell, where I was a professor at large last week. A friend of mine was in the back of the room, and later told me: “A bunch of students were on their laptops, but only about half of them were looking at Instagram and the other half were looking at information about you and your books!”

He meant this to be a complement but my initial reaction was — what? Students were looking at their laptops while we were talking? I had no idea. In my mental construction of the event, both while it was happening and afterwards, every eye in the room was on me.

And this made me reflect — when I am giving a talk, or even a classroom lecture, I’m inclined to always think I’m being listened to. But of course that’s not true! It couldn’t be true!

There are limits, of course. If I’m lecturing and I’ve lost the whole room, I see their eyes die and I notice it. I stop and regroup and change course. But if half the kids are tuned out? I’m just gonna be honest, I probably don’t notice that.

Now you can read this as saying I’m a huge egotist who relies on unrealistic assessments of how interesting I’m being, and thanks to this reliance am failing to engage the class. Or you could say it’s very, very hard to teach class in such a way that there’s not some notable proportion of students tuned out at any given moment, and that it would be even harder to teach class well if you were constantly aware of which students those were. And as a counterpoint to that sympathetic assessment, you could say it’s not a random and constantly shifting sample of students who are tuned out; there might be a notable proportion who are almost tuned out and who I’m allowing myself to fail, or rather to not even try, to reach.

I don’t really know!

I have a new paper up on the arXiv today with TriThang Tran and Craig Westerland, “Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields.”

There’s a bit of a story behind this, but before I tell it, let me say what the paper’s about. The main result is an upper bound for the number of extensions with bounded discriminant and fixed Galois group of a rational function field F_q(t). More precisely: if G is a subgroup of S_n, and K is a global field, we can ask how many degree-n extensions of K there are whose discriminant is at most X and whose Galois closure has Galois group G. A long-standing conjecture of Malle predicts that this count is asymptotic to c X^a (log X)^b for explicitly predicted exponents a and b. This is a pretty central problem in arithmetic statistics, and in general it still seems completely out of reach; for instance, Bhargava’s work allows us to count quintic extensions of Q, and this result was extended to global fields of any characteristic other than 2 by Bhargava, Shankar, and Wang. But an asymptotic for the number of degree 6 extensions would be a massive advance.

The point of the present paper is to prove upper bounds for counting field extensions in the case of arbitrary G and rational function fields K = F_q(t) with q prime to and large enough relative to |G|; upper bounds which agree with Malle’s conjecture up to the power of log X. I’m pretty excited about this! Malle’s conjecture by now has very robust and convincing heuristic justification, but there are very few cases where we actually know anything about G-extensions for any but very special classes of finite groups G. There are even a few very special cases where the method gives both upper and lower bounds (for instance, A_4-extensions over function fields containing a cube root of 3.)

The central idea, as you might guess from the authors, is to recast this question as a problem about counting F_q-rational points on moduli spaces of G-covers, called Hurwitz spaces; by the Grothendieck-Lefschetz trace formula, we can bound these point counts if we can bound the etale Betti numbers of these spaces, and by comparison between characteristic p and characteristic 0 we can turn this into a topological problem about bounding cohomology groups of the braid group with certain coefficients.

Actually, let me say what these coefficients are. Let c be a subset of a finite group G closed under conjugacy, k a field, and V the k-vectorspace spanned by c. Then is spanned by the set of n-tuples (g_1, … , g_n) in c^n, and this set carries a natural action of the braid group, where twining strand i past strand i+1 corresponds to the permutation

So for each n we have a representation of the braid group Br_n, and it turns out that everything we desire would be downstream from good bounds on

So far, this is the same strategy (expressed a little differently) than was used in our earlier paper with Akshay Venkatesh to get results towards the Cohen-Lenstra conjecture over F_q(t). That paper concerned itself with the case where G was a (modestly generalized) dihedral group; there was a technical barrier that prevented us from saying anything about more general groups, and the novelty of the present paper is to find a way past that restriction. I’m not going to say very much about it here! I’ll just say it turns out that there’s a really nice way to package the cohomology groups above — indeed, even more generally, whenever V is a braided vector space, you have these braid group actions on the tensor powers, and the cohomology groups can be packaged together as the Ext groups over the quantum shuffle algebra associated to V. And it is this quantum shuffle algebra (actually, mostly its more manageable subalgebra, the Nichols algebra) that the bulk of this bulky paper studies.

But now to the story. You might notice that the arXiv stamp on this paper starts with 17! So yes — we have claimed this result before. I even blogged about it! But… that proof was not correct. The overall approach was the same as it is now, but our approach to bounding the cohomology of the Nichols algebra just wasn’t right, and we are incredibly indebted to Oscar Randall-Williams for making us aware of this.

For the last six years, we’ve been working on and off on fixing this. We kept thinking we had the decisive fix and then having it fall apart. But last spring, we had a new idea, Craig came and visited me for a very intense week, and by the end I think we were confident that we had a route — though getting to the present version of the paper occupied months after that.

A couple of thoughts about making mistakes in mathematics.

I don’t think we really handled this properly. Experts in the field certainly knew we weren’t standing by the original claim, and we certainly told lots of people this in talks and in conversations, and I think in general there is still an understanding that if a preprint is sitting up on the arXiv for years and hasn’t been published, maybe there’s a reason — we haven’t completely abandoned the idea that a paper becomes more “official” when it’s refereed and published. But the right thing to do in this situation is what we did with an earlier paper with an incorrect proof — replaced the paper on arXiv with a placeholder saying it was inaccurate, and issued a public announcement. So why didn’t we do that? Probably because we were constantly in a state of feeling like we had a line on fixing the paper, and we wanted to update it with a correct version. I don’t actually think that’s a great reason — but that was the reason.When you break a bone it never exactly sets back the same way. And I think, having gotten this wrong before, I find it hard to be as self-assured about it as I am about most things I write. It’s long and it’s grainy and it has a lot of moving parts. But we have checked it as much as it’s possible for us to check it, over a long period of time. We understand it and we think we haven’t missed anything and so we think it’s correct now. And there’s no real alternative to putting it out into the world and saying we think it’s correct now.