Giants 15, Brewers 1

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.

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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 they are typically graphs we hope are expanders.) Nor did “superduperstrong approximation” catch on.

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Learning a made-up language, III

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.

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Lecture and egotism

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 compliment 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!

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Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields: a new old paper

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 V^{\otimes n} 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

(g_1, \ldots, g_n) \rightarrow (g_1, \ldots, g_{i+1}, g_{i+1}^{-1} g_i g_{i+1}, \ldots, g_n).

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

\dim H^i(Br_n, V^{\otimes n})

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.
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I’d like to make a request, II

In re my last post about WIBA Madison’s Classic Rock; a couple of days later I was listening again and once again the DJ was taking listener calls, but this time it was because he was angry that McDonald’s was using Cardi B as a spokeswoman; he wanted the listener’s opinion on whether Cardi B indeed represented, as McDonald’s put it, “the center of American culture” and if so what could be done about it. Nothing, the listeners agreed, could be done about this sad, the listeners agreed, state of affairs. It has probably been 20 years since I heard the phrase “rap music” uttered, certainly that long since I heard it uttered so many times in a row and with such nonplus.

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I’d like to make a request

I was listening to WIBA 101.5 Madison’s Classic Rock in the car while driving home from an east side errand and heard something that startled me — the DJ taking requests from listeners calling in! Now that startled me — why wait on hold on the phone to talk to a DJ when in 2023 you can hear any song you want at any time, instantly?

And then I thought about it a little more, and realized, it’s not about hearing the song, it’s about getting other people to hear the song. Like me, in the car. 2023 is a golden age of listening to whatever you want but is an absolute wasteland for playing music for other people because everybody is able to listen to whatever they want! So there’s much less picking music for the whole room or picking music for the whole city. But at WIBA they still do it! And so listeners got to play me, in my car, this song

and this song

neither of which was really my cup of tea, but that’s the point, radio offers us the rare opportunity to listen to not whatever we want.

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“The pandemic made things worse.”

The Hill reports on a Pew Research study showing high proportions of Americans without romantic partners:

Recent years have seen a historic rise in “unpartnered” Americans, particularly among the young. The pandemic made things worse.

Here’s the Pew report the Hill piece draws on, which says:

Roughly six-in-ten young men report being single. Overall, three-in-ten U.S. adults are single, meaning they are not married, living with a partner or in a committed romantic relationship. This share has not changed since the Center asked this question in 2019.

Always click the links to see what the study really says!

As for me, I’d estimate I was single somewhere between 50-60% of the time between the ages of 18 and 29.

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Learning a made-up language, II

Following up on yesterday’s post: one could also ask whether GPT-3 can compose in a made-up human language. For instance, you could take English and modify it a bit:

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.”

In other words, the new rules are that verbs go to the end of the sentence, all verbs take an auxiliary verb indicating tense, adverbs go between auxiliary verb and main verb. Is this short text enough for GPT-3 to locate context in a language that isn’t one of the languages in its training set but rather “English with modifications inferred from prompt?”

Not yet:

Write more dialogue where Henderson tells you about his day.

GPT-3: “I then went to the park and did some reading. The sun was shining and it made me feel so relaxed. I saw some birds flying about, and it was quite a sight. After that, I went to the beach and enjoyed the sound of the waves. I even went for a swim! Finally, I returned home and watched some television before going to bed.”

It’s totally possible that more prompting would make this work, though – I didn’t have the patience to manually type in a bunch more Gravinic-inflected English. I’m not actually sure what kind of context GPT-3 is placing itself in with this response.

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Learning a made-up language

LLMs seem to be really good at swift, reliable context location — e.g. it takes only a little bit of Python or romance novel or instruction manual for the LLM to recognize a context which requires specific choices of syntax and diction. I wondered though whether GPT-3 was recognizing the context of “Python” or a more general context of “programming language” which is further fine-tuned to a subcontext of Python. It seems like it’s the latter! Here was my way of testing. I gave it the prompt

The following is code in a programming language called Ranger that computes the first n triangular numbers.

ask('Please enter a number: '),n)

#initialize the triangular number

triangular_number : 0

as i ranges to n+1
triangular_number : triangular_number + i
write i then triangular_number

Ranger doesn’t exist: I just made something up where for loops are written “as i ranges to” and assignment of variables is done with a : and you don’t enclose the argument of a print (or “write”) statement in parentheses. To be fair, it’s basically python with some dictionary changes. Still, I thought I would see if GPT-3 can write code in Ranger even though there’s no Ranger code in its training set and so no actual “Ranger context” to locate. And it can!

Write a program in Ranger to compute the 10th prime number.

#initialize the prime number
prime_number : 1

#initialize the count

count : 0

#iterate from 2 to infinite

as i ranges to infinite
is_prime : true

as j ranges to i-1
    if i % j is 0
        is_prime : false

if is_prime is true
    count : count + 1

    if count is 10
        prime_number : i

write prime_number

So you can see that GPT-3 correctly infers that the right context is “C/Python-like code” and correctly recognizes from the prompt that : means assignment, “as … ranges” is how you do a for loop, etc. It’s interesting that it’s way of testing primality is to check divisors of i all the way up to i; surely there isn’t a lot of existing code in the training set that does it in this massively inefficient way!

To be fair, this is an easy problem, and if you give it a somewhat harder one, it does this:

Write a program in Ranger to count the number of binary strings of length 10 which have no occurrence of the substring ‘0101’.

# Initialize the number of binary strings
num_binary_strings : 0

# Iterate over all binary strings of length 10
as binary_string ranges to 2^10
    # Check if the binary string contains the substring '0101'
    if binary_string does not contain '0101'
        num_binary_strings : num_binary_strings + 1

# Print the result
write 'Number of binary strings of length 10 which have no occurence of the substring "0101": ', num_binary_strings

I guess this is sort of pseudocode? It doesn’t really know how to iterate over binary strings but knows there are 2^10 of them so it just “ranges” to that. Nor does it know how to check string containment in Ranger (how could it?) so it switches to English. Not a bad answer, really!

It would be interesting to try something like this where the invented language is a little more different from existing languages than “Python with some 1-for-1 word and symbol changes.”

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