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 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.

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.

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.

“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.

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.

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
        break

if is_prime is true
    count : count + 1

    if count is 10
        prime_number : i
        break

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.”

Uncontested touchdown

The Chiefs beat the Eagles in the Super Bowl and even people who liked the outcome didn’t like the ending. With a little under two minutes left, the score tied, and the Eagles out of time outs, running back Jerick McKinnon broke free and headed for the end zone; but he stopped at the 2 yard line and took a knee, setting up the opportunity for the Chiefs to run down the clock to nothing and then kick a chip-shot game-winning field goal. Had McKinnon scored a touchdown, the Eagles would have been 7 points down, but Jalen Hurts would have had the chance to try to make it back up the field, with no time outs left, and even the score.

That’s football! That’s what people want to see! Instead, we got the Chiefs taking a knee.

What if McKinnon weren’t allowed to do what he did? That is — what if a touchdown play that the defense chose not to contest simply counted as a touchdown? Or, more simply — what if the defense, at any time, were allowed to concede a touchdown as “uncontested,” give up 6 points and the conversion, and force a kickoff?

It would be kind of like the intentional walk of football. Or even more specifically the intentional bases-loaded walk, where you give up points on purpose to achieve a larger strategic goal. But the uncontested touchdown rule, instead of avoiding a thrilling faceoff between the tiring pitcher and the fiercest slugger, would avoid… the weak fart of an ending we just saw.

I think teams would hardly ever do this. In fact, I can’t really think of any situation where a team would do this other than the exact situation that came to pass in this Super Bowl. And I think the Super Bowl would have been better football if the rules had given Philadelphia this option.

Bounce

I was vacationing with the kids in San Francisco and it turned out for transit reasons to improve our day a lot to be able to store our suitcases somewhere in the city for the whole day. There is, as they say, an app for that, called Bounce. It’s a pretty clever idea! You pay $7.50 a bag and Bounce connects you with a location that’s willing to store luggage for you — in our case, a hotel (a budget option which is apparently famous for having the toilet just being out there openly in the room, to save space) but they use UPS locations and other stores too. A luggage locker at the train station would be cheaper, but of course that would mean you have to go to the train station, which might be out of your way.

Now the question is this — could I have saved some money and just shown up at a random hotel, handed the bellhop a twenty, and asked him to keep four bags in the back room for the day? Seems kind of reasonable. On the other hand, I can imagine hotels being under insurance instructions not to store bags for unknown non-guests. But why wouldn’t the same insurance caution keep them from signing up with Bounce? Maybe Bounce has taken on the liability somehow.

Anyway, this is not a service I anticipate needing often, but in a moment when it was exactly what I needed, it did exactly what I wanted, so I recommend it.

PS: My kids are now extremely into San Francisco.

Little did I know (real analysis edition)

I just finished teaching Math 521, undergraduate real analysis. I first took on this course as an emergency pandemic replacement, and boy did I not know how much I would like teaching it! You get a variety of students — our second and third year math majors, econ majors aiming for top Ph.D. programs, financial math people, CS people — students learning analysis for all kinds of reasons.

A fun thing about teaching outside my research area is encountering weird little facts I don’t know at all — facts which, were they of equal importance and obscurity and size and about algebra, I imagine I would just know. For instance, I was talking about the strategy of the Riemann integral, before launching into the formal definition, as “you are trying to find a sequence of step functions which are getting closer and closer to f, because step functions are the ones you a priori know how to integrate.” But do Riemann-integrable functions actually have sequences of step functions converging to them uniformly? No! It turns out the class of functions which are uniform limits of step functions is called the regulated functions and an equivalent characterization of regulated functions is that the right and left limits f(x+) and f(x-) exist for any x.

Compactness as groupwork

Compactness is one of the hardest things in the undergrad curriculum to teach, I find. It’s very hard for students to grasp that the issue is not “is there a finite collection of open sets that covers K” but rather “for every collection of open sets that covers K, some finite subcollection covers K.” This year I came up with a new metaphor. I asked the students to think about how, when a professor assigns a group project, there’s always a very small subgroup of the students who does all the work. That’s sort of how compactness works! Yes, the professor assigned infinitely many open sets to the project, but actually most of them are not really contributing. And it doesn’t matter if some small set of students could do the project; what matters is that some small group among the students assigned to the project is going to end up doing it! And this seems to happen — my students nodded in agreement — no matter how the professor forms the group.