Crystalline prisms

Week of Monday, June 5, 2023 #

This week I've been stuck trying to prove something that I left til the last stages of writing my paper, because I thought it would be very quick to prove, and it's turning out to be extremely difficult.

Roughly speaking, I have these two axes, and I have this operation that pulls you towards the $x$-axis. If you're already on the $x$-axis, it does nothing. And I need to show that, although this operation pulls you toward the $x$-axis, it can't pull you in if you're not already there.

(Math ppl: I want to show that a prism is crystalline if and only if is crystalline)

The reason I'm doing this is that there's these mathematical objects called prisms that were invented in 2019, and have been pretty revolutionary. Many problems that were considered untouchable for decades have been solved just in the past few years, largely coming out of this circle of ideas. I'm developing a different perspective on prisms, which is somehow based on the rotational symmetries of the circle.

In order to justify my perspective, I have to explain how to see various aspects of the theory of prisms in my language—actually, not just show that it's possible, but that it's natural to do so. In the diagram above, I have a natural candidate for prisms which "lie on the $x$-axis" (called crystalline prisms), but it turns out to actually correspond to things whose twist ends up on the $x$-axis. So I have to show that this is actually the same as already being on the $x$-axis.

For comparison, there wasn't any existing concept in the "symmetry world" to correspond to prisms on the $y$-axis (called transversal prisms). But I invented (or discovered) something to correspond to these, and these turned out to be something extremely fundamental that no one had noticed before, and which massively simplify a lot of things in "symmetry world".

Wednesday, June 7, 2023 #

Anyway, I've been trying and failing to prove this for several days. I messaged Ben on Discord to explain my problem and ask if he had any ideas. Ben suggested I first try to prove this for transversal ($y$-axis) prisms, then extend from that case.

This is a standard technique: often you can use the transversal guys to control all the others. But it doesn't work in this case, basically because the only thing which is on both the $x$-axis and the $y$-axis is $(0,0),$ and that's boring. So I told Ben "no, that will never work", he said "oh yeah, whoops, sorry that was silly", and I said "no worries, I totally tried the same thing before realizing it was nonsense".

Now, there's a prime number involved in all of this, and I have managed to make this work for the prime number $2.$ There's a bunch of stuff in this field that works for any prime number except $2,$ so maybe I could use that to deal with the remaining primes? But that stuff goes over my head, so I spent a few hours trying to understand it better.

(Math ppl: I'm referring to obtaining prismatic cohomology from -de Rham cohomology, e.g. Proposition 4.8.8 of Absolute prismatic cohomology)

Thursday, June 8, 2023 #

The thing I'm trying to prove is obviously false. Like, one of the most basic examples of a prism is a counterexample. Jesus I'm dumb.

(Math ppl: I'm referring to the Hodge-Tate point of , coming from , which is a counterexample since .)

Now, if the answer to "what do crystalline prisms correspond to?" isn't what I expected, then that question becomes much more interesting. It's interesting, but it's a lot more work, and I just want this ****ing paper to be done.

Wait—actually maybe this isn't a counterexample. In this subject, there are static things and animated things. This counterexample is one of the standard examples of an animated prism, but the question I'm working on is really only about static things. So I did some googling, and yes, this counterexample will never be static. So we're back in business!

Friday, June 9, 2023 #

SUCCESS!!!

I proved what I wanted. It was a little bit subtle: it seemed like you should be able to do the proof in either of two ways, but actually one way was slightly better, and that was exactly what I needed.

I excitedly messaged Noah on Discord, and said "you do this and this and this, and after sufficiently many shenanigans, you get what you want." And Noah said, "hmm, I'm not seeing the shenanigans", and I told him, "it's a bit sneaky, I'll write something up tonight".

But it was time to celebrate, so I biked over to the Brooklyn Society for Ethical Culture for board game night. My team won at Secret Hitler (go Fascists!), and then at midnight we went to Union Hall for karaoke. I got home around 3:30am, so I didn't get back to Noah.

Saturday, June 10, 2023 #

Woke up sometime after noon, started writing up the proof to send to Noah. That thing where it seemed like the proof should work in either of two ways, but actually it only worked in one? Yeah it actually just doesn't work. I'd gotten my head so tangled up in this problem that I ended up making a circular argument.

There's no happy ending to this story. I put out the paper in September, and I just left this as a conjecture.