Reflections on YTM

Epistemic status: low. I wrote this in an airport, in a fury, straight after YTM 2026 (the Young Topologists Meeting), most of which I spent nodding along to things I didn’t understand and suspect nobody else did either. Mostly it’s me rambling about a bad week, with maybe one real idea buried in it.

The setup

Imagine you’re a musician at a music festival. Everyone’s supposed to share their own composition, so that the others can appreciate it.

(∗ 1st talk begins ∗: I apologize, I won’t have time to explain everything, but I’m happy to answer questions afterwards.)

So, the main idea is that certain music scales, studied by Pseudo-famous person 1 and Pseudo-famous-person-only-in-a-very-niche-context 2 (both dead, or retired), have been studied for fifteen years now. For no reason other than that someone spent time and effort on them, and nobody else ever bothered to ask why.

Let \(=\ell^{\sharp}\). Then we can use the shifts \(\ell^{\sharp}[i]\) to construct the music scale in 8-TET, using a very important tool called cool-name #1, that we abbreviate with \(C_1^{\text{cool}}\).

(not fixing the other references, it’s obvious)

The theorem

So, if we piece it together (there’s not enough time to expand this):

\[ \Bigl(C_2^{\text{cool}}\bigl(C_1^{\text{cool}}(\ell^{\sharp}[i])\bigr)^{t}\Bigr) \;\oplus\; C_1^{\text{cool}}\bigl(lll(\ell^{\sharp}[6])\bigr)[m] \]

these are the 7 cones for [NHARTPOTM]¹.

Here \(n\) is a very important number that decides the pitch of the composition. This construction is very beautiful, because it lets us describe any other cohomology-pseudo, cool-name-1, cool-name-2, random-name-that-I-just-came-up-with-because-it’s-cute. This whole collection was used by Random-name #2 to compose the national anthem (1992) of a state that doesn’t exist anymore. They say it was so beautiful that people used to cry every time, and end up hugging each other in an eternal sense of community, happiness and hope.

But let’s focus on the punchline now. If we set

\[ \circledast \;\equiv\; \operatorname{glzt}(C_{12})^{lll} \]

then

\[ \operatorname{glzt}(C_{12})^{lll-1} \;=\; \operatorname{glzt}(C_{12})[-6] \]

is the generally recursive pattern of the sequence. QED.

Questions

Question. I don’t know the work of Random person #1, and I haven’t read [NHARTPOTM], but I was wondering if you could share some of your thoughts on why \(\operatorname{glzt}(C_{12})^{lll}\) sounds the way it does, for a given \([m]\)?

Answer (branch 1). So, in practice this is very difficult to say. \(n\) varies in a log-\(t\) fashion with respect to a 197-TET note, chosen from the 7th quartile of the distribution of 18th equivariant triples. So, yeah. It’s a hard problem. But thanks for the question.

Answer (branch 2). Yeah, so, I know this has been applied in ornithology, to study a certain species of hummingbird: when they mate, they produce a similar sound, close enough to what [NHARTPOTM] (very vaguely) describes. But I don’t know, I don’t study birds.

Question 2. Is the composition \(\operatorname{glzt}(C_{12})^{lll}\) written for a string concerto, or a piano concerto?

Answer. So, it really depends on \([m]\). If \(n \geq (7+8k)^{lll}\), then yes, but only for violas that don’t have a \(\sharp\) string. If \(n \leq ((lll))^{lll}\), then the composition doesn’t fit any instrument at all. I know that’s kind of a big problem, but we’re working to solve it, at least for

\[ lll = lll\cdot\bigl((\wedge^{\sharp\cdot\sharp})_{b}\bigr)^{d} \qquad \text{in 88-TET.} \]

But seriously now

I think the point I was trying to make, through all this self-play, is that we’re so stranded we don’t even know how lost we are anymore. It’s like waking up on a life-raft in the middle of the ocean, and not being able to conceptualize what “lost” even means.

We can’t find ourselves in what we thought we knew. There’s nothing underneath. I guess this is the true meaning of lost: it’s about the void it leaves behind, not knowing what it was.

What is this helplessness that conferences like these ones leave on my skin?

It’s about losing grip on what the concepts really mean. Not their definitions, not quite their epistemological justification,² but their epistemological pedigree: what followed what, why \(B\) instead of \(B’\) or \(B’’\), what the design choices were, why they were necessary or even interesting.

\[ A \longleftarrow B \longleftarrow C \longleftarrow \cdots \]

And it’s about the why. Why we’re even solving the problem in the first place. About the fact that we don’t acknowledge the human biases, not even slightly, taking for granted that specifying the syntax is enough.

We stand on the shoulders of giants, but we never bother understanding the giants. We generalize and optimize things that don’t need to exist, when we can’t even justify them.

This bothers me. And it doesn’t seem to bother the others.

If I take this stance openly, it looks like I’m villainizing the people who keep this system of useless, auto-confined criticism running. But I couldn’t care less. I’m sure they’re nice, and passionate, and lovable people. But you cannot grow a plant with just love. And you cannot have a good community if you don’t foster healthy criticism.

The representability problem: in a place where you can replace anything with anything else, justifying the choice of representation should be mandatory.