hallasurvivor

@hallasurvivor@sunny.garden

math/music. they/them.

I'm a PhD student interested in the intersection of algebra, geometry and logic. An intersection often made clearer with the language of category theory.

Leftist

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hallasurvivor, to math

It's been a second since I posted a little . So here's a cute one:

It's "easy to see" that

∑ 1/(a^2 + b^2)

diverges (here a,b ∈ ℕ).

(If this isn't easy to see, consider this the puzzle!)

What happens if you restrict attention to coprime a,b?

Does this sum converge? If yes, can you say what it converges to (maybe using the riemann zeta function)? If no, can you say how quickly it diverges?

I'll post an answer in the next few days, but I'm excited to see what people try! Feel free to boost so more people can get involved ^_^

hallasurvivor, to music

Achievement unlocked!

I was messing around at a piano, and a harmonic move (which I know theoretically and practically should sound fine) sounded weird.

Turns out I had voiced it with parallel fifths, and playing it again with a different voicing totally solved the problem!

I've always known some arguments for avoiding parallel fifths (only in certain settings, ofc), but this is the first time I've played something and really HEARD an issue with it.

hallasurvivor, to random

I wish this guy had just kept listing names, lol.

hallasurvivor, to math

New blog post is out!

It turns out that 37 is the "median second prime factor"!

By this I mean, if you look at the second smallest prime factor of a large number N, it is < 37 about half the time, and > 37 about half the time!

This fact stunned me when I first read it, so I had to figure out why it was true. If you're also curious, you can read all about it here:

https://grossack.site/2023/11/08/37-median

#math

hallasurvivor,

@ellie

It could be neither! It's an incredibly surprising fact, and I think it's getting a lot of attention precisely because it pushes up against a lot of peoples intuition

hallasurvivor,

@johncarlosbaez @ellie

What a fun fact!! The computer search, more than 2 :P.

I was recently thinking about how easy it is to find a large gap between primes, because someone on mse was asking about a subsetsum variant where you want to sum to any prime. It's not obvious how to reduce classical subsetsum to this problem without a big prime gap. But those are less than polynomially common, so my idea didn't work!

hallasurvivor,

@johncarlosbaez @gjm

Yeah I'm sure that's right. For each prime p you can calculate what fraction of numbers have p as their kth prime factor. Call this λₖ(p).

Then you can ask for the median, which is the first prime p so that

λₖ(2) + λₖ(3) + λₖ(5) + ... + λₖ(p)

crosses 1/2.

You can also ask which prime occurs with highest density. That is, which p maximizes λₖ(p). Apparently 13 answers this "maximizing" question for λ₄, with 31/5005 being the fraction of numbers whose 4th prime is 13. Every other prime accounts for a smaller fraction of numbers!

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

A grasshopper lands at a random point on a planar lawn of area one. It then jumps once, a fixed distance 𝑑, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper lands on the lawn again?

Surprisingly, the lawn should never be shaped like a disk! Here's what it should look like for various choices of 𝑑. For larger values of 𝑑 it gets even weirder.

• Olga Goulko and Adrian Kent, The grasshopper problem, https://arxiv.org/abs/1705.07621.

Follow @esoterica to learn more surprising results in mathematics!

hallasurvivor,

@johncarlosbaez @esoterica

For large enough d, is your region ever disconnected?

hallasurvivor, to math

You know her and love her, it's the Prototypical Pumpkin Domain!

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

I'm really bad at programming... so:

Could someone work out the continued fraction expansion of ln(3)/ln(2) and use it to generate a bunch of good rational approximations of this number? Wolfram Alpha easily gives me the continued fraction expansion below, but for music theory I want the rational approximations it gives.

Like this:

1 + 1/1 = 2
1 + 1/(1 + 1/1) = 3/2
1 + 1/(1 + 1/(1 + 1/2)) = 8/5

The last one means that in a 5-tone equal-tempered scale, going up 8 notes is close to going up an octave and a fifth. This is one reason the 5-tone equal-tempered scale, called a pentatonic scale, is popular in places like Bali.

1 + 1/(1 + 1/(1 + 1/(2 + 1/2))) = 19/12

This means that in a 12-tone equal-tempered scale, going up 19 notes is close to going up an octave and a fifth. This is one reason the 12-tone equal-tempered scale is popular in the West - and by now everywhere!

What comes next? Well, I worked out the next one, but I'd really like to see dozens more of these fractions.

Also: if you're a number theorist, maybe you know theoretical results about how good these approximations get, i.e. how close to ln(3)/ln(2) a fraction with denominator N can possibly be. I fear only the usual facts are known, e.g. there are infinitely many fractions M/N in lowest terms such that

|ln(3)/ln(2) - M/N| < 1/N²√5

This is true for every irrational number, not just ln(3)/ln(2):

https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)

By the way, the continued fraction expansion I'm getting from Wolfram Alpha is

[1; 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, 4, 3, 1, 1, 15, 1, 9, 2, 5, 7, 1, 1, 4, 8, ...]

in the usual shorthand. So, when we get to 23 and 55 we should get equal-tempered scales with really accurate fifths.

hallasurvivor,

@johncarlosbaez

Sagemath will do this for you easily. Here's some code:

cf = continued_fraction(ln(3) / ln(2))

# get the first 10 rational approximations  
for n in range(1,10):  
 print(cf[:n].value())

You can copy/paste this (without the ```s) into a sagecell here, for instance, to play around with it without having to download sage:
https://sagecell.sagemath.org/.

In particular, you can see that cf[:10] evaluates to
[1; 1, 1, 2, 2, 3, 1, 5, 2, 23]
so we would expect cf[:10].value() to be a very good approximation. Sage says it's equal to
24727 / 15601
which sage says is only 1.68 x 10^{-9} off the true value!

hallasurvivor,

@johncarlosbaez

I use it all the time! But I "grew up" with it and used it throughout my undergrad to work on problems (and to save me having to do linear algebra by hand, haha)

hallasurvivor,

@johncarlosbaez

I wonder how easily the human ear can distinguish notes that close to each other... Even a 10 cent difference is pretty hard to hear. I have to imagine that's a bigger interval than a 1/15601th of a perfect 12th, but I haven't checked how the two sizes compare.

Edit: Unless I did my math wrong (which I'm completely willing to believe) it looks like one 15601-step should be like 0.07 cents, which for all intents and purposes is the same note, haha.

hallasurvivor, to random

Do you ever think about the fact that there's no complex structure on the 4-sphere? Wild. More generally there's no complex structure on any sphere but the 2-sphere (and possibly the 6-sphere, for which I think this is open).

This makes spheres rather tricky to study from the perspective of algebraic geometry. This is weird to me because like... they're spheres, haha. Their real locus is extremely easy to describe algebraically, but for some reason they're more difficult than one would first expect. See here for instance:

https://mathoverflow.net/questions/115657/what-is-the-algebraic-geometry-version-of-the-spheres

hallasurvivor,

I haven't thought about complex structures on spheres, or the fact that spheres aren't a common algebo-geometric example (we opt for projective space instead) until tonight. And I 𝘤𝘦𝘳𝘵𝘢𝘪𝘯𝘭𝘺 haven't thought about "motivic spheres", whatever those are.

But this seems like something @johncarlosbaez has likely thought about...

hallasurvivor,

@johncarlosbaez

I feel like i've read a lot of blog posts about homotopy groups of spheres, and about the nonexistence of lie group structures on various spheres. I guess I mistakenly thought you'd also written about (the nonexistence of) complex structure on spheres!

I will say, the proof isn't that hard if you're willing to say "not a complex variety" instead of "no complex structure":

Every (projective) complex variety is symplectic, so [ω] ≠ 0 ∈ H²(X). But most spheres have H²(Sⁿ) = 0!

hallasurvivor,

@johncarlosbaez

Everything to do with the word "motivic" scares me, haha. Someday I'll get around to thinking about motives (since they look interesting and useful, and they're closely related to a lot of stuff that I have thought about), but I should probably spend some time thinking about the usual etale cohomology first!

hallasurvivor, to random

@johncarlosbaez

I just read your blog post about the "free 2-rig" and schur functors, and I can't help but laugh at the timing. At the end of the post you show that (for a 2-rig R) every natural endofunctor R --> R is a (schur) polynomial, and you compare this to the decategorified case of every natural function R --> R (for a ring R, now) is a polynomial.

(Also, I'm sure you know this, but for other readers, in both cases here "natural" is meant in the technical sense of "defined uniformly for all 2-rigs (resp. rings) R")

The timing is funny because @tao recently posted a blog post of his own working out the decategorified version in detail!

And for anyone who hasn't seen them yet, here are the two posts I'm referencing:

https://golem.ph.utexas.edu/category/2023/10/the_free_2rig_on_one_object.html

https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-an-identification-of-form-and-function-the-case-study-of-polynomials/

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

If you draw all roots of all polynomials whose coefficients are ±1, you get an amazing picture that raises lots of challenging puzzles!

I really hope someone reads our short article:

https://www.ams.org/journals/notices/202309/rnoti-p1495.pdf

and solves the main puzzle: why do the fractal regions of this set look so much like "dragon sets"? We have a good heuristic explanation, but no proof yet.

People sometimes get excited about math when they learn about fractals, and then disappointed when they discover rather few professional mathematicians prove theorems about fractals. If you ever wanted to prove a cool theorem about fractals, this could be your chance!

If you read part 2, I'll show you what I'm talking about.

(1/2)

hallasurvivor,

@johncarlosbaez

This is super exciting! When I was an undergrad the BIG theorem I was trying to prove was related to roots of polynomials with ±1 coefficients. In particular, I needed to show that for every polynomial p of a certain form there was a ±1-polynomial congruent to -1 mod p.

I actually had sage draw a TON of pictures that looked exactly like your "dragon sets", and the fractal tiles from your second post look like something Lagarias and Wang were working on that my advisor turned me onto.

Anyways, thanks for the throwback! Here's a picture I made from that era. I completely forget what it's supposed to mean:

hallasurvivor,

@johncarlosbaez

Destiny is funny like that! I don't remember, but if this page (or something like it) existed in 2018 the odds are VERY high that I saw it back then!

Also I assume that's @jdchristensen who originally made the pictures? What a small world, since we ended up meeting through HoTT.

hallasurvivor, to ComputerScience

Just watched a characteristically great talk by @jonmsterling on Synthetic Domain Theory. It's extremely clear, and if anyone is looking for a great reason for people who care about to care about theory, I highly recommend it (though maybe people who follow me on mastodon are more likely to already have good reasons of their own, haha)

https://www.youtube.com/watch?v=lLvweTSmR40

hallasurvivor, to random

Literally every time I feel crummy I think "I always feel better after I play music, so I guess I'll go play music". Somehow I never expect it to work and it always does.

It's like magic how quickly it turns my mood around, haha. Less than 5 minutes at the piano and I'm IN IT and I leave feeling so much better

hallasurvivor, to random

Oh shit. I had no idea we'd made it so precise!

hallasurvivor, to random

Here's a fact I just learned that's obvious in hindsight (for a certain definition of "obvious"):

A G-set is the same thing as a (set) bundle over BG!

I've actually made this same observation a few times before, with different language each time. For instance, since a "sheaf/bundle on X" is an element of Sh(X), this is the same observation as viewing the topos of G-sets as the topos of sheaves on BG.

hallasurvivor, to math

Huh. I've never seen a word cloud in a paper before.

hallasurvivor,

@johncarlosbaez

I don't know the Abrose-Singer theorem, but is this a case of modern language making an old result look easier than it is?

You can view a bundle E over X as a function sending each point x to the fibre E(x) over that point. Then a connection gives you a way of turning a path α : x → y into an isomorphism E(α) : E(x) ≅ E(y). As I understand it, the definition of "flatness" says exactly that E(α) = E(β) for homotopic paths α and β!

So in case E has a flat connection, E(-) becomes a functor Π₁ → Vect from the fundamental groupoid, and provided X is connected, Π₁ is equivalent to the fundamental group π₁(X,x) for your favorite basepoint x, so that E(-) is equivalent to an action of π₁(X,x) on E(x).

Maybe there's some analytic technicalities that I'm missing out on that shows this "really works"?

highergeometer, to random
@highergeometer@mathstodon.xyz avatar

I'm looking forward to finding out what Scholze wants to talk about in:

Title: What does Spec Z look like?
Abstract: "Mazur observed that the etale cohomology groups of Spec Z indicate that it looks like a 3-manifold, and for each prime p the closed subset Spec F_p​ of Spec Z looks like a circle. Deninger suggested that moreover, there should be an action of the reals on Spec Z, with the periodic orbits being precisely those Spec F_p​'s, each one becoming an orbit of length log⁡plogp. More recently, there is growing evidence that there should be a version for Spec Z of the function field theory of shtukas, which should relate to the Langlands correspondence and yield a cohomology theory close to the theory of motives. While all of this remains unrealized, I will indicate a geometric framework which at least has the potential to allow for a rather faithful realization of this picture; in particular, Deninger's expected action of the real numbers is realized naturally. Restricted to the p-adic or real part of Spec Z, it can be made rather precise, and I will indicate relations to the Fargues-Fontaine curve, the twistor projective line, as well as p-adic and complex Hodge (or twistor) theory, and related theories of local shtukas and geometric versions of local Langlands correspondences."

https://www.mpim-bonn.mpg.de/node/12330

hallasurvivor,

@highergeometer @johncarlosbaez @francisbrb

Thanks for the link!

Around the 25m mark, Scholze mentions that this winter he and Clausen will be coteaching a course (which will be livestreamed!) on the main technical tool of the talk ("analytic stacks").

I wonder if those lectures will be recorded too. I know I would love to watch them, and I'm sure others would too.

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