mc

@mc@mathstodon.xyz

The Twitter account formerly known as https://mathstodon.xyz/@mattecapu.
Expat 🇮🇹→🏴󠁧󠁢󠁳󠁣󠁴󠁿, PhD @mspstrath, mathematician (category theory for general systems theory & cybernetics), socialist, aspiring psycohistorian, meme enjoyer (shitpost is not endorsement).
Custodian of CT Zulip, email me for an invite.

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

You can win $1,000,000 for proving the Hodge Conjecture - it's one of seven Millennium Prize problems.

But if you want to become a millionaire, this is one of the hardest ways. So it's better to work on this for the love of math. Indeed, the only person who has won a Millennium prize so far turned it down, and still lives in his mom's apartment!

So what's the Hodge conjecture? It says roughly that for a smooth complex projective variety, all the rational homology classes that could possibly be represented by linear combinations of subvarieties actually 𝑎𝑟𝑒.

(Do you feel that million-dollar prize slipping further away already?)

I could explain what this means a bit better, but Frank Calegari does a great job here:

• Motives and L-functions, Section 4, https://www.intlpress.com/site/pub/pages/journals/items/cdm/content/vols/2018/0001/a002/

So I'll just rhapsodize on what it means that the Hodge Conjecture is still unsolved. It means the human race is profoundly ignorant about how polynomials are connected to topology. For smooth manifolds - the playground for differential geometry - we know a shitload about which homology classes can be represented by smooth submanifolds. But for projective varieties - the playground of algebraic geometry - we are comparatively clueless about the analogous question.

And in sense, the reason is that polynomial functions are a lot less flexible than smooth functions. You can't bend a polynomial in just a small region while leaving it alone elsewhere. So algebraic geometry is a lot further from topology than differential geometry is. It imposes a lot of extra constraints, and we don't fully understand the implications of those constraints.

Sorry, no really serious math in this post, just chat....

mc,

@johncarlosbaez Is Hodge's conjecture considered plausibly true or false?

mc,

@johncarlosbaez hehe Tarski would laugh his ass off reading the first two sentences

really cool, I didn't know Hodge's conjecture had such a clear intuition btw. new fav millenium problem as far as I'm concerned.

mc, to random

there must be something really profound hiding in the fact no good definition of 0*oo is possible, as if the structure of the extended reals wanted to be something more

mc,

what bugs me is that both [0,1] and [1,oo] are closed monoidal and they seem they'd like to join to form a compact monoidal (group) [0,oo] but instead they don't because there's no way to define 0*oo..!
it seems like the two possible results, 0 and oo, want to be identified, but of course if you do this you end up with identifying everything because of the transitivity of the order.
it feels as if the group structure of [0,oo] is only 'locally' defined in some sense... maybe @johncarlosbaez knows somethis about this problem?

mc,

@johncarlosbaez I guess the answer to your question should be the linear relation embedding the origin. that's interesting... I contemplated the idea that when we say 0*oo is 'undefined' we literally mean there is an extra point of [0,oo], which is its own connected component in the order. I disliked this because it feels ad-hoc and clashes with the nice feeling of completeness of the projective line.
but maybe I can like it more if I motivate it in the way you explain: 'undefined' is actually the 0 relation R->R 🤔

mc,

@johncarlosbaez I wonder if Shulman's paper on *-autonomous envelopes is what I'm looking for: https://arxiv.org/abs/2004.08487
Shulman describes the universal way to make a closed symmetric monoidal category with distinguished limits, colimits and powers compact, and this seems to be my problem
I'll see if I can parse the Chu construction he uses

mc, to random

type theory frees us from the prison Cantor built for us

mc,

@johncarlosbaez I agree! The OP was a tongue-in-cheek joke.
As you and Lawvere said, category theory is more a way to work with things, such as sets, than a replacement. I too don't like going around saying sets are inherently sinful (fun fact, same for prisons!). However, there is a different, but related sense in which sets are limiting though: they lack any geometry to them and freeing us from the Cantor paradise unlocks a much more beautiful paradise.

mc, to random

Quantitative logic is taking off
https://arxiv.org/abs/2402.03543

mc, to random

american psycho business card scene but it's mathematicians comparing latex typesetting

mc, to random

Reflections from factorization systems: https://www.localcharts.org/t/reflections-from-factorization-systems/13111

mc,

@johncarlosbaez :) thanks!

mc, to random

I've written (and drawn!) up an explanation of a cute fact @dylan and me worked on a while ago, namely that Tambara modules are literally modules for the action of a monoidal profunctor:
https://www.localcharts.org/t/tambara-modules-are-modules/12745

mc,

@BartoszMilewski @dylan Yeah that's what I say, in passing, in the post too. Thanks for spelling it out! I'll link back to this toot of yours.

mc,

@BartoszMilewski @zanzi @dylan I'm almost illiterate in Haskell but I guess replacing (m, b)-> c in the above with somehting like (act m b) -> c where act is a function implementing the action should make it more general that just the cartesian self-action of Hask?

mc, to random

"why is it so dark"
"i'm in scotland"
"oh"

mc,

@johncarlosbaez in the dark, the only thing we have is each other

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

The headline screams

"𝗦𝗶𝗻𝗴𝘂𝗹𝗮𝗿𝗶𝘁𝗶𝗲𝘀 𝗱𝗼𝗻’𝘁 𝗲𝘅𝗶𝘀𝘁,” 𝗰𝗹𝗮𝗶𝗺𝘀 𝗯𝗹𝗮𝗰𝗸 𝗵𝗼𝗹𝗲 𝗽𝗶𝗼𝗻𝗲𝗲𝗿 𝗥𝗼𝘆 𝗞𝗲𝗿𝗿

What's up with that?

As usual, it's less of a big deal than it looks. But Kerr is smart: he's the guy who first found solutions that describe rotating black holes. And the article is nice and clear. So it's more interesting than the usual nothingburger we've come to expect from screaming headlines about fundamental physics.

Kerr is talking about Penrose's famous singularity theorem, the one Penrose won the Nobel prize for. This says roughly that if space is infinite in extent, and light becomes trapped inside some bounded region, and no exotic matter is present to save the day, general relativity predicts that either a singularity or something even more bizarre must occur.

Of course I'm not stating the theorem precisely here. Each of the vague terms I just used must be made precise. But what's Kerr claiming?

Kerr isn't claiming Penrose's result is false. Instead, he's doing two things.

First, he's pointing out that the definition of "singularity" used in the theorem is not the only definition possible. A "curvature singularity" is - very roughly - a place where the curvature of spacetime approaches infinity. But that's not the kind of singularity that Penrose was talking about! Instead, he was talking about a place where the path of a particle can suddenly end. This may or may not be a curvature singularity.

This is not news.

Second, Kerr is arguing that rotating black holes have some singularities in Penrose's sense that aren't curvature singularities. Unfortunately this argument seems to be wrong.

(1/3)

https://bigthink.com/starts-with-a-bang/singularities-dont-exist-roy-kerr/

mc,

@johncarlosbaez "A "curvature singularity" is - very roughly - a place where the curvature of spacetime approaches infinity. But that's not the kind of singularity that Penrose was talking about!"

Reading this, I felt like scicomm failed me a bit. I guess it's not to be expected everyone has the math background to understand the difference between these two kinds of singularities, but engraving the image of black holes as bottomless pits in everyone's minds is misleading in a way that can probably be avoided.

mc,

@johncarlosbaez oh, whoops! I feel better then :)

mc, to random

real-life bootstrap causality paradox is when you misunderstand someone as giving you a great insight and then it turns out they were say something else and you end up explaining them what you thought was their idea

mc, to random

If you're learning category theory, you might have stumbled on the adage 'categories are monads in spans'. Well, here's a detailed explanation of this fact: https://www.localcharts.org/t/categories-are-monads-in-spans/11397

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

David Bennett has a great video reminding us that outside the English-speaking world, most people don't call musical notes "C D E F G A B C". Instead they call them "do re mi fa so la si do", which is a system called "solfège" - or they use other systems.

But I was a bit shocked to see his map showing "no data" on what people do in most of Africa, or southeast Asia, or Korea, or Mongolia, or Pakistan, or Turkmenistan, or Uzbekistan.

Mind you, I don't hold it against him. He's an independent video-maker struggling to earn a living explaining music theory. He probably can't afford to spend hours and hours researching every country. But it saddens me to see this map.

I bet some of you here know the systems for naming notes in these greyed-out countries! Please tell us! And if you can find or make a better map, please do.

David Bennett's video is here:

https://www.youtube.com/watch?v=MVA8bgSBt5A

It has some nice info on the history of the solfège system and how it's used differently in English-speaking countries.

mc,

@johncarlosbaez what I don't understand is why A corresponds to La, which is the fourth note in solfege

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

Here you see three planets. The blue planet is orbiting the Sun in a realistic way: it's going around an ellipse.

The other two are moving in and out just like the blue planet, so they all stay on the same circle. But they're moving around this circle at different rates! The green planet is moving faster than the blue one: it completes 3 orbits each time the blue planet goes around once. The red planet isn't going around at all: it only moves in and out.

What's going on here?

The answer goes back to 1687, when Newton published his Principia Mathematica. This book is famous, but in Propositions 43-45 of the first book he did something that people never talked about much — until recently.

He figured out what additional force, besides gravity, would make a planet move like one of these weird other planets. It turns out an extra force obeying an inverse cube law will do the job!

This may seem bizarre and irrelevant to real-world physics. But it's not: the centrifugal force obeys an inverse cube force law!

That is: if you sit at the Sun and spin around, you'll seem to see planets rotating around the Sun faster, or slower, than they usually do. And they will seem to be pulled away from the Sun by a force. This is the centrifugal force: a fictitious force caused by a rotating coordinate system.

Puzzle: can the weird motions of both the red and green planets be explained by an additional force pulling the planets outward, like the centrifugal force — or do we need to invoke an inward-pointing force in one case? If so, which case?

For more details — but not the answer to this puzzle — go here:

https://math.ucr.edu/home/baez/inverse_cube.html

mc,

@johncarlosbaez @BartoszMilewski somehow I made it past three different mathematical physics courses without realizing the meaning of this effective potential

mc,

@johncarlosbaez @BartoszMilewski Indeed, I realized it now. Thanks! :)

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

Though part of me - the worst part - would like to join the clever crowd who endlessly pontificate and interview each other, I'm held back by my intense aversion to publicly talking about:

  1. consciousness
  2. free will
  3. string theory and other theories of everything
  4. are mathematical objects real?
  5. is reality a simulation?
  6. interpretations of quantum mechanics
  7. quantum computers
  8. large language models, machine learning, AI

and most other topics that the "digiterati", the "intellectual dark web", and other quasi-scientific talking heads enjoy bloviating about. I'd much rather curl up with a good solid book on the life cycle of lichens, or the organizational structure of car repair shops.

mc,

@johncarlosbaez lmk what you find out about the organizational structure of car repair shops, logistics of businesses is something I often daydream about

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