Mathematicians find hidden structure in a common type of space | Hacker News Reader

Mathematicians find hidden structure in a common type of space

quantamagazine.org

107 pointspseudolus15 comments

An engineer, a physicist and a mathematician are sleeping in a hotel.

In the middle of the night a power surge happens and all three of their room kettles catch fire.

The engineer quickly locates the fire extinguisher and puts out the fire.

The physicist remembers from his college years that combustion feeds on oxygen, so he runs to the bathroom and covers the kettle with towels, putting out the fire.

The mathematician wakes up, sees the fire extinguisher in the corner and promptly goes back to sleep: he just proved that a solution exists.

More like a chemist.

I was just pondering whether this result would eventually be useful to physics or science generally in say ways like Galois' ideas and Group Theory eventually became useful. Just a thought.

Pondering as well... Machine learning should have some applications

The article read like they did the StarTrek game called Kal-To, where the goal is to find structure in randomness.

Not time to ponder this properly but have you ever played Dobble? it feels related to the way those card sets are constructed

Dobble works because the players are constructing the finite projective plane (P2) over a finite field (F7), so it is a (fun?) application of finite geometry. This paper seems to be an important result in finite geometry, so you're exactly right!

The paper is way over my head, but it seems to be an important result in the field of finite geometry, i.e. linear algebra over finite fields. Finite geometry often provides the mathematical foundations for error-correcting coding schemes that are found in many applications where information needs to be transferred efficiently and robustly. So, I could guess that this result would have practical applications in at least that domain. However, since that domain is so important (e.g. packet transfer underlying the internet), this could be a very important paper indeed, even for "real-world" applications.

That's a good question. I certainly am not willing to rule it out. If you had asked me 20 years ago if I thought algebraic topology was going to be of any use to physics, I'd have probably just laughed the question off. In retrospect, that seems a bit naive, since the Standard Model, mathematically speaking, is just setting up a Lie group and its corresponding group algebra, then turning some algebra cranks see what comes out, but I'm not so sure I knew that back then.

Someone once told me that mathematicians are frequently doing "the science of 100 years from now." I hope we don't have to wait that long to find out if design theory has a place in the Theory of Everything.

https://physics.stackexchange.com/questions/1603/application...

> since the Standard Model, mathematically speaking, is just setting up a Lie group and its corresponding group algebra, then turning some algebra cranks see what comes out, but I'm not so sure I knew that back then.

(Emphasis mine)

You make it sound as if the Standard Model of particle physics were mathematically well-defined but it's anything but, see

https://en.wikipedia.org/wiki/Haag%27s_theorem

https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_t...

https://mathoverflow.net/questions/19495/mathematics-of-path...

TL;DR up top because this got a bit long: there really aren't a lot of huge, apparent mathematical issues that I'm aware of (though, not being a physicist or someone who deals with this particular Lie algebra on a frequent basis, I could be wrong.) And, there's at least some hope that the hoped for general relativity + gravity unification with the standard model will of necessary save us from any real issues.

---

Ok, so, when you say "mathematically well-defined," you do have a bit of a point there. But, it's a misleading phrase to say the least, because the actual problems that you need to worry about more or less amount to:

> "We have a real world over here, and a mathematical model over here. The model has had a lot of success predicting and explaining things, but we're not exactly sure if they match up exactly."

That is a very valid concern, if you're a physicist. However, what this is explicitly saying is that the concerns about the model are not mathematical per se. I would, of course, put a rather large asterisk on that statement, because the unitary gauge field theory of SU(3) × SU(2) × U(1), while it is relativistic in the sense of incorporating special relativity, it does not incorporate general relativity at all.

Again, if you're a physicist, that's pretty bad, at least in a principled, theoretical sense. But, going back to this notion that the SM has been a very practically successful theory, I would bet that one could have a nice career as a theoretical physicist without stepping too hard on those types of concerns. Not being a physicist, I may very well be wrong about that, but it seems like a truthy statement to me, at least.

This "relative" disparity (pun intended) becomes an issue at the point where gravitons are expected to appear. That carries its own extra special bag of problems, because GR is not normalizable, so you're going to get divergence to infinity in places you would really rather not have them. So, that's kinda bad, but, the good news is that gravity is the weakest force, which means that until the point where all four forces are unified, we're pretty okay from a physical PoV. That is, unless you happen to be very near the event horizon of a black hole, in which case you have weightier problems (pun again intended lol).

The overall expectation seems to be that a true quantum theory of gravity must almost by necessity solve all or most of these issues. So, again, not so much of a problem mathematically, not much of a big deal until you're getting up close and personal at distances where the other 3 forces really reign supreme.

I think we're completely talking past each other.

All the issues you've mentioned are valid issues but they're about the physics. They are relevant and important but, as you say, the Standard Model and QFT have been very successful at describing a large number of physical phenomena at a certain scale, so the model is fine. Its applicability might be limited to certain scales or situations but that's not really different from any other theory in physics.

The problem with quantum field theory from a mathematical point of view, however, is that many of the mathematical objects that the symbols in the equations in physics text books represent (e.g. the "quantum fields", operators, inner products, integrals etc.) don't even exist in any (mathematically) meaningful way. Physicists pretend they do, manipulate them following very questionable rules (The limit N → ∞ doesn't exist? Doesn't matter, we'll apply it anyway and commute it with the integral!) and only the fact that, somehow, when the dust settles, those equations produce numbers that we actually see in experiments, saves the theories.

Take, e.g. the path integral for any of the field theories in the standard model: Physicists derive predictions from it. Yet it doesn't exist[0]: The integral doesn't exist. The measure doesn't exist. The sigma algebra doesn't exist. Etc. etc.

[0]: Or at least the question of its existence has eluded entire armies of mathematicians. So at the very least it is undefined.

I'm not good enough at maths to understand this... just wondering does this have anything to do with the "structures beyond time and space" which according to Donald Hoffman were identified in recent years?

I'm assuming that each "vortex" is unique. So for a collection of n vortices, given constraints (probably botching the jargon, I'm a recovering XQuery guy bear with me) of sets of k size, how do you prove a subset of t is always unique throughout the collection.

And I guess the proof they proffer extends to infinity? A perspective given in the article was sort of, if there's no evidence to the contrary, it's probably true, but my take-away is these 3 folks proved it?

I'm sure I could read the papers, though I doubt I'd understand them. But it seems so straightforward. It's combinatorical, so gets out of hand easily I'm sure, but what am I missing here?

Loved the write-up though. The carpet example made me giggle.

A great found nonetheless!

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