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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 b

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.

pseudostem 4y ago
Pondering as well... Machine learning should have some applications
prox 4y ago
The article read like they did the StarTrek game called Kal-To, where the goal is to find structure in randomness.
sideshowb 4y ago
Not time to ponder this properly but have you ever played Dobble? it feels related to the way those card sets are constructed
- sthomer 4y ago
  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!
sthomer 4y ago
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.
Paul-Craft 4y ago
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/applications-of-algebraic-topology-to-physics https://physics.stackexchange.com/questions/1603/application...
- codethief 4y ago
  > 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/Haag%27s_theorem https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_theories_that_satisfy_the_axioms https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_t... https://mathoverflow.net/questions/19495/mathematics-of-path-integral-state-of-the-art https://mathoverflow.net/questions/19495/mathematics-of-path...
  - Paul-Craft 4y ago
    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 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.

pseudostem 4y ago

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

prox 4y ago

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

sideshowb 4y ago

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

sthomer 4y ago
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!

sthomer 4y ago

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.

Paul-Craft 4y ago

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/applications-of-algebraic-topology-to-physics https://physics.stackexchange.com/questions/1603/application...

codethief 4y ago
> 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/Haag%27s_theorem https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_theories_that_satisfy_the_axioms https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_t... https://mathoverflow.net/questions/19495/mathematics-of-path-integral-state-of-the-art https://mathoverflow.net/questions/19495/mathematics-of-path...
- Paul-Craft 4y ago
  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.