Just to add a comment on "why this idea is cool" from my perspective (I'm a mathematician).
The situation being studied is: C is a curve in the plane (as another commenter pointed out, the z variable can essentially be ignored and set to z=1), described by a horrendous equation f(x,y) = 0 with very few rational solutions.
Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly exists a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)
But, it seems completely hopeless to try to find the equation g(x,y) in practice, other than by first finding all the rational points on C by other means, and then just writing down a different curve passing through them.
So what's special here is that this "Selmer variety" approach provides a method, partly conjectural, for constructing C' directly from C. And the paper being described has successfully applied this method to prove that, at least in this one case, C' intersects C at precisely the rational points. (And once you have the two equations, it's easy to solve for the intersection points -- we now have two equations in two variables).
PS: Part of what's special here is the connection between number theory and geometry. A Diophantine equation has infinitely-many solutions if you allow x and y to be real numbers -- there's the entire curve. It's usually an extremely delicate number theory question to analyze which solutions are rational. But here, we're converting the problem to geometry -- intersecting two curves (much easier).
Follow-up question: is there any practical significance of rational solutions? I can understand why one might be looking for integer solutions to an equation. Can you provide an example where rational solutions correspond to something interesting in the context modeled by the equation – for example the "path travelled by light" thing hinted at in the article?
> Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly exists a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)
Ok, any finite set of points can be interpolated by a curve.
Why is it obvious that one of these interpolated curves will necessarily avoid intersecting C at any other point?
> Drawing inspiration from physics, he thinks of rational solutions to equations as being somehow the same as the path that light travels between two points.
What does this even mean? This looks like fancy words for "Kim used lines in his solution".
The earlier article that the article links to says a bit more about this. It's still pretty far from concrete, but it does say a bit more. When they talk about the path that light travels between two points, they're talking about the principle of least action.
Going by the earlier article, the idea seems to be roughly like the following. You associate to each point the fundamental group based on that point. All these fundamental groups then live in some larger space, and the ones based at rational points will minimize some quantity analogous to action (or, if we're thinking of light, time).
That's basically the best I can figure just based on the earlier article.
It's something like, (and please don't try too hard to understand my description here, I watched the video over a year ago, and you've got Feynman at the link) imagine a photon is a particle going in all directions, and has an arrow inside it spinning at a rate of 10^15 revolutions per second, and the probability that light goes from its origin to a particular destination is the amplitude of the sum of all the internal arrows of the all the possible paths between the origin and the destination.
Even if this is not what the article is referring to, its a reminder that for a lot of people, light doesn't move in straight lines.
Finally, the series linked series of lectures are amazing, if you have a layperson's interest in Physics then this will take your knowledge to a new level.
Maybe it means something along the lines of "instead of thinking of those equations as curves in a flat space, he thinks of them as lines in a curved space". Disclaimer, I know nothing about number theory.
It would be nice if the had drawn the damn curve (and the famous 7 points on it). Not all of us are capable of plotting a fourth degree curve on our heads.
It's not straightforward to get a useful visual plot, but the article does include a somewhat artistic one (w/o labeled axes) in the banner image at the top. It's a polynomial in 3 variables, so plotting the polynomial itself is 4-dimensional, and the roots are in 3 dimensions. The visualization at the top of the article is a 3d plot of the locations of the roots.
The seven rational solutions are given on p. 30 of the paper, and are: (x,y,z) = (1,1,1), (1,1,2), (0,0,1), (-3,3,2), (1,1,0), (0,2,1), (-1,1,0). Those were already known though. The new result of the paper is to prove that there aren't any others.
Could someone please point out the benefits and implications of this? Does this bring us closer to some very important solution or does it have some real world applications?
Edit: I suppose this is a related article talking about the same problem from December 2017. What changed since then?
The benefits are that mathematicians think it is interesting, and this allows us to answer mathematical questions we could not answer before.
There might be practical applications I do not know of, but that's not the point of fundamental mathematical research.
For a thorough defense of mathematics, consider 'A mathematicians apology' by G.H. Hardy or [1], a modern response to that.
To summarize [1], the argument is that the main product of mathematics is an environment that creates mathematicians. Those mathematicians can the use their problem solving skills in practical applications.
I have an applied mathematics background and I find it quite amusing that the descriptions of this article are so opaque and inscrutable that the ten-plus comments here are all penned by people that are absolutely fixed by what concept the article is trying to express (including myself, incidentally, even after checking the original paper because it’s a totally different domain and level of sophistication compared to my zone).
There’s a real dearth of summaries for papers that are aimed at those who have taken a reasonable amount of math: say calculus, some number theory, linear algebra. It always either ends up being some sort of terrible analogy to a real-world phenomenon that has no math in it or something that’s about as complicated as the paper itself.
[+] [-] jlev1|8 years ago|reply
The situation being studied is: C is a curve in the plane (as another commenter pointed out, the z variable can essentially be ignored and set to z=1), described by a horrendous equation f(x,y) = 0 with very few rational solutions.
Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly exists a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)
But, it seems completely hopeless to try to find the equation g(x,y) in practice, other than by first finding all the rational points on C by other means, and then just writing down a different curve passing through them.
So what's special here is that this "Selmer variety" approach provides a method, partly conjectural, for constructing C' directly from C. And the paper being described has successfully applied this method to prove that, at least in this one case, C' intersects C at precisely the rational points. (And once you have the two equations, it's easy to solve for the intersection points -- we now have two equations in two variables).
PS: Part of what's special here is the connection between number theory and geometry. A Diophantine equation has infinitely-many solutions if you allow x and y to be real numbers -- there's the entire curve. It's usually an extremely delicate number theory question to analyze which solutions are rational. But here, we're converting the problem to geometry -- intersecting two curves (much easier).
[+] [-] skosch|8 years ago|reply
Follow-up question: is there any practical significance of rational solutions? I can understand why one might be looking for integer solutions to an equation. Can you provide an example where rational solutions correspond to something interesting in the context modeled by the equation – for example the "path travelled by light" thing hinted at in the article?
[+] [-] thaumasiotes|8 years ago|reply
Ok, any finite set of points can be interpolated by a curve.
Why is it obvious that one of these interpolated curves will necessarily avoid intersecting C at any other point?
[+] [-] soVeryTired|8 years ago|reply
[+] [-] saagarjha|8 years ago|reply
What does this even mean? This looks like fancy words for "Kim used lines in his solution".
[+] [-] Sniffnoy|8 years ago|reply
Going by the earlier article, the idea seems to be roughly like the following. You associate to each point the fundamental group based on that point. All these fundamental groups then live in some larger space, and the ones based at rational points will minimize some quantity analogous to action (or, if we're thinking of light, time).
That's basically the best I can figure just based on the earlier article.
[+] [-] AndrewOMartin|8 years ago|reply
https://www.youtube.com/watch?v=kMSgE62S6oo
It's something like, (and please don't try too hard to understand my description here, I watched the video over a year ago, and you've got Feynman at the link) imagine a photon is a particle going in all directions, and has an arrow inside it spinning at a rate of 10^15 revolutions per second, and the probability that light goes from its origin to a particular destination is the amplitude of the sum of all the internal arrows of the all the possible paths between the origin and the destination.
Even if this is not what the article is referring to, its a reminder that for a lot of people, light doesn't move in straight lines.
Finally, the series linked series of lectures are amazing, if you have a layperson's interest in Physics then this will take your knowledge to a new level.
[+] [-] vinchuco|8 years ago|reply
[+] [-] fwdpropaganda|8 years ago|reply
[+] [-] enriquto|8 years ago|reply
[+] [-] _delirium|8 years ago|reply
The seven rational solutions are given on p. 30 of the paper, and are: (x,y,z) = (1,1,1), (1,1,2), (0,0,1), (-3,3,2), (1,1,0), (0,2,1), (-1,1,0). Those were already known though. The new result of the paper is to prove that there aren't any others.
[+] [-] chowells|8 years ago|reply
[+] [-] unknown|8 years ago|reply
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[+] [-] mraison|8 years ago|reply
[+] [-] johnhenry|8 years ago|reply
[+] [-] kungito|8 years ago|reply
Edit: I suppose this is a related article talking about the same problem from December 2017. What changed since then?
https://www.quantamagazine.org/secret-link-uncovered-between...
[+] [-] rocqua|8 years ago|reply
For a thorough defense of mathematics, consider 'A mathematicians apology' by G.H. Hardy or [1], a modern response to that. To summarize [1], the argument is that the main product of mathematics is an environment that creates mathematicians. Those mathematicians can the use their problem solving skills in practical applications.
[1] https://ldtopology.wordpress.com/2017/03/18/a-new-mathematic...
[+] [-] sesqu|8 years ago|reply
[+] [-] behindmyscreen|8 years ago|reply
This could have impacts on computational encryption....it could also just make mathematics in this area easier to work through.
[+] [-] qubex|8 years ago|reply
[+] [-] saagarjha|8 years ago|reply
[+] [-] eshwar37|8 years ago|reply
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[+] [-] kichu37|8 years ago|reply
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