This is a very phenomenal result and everyone in the field is excited about this! Josh and Hong both gave talks about this at a conference 2 weeks ago in Berkeley, and the videos are online: [1] [2]. Josh's talk (at least from my perspective of someone who is adjacent to this field) is quite approachable, whereas Hong talks more about the induction scheme on Guth's grains decomposition which is quite a bit more technical.
I visited Josh at UBC last year around this time and I recall asking him if he thought Kakeya in dimension 3 would be solved soon. I remember that he believed it would be (though perhaps he was worried by someone other than Hong and himself). In the end they were able to complete the proof themselves.
Hong is probably quite a serious candidate for the fields medal because of this. She has already made impressive progress on problems in harmonic analysis and geometric measure theory and she is one of few people has a firm foothold in both fields at the same time.
The quanta article talks about a 'tower of conjectures' in harmonic analysis; at the top of the tower is the so-called "local smoothing conjecture" (a conjecture about how much waves, such as 'idealized' sound waves, can amplify from some initial configuration when averaged over time). A Kakeya set is a certain type of geometric obstruction to local smoothing; resolving the full conjecture also requires handling so-called 'oscillatory' obstructions. In dimension 2 + 1 (2 spatial and 1 time dimension) the local smoothing was only recently resolved (also by Hong and co-authors [3]); even though the corresponding result for Kakeya sets in dimension 2 has been known for over 40 years.
The animation in the article looks like a simplified diagram of a rotary engine or a drill that will cut polygon-shaped holes instead of round holes.
The article mentions a connection to the Fourier transform, which makes sense because nested rotations are essentially summed sine waves in a different coordinate space.[1]
Is there more of a connection than that between the Kakeya conjecture, physical machines like rotary engines, and additive wave synthesis, or are they all fairly different branches of "interesting things one can do with nested rotations / summed sine waves"?
I'm curious if proving the conjecture opens up new possibilities in mechanical engineering or sound/EM wave synthesis/analysis, in other words.
[1] Apologies in advance if I mauled this description.
"Consider a pencil lying on your desk. Try to spin it around so that it points once in every direction, but make sure it sweeps over as little of the desk’s surface as possible."
I'm really stuck at the start here - moving a pen so that it pointing in all directions is basically impossible - the space of directions is two-dimensional and you can only trace out a one-dimensional curve (or pair of curves).
Ok, wikipedia makes it clearer:
"In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."
Quanta writers are generally very good at explaining things, but wikipedia wins hands down in this case...
It’s not pointing in all directions at once, it’s pointing once in each direction. So if you spin the pencil so it does exactly one complete rotation that works, doesn’t it?
I think OP is thinking about covering the sphere of directions in 3D space, not just directions in a 2D plane. No matter how hard you spin the pencil, you're drawing a one-dimensional curve that has no area, so any finite amount you draw will cover zero percent of the area of the two-dimensional sphere surface.
Yes, and spinning the pencil on its centre like that shows that the circle (of pencil length diameter) is such a set. (I think you're thinking about it the wrong way around: it's which containing shapes allow this, not how can it be done at all.)
> "In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."
Is that definition correct/complete? It leaves open the option that such a set isn’t connected. I think there’s an additional requirement that, for any two directions D and E, you can move a line segment oriented in direction D so that it’s oriented in direction E without any point on it ever leaving the set.
Animation looks like how a skilled fighter might used a double barbed weapon. Maybe there is a connection? Minimise movement (time, energy use) while maximising agility?
I can tell a HN article is from Quanta from the obnoxious clickbaity title. I'm shocked that they have to do this given their target audience (because I'd rather not believe they just choose to)
Each area has its own “most important” conjecture(s), so Quanta is in no danger of running out of proofs to write about that somebody legitimately thinks are super important.
(Alas, while “most important” conjectures are a renewable resource, lay reader tolerance for such headlines may not be.)
The study of Kakeya sets is over 100 years old — so if you think this generalized conjecture is the most important one in their subfield, you can see why somebody would describe it this way.
Just clicked on this and discovered that it's in German - not that I have an issue with people speaking German, but more that I have trouble understanding it.
Readit News
I visited Josh at UBC last year around this time and I recall asking him if he thought Kakeya in dimension 3 would be solved soon. I remember that he believed it would be (though perhaps he was worried by someone other than Hong and himself). In the end they were able to complete the proof themselves.
Hong is probably quite a serious candidate for the fields medal because of this. She has already made impressive progress on problems in harmonic analysis and geometric measure theory and she is one of few people has a firm foothold in both fields at the same time.
The quanta article talks about a 'tower of conjectures' in harmonic analysis; at the top of the tower is the so-called "local smoothing conjecture" (a conjecture about how much waves, such as 'idealized' sound waves, can amplify from some initial configuration when averaged over time). A Kakeya set is a certain type of geometric obstruction to local smoothing; resolving the full conjecture also requires handling so-called 'oscillatory' obstructions. In dimension 2 + 1 (2 spatial and 1 time dimension) the local smoothing was only recently resolved (also by Hong and co-authors [3]); even though the corresponding result for Kakeya sets in dimension 2 has been known for over 40 years.
[1] https://player.vimeo.com/video/1062254156
[2] https://player.vimeo.com/video/1063428579
[3] https://annals.math.princeton.edu/2020/192-2/p06
The article mentions a connection to the Fourier transform, which makes sense because nested rotations are essentially summed sine waves in a different coordinate space.[1]
Is there more of a connection than that between the Kakeya conjecture, physical machines like rotary engines, and additive wave synthesis, or are they all fairly different branches of "interesting things one can do with nested rotations / summed sine waves"?
I'm curious if proving the conjecture opens up new possibilities in mechanical engineering or sound/EM wave synthesis/analysis, in other words.
[1] Apologies in advance if I mauled this description.
I'm really stuck at the start here - moving a pen so that it pointing in all directions is basically impossible - the space of directions is two-dimensional and you can only trace out a one-dimensional curve (or pair of curves).
Ok, wikipedia makes it clearer:
"In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."
Quanta writers are generally very good at explaining things, but wikipedia wins hands down in this case...
[1] https://en.wikipedia.org/wiki/Kakeya_set
> "In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."
Is that definition correct/complete? It leaves open the option that such a set isn’t connected. I think there’s an additional requirement that, for any two directions D and E, you can move a line segment oriented in direction D so that it’s oriented in direction E without any point on it ever leaving the set.
(Alas, while “most important” conjectures are a renewable resource, lay reader tolerance for such headlines may not be.)