Sphere packings in 8 dimensions (after Maryna Viazovska)
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- Опубликовано: 20 окт 2024
- The is a math talk about the best possible sphere packing in 8 dimensions. It was an open problem for many years to show that the best 8-dimensional sphere packing is given by the E8 lattice. We describe the solution to this found by Maryna Viazovska, building on work of Henry Cohn and Noam Elkies.
Typo around 13:40: f(0) and f^(0) should be exchanged in the fractions.
The paper by Henry Cohn and Noam Elkies is arxiv.org/abs/...
The paper by Maryna Viazovska is at arxiv.org/abs/...
This talk did not cover the proof that the Leech lattice is optimal in 24 dimensions, but this can be found in arxiv.org/abs/...
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Recently, there was beautiful work done by Hartman, Mazáč, and Rastelli (1905.01319 on the arxiv) showing a connection between the sphere packing problem and the modular bootstrap program in quantum field theory. In particular, the magic functions from this lecture happen to have a physical interpretation.
Favorite channel. Thanks!
why do modular forms keep coming up everywhere? I don‘t know much of any mathematics connected to them (which probably shows that I don’t know much of any mathematics), but they seem like a very niche subject on the surface. Their structure doesn’t seem to convey anything fundamental (like say, in fourier series) nor do they seem like a candidate for unifying mathematics (like category theory) or something off of which all maths can be built in a quirky but understandable manner (like the natural numbers).
Is it known why modular forms are seemingly connected to everything (Fermats Last Theorem, Sphere packing, monstrous moonshine)? It seems to me like the connections to modular forms often pop out of nowhere.
The fact that modular forms turn up so often is one of the big mysteries of mathematics. I don't know of any good explanation.
Aren't they related to number theory? If they're related to number theory, it makes sense that they would pop up all over the place. I'm in the process of learning about them, and specifically I'm interested in finding out if they can be done in the quaternionic and possibly octonionic setting (if not with the octonions then maybe with a clifford algebra that can be constructed from octonions). Octonions are a candidate for unification of physics, so if you can do modular forms with octonions or a structure similar to them, then it seems possibly connected to unification.
Maryna Viazovska won the Fields medal in 2022 for this work!
Congratulations to Prof. Viazovska for the Fields medal!
Oh, thank you, this is one of my favorite topics
It seems there's a typo at 13:37. Should |L| >= f(0)/f_hat(0) be |L| >= f_hat(0)/f(0)? Referencing the Cohn/Elkies paper, this does appear to be the case.
Maybe a moral can be that integrating modular forms is nice. Probably because the modularity allows you to do nice change of variables which become functional equations, as for example happens for the zeta function whose functional equation follows after vizualising it as an integral of the theta function.
small typo at 13:40, it should be \hat{f}(0)/f(0) in the final inequality (the fraction has to be inverted)
Great video, but it leaves unanswered the question of how the Leech lattice was shown to be optimal without the Elkies Cohn bound. It seems hard to prove upper bounds in any other way.
The proof that the Leech lattice is optimal can be found in arxiv.org/abs/1603.06518
It used the Cohn-Elkies bound.
In 3 dimensions, why would face centered cubic ABCABC... be more efficient than ABAB... or any other sequence? I am grateful for any clue.
Please keep up the good work!
Can you let me know about relation with Moonshine number?
In six dimensions there's 3 positive numbers and symmetrical would be negative, meaning using x,y,z in third angle projections. This 8th dimensiona so four positive and four negative but I think Maryna used matrix and Gaussian with Euler to solve this .
There's a lot of the number 1728 at 25:10. Should this remind me of Ramanujan and taxicab numbers?
1728 is relevant here because it's 12^3. I don't think it's relevant here that when you add 1=1^3 to it you get a number that can be written as a sum of two other cubes.
The graph that was drawn for g(x) has a simple zero at x=\sqrt{2}. But it appears, from the way you defined the function g(x), that there is a zero of order two at x=\sqrt{2}. Does the Laplace transform part has a simple pole at x=\sqrt{2}?
The function was only defined for r>√2. He said that you had to define it differently for r
Great videos! Just subscribed
yeeeeeeee
Yay!! Fields medal!
8 and 24 can be explained easily enough in chaos phase space after considering the probability of perception