Timeline for Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
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18 events
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| May 26, 2023 at 17:53 | comment | added | Turbo | @HenryCohn What is the reading methodology to understand the breakthrough work(s) in Sphere Packing and its applications to potentially (but not yet studied) packing in other spaces coming from information theory such as packing related to zero error information theory? | |
| May 23, 2020 at 13:23 | comment | added | VS. | Why is methods from sphere packing have anything to do with uncertainty principle in link.springer.com/article/10.1007/s00222-019-00875-4? The problems seem philosophically different? Is there anything that relates both? Perhaps there is a better elucidation of the relation. I can post a new question if needed. | |
| Apr 23, 2020 at 10:58 | comment | added | VS. | In your opinion is there any other application of modular forms to traditional computational number theory or is the only known use case is in packing problems? | |
| Feb 14, 2020 at 19:14 | comment | added | VS. | Could working with Epstein Zeta function directly indicate symmetries of the function necessary to get best packing? | |
| Feb 13, 2020 at 20:37 | comment | added | Henry Cohn | No direct connection; they just both involve trying to figure out what deeper meaning there might be behind numerical relationships. | |
| Feb 13, 2020 at 20:34 | comment | added | VS. | Sorry what the connection between $n\in\{3,8,24\}$ and "Monstrous Moonshine"? | |
| Feb 13, 2020 at 20:12 | comment | added | Henry Cohn | For the second question about conformal field theory and packing in other settings, I don’t know of a relationship, but I have no grounds for ruling one out. | |
| Feb 13, 2020 at 20:09 | comment | added | Henry Cohn | No in the second case: the 240 vectors form the unit integral octonions, but they aren’t associative so the question of whether they act on the Leech lattice doesn’t even make sense (they don’t even act on themselves, let alone other things). Nevertheless, there are octonionic constructions of the Leech lattice that are a little more subtle but explain the divisibility by 240. So sometimes things work the way you’d hope, but sometimes we need more subtle or sophisticated explanations. | |
| Feb 13, 2020 at 20:07 | comment | added | Henry Cohn | Unfortunately I don’t know of an overall reference. “Monstrous moonshine” is probably the most dramatic and important case, but there’s a lot of other numerology scattered in the literature. Figuring out what’s going on can be tricky. For example, suppose we’re counting minimal vectors in lattice. For D_4, E_8, and the Leech lattice, we get 24, 240, and 196560, with each number divisible by the previous one. Is there an explanation in terms of group actions? Yes for the first case: the 24 vectors form the units in the Hurwitz integral quaternions, and they act on E_8. | |
| Feb 13, 2020 at 20:04 | comment | added | VS. | 'Regarding 3, 8, and 24, there are a lot of numerical relationships in this area, some of which seem deeply meaningful and others less so, and it is hard to say.'. 1. Is there an useful reference? 'Conformal field theory builds in modular invariance (which amounts to conformal invariance on tori),'. 2. Is there some theory of 'conformal form' applicable to packing problems on other spaces such as tori? Just wondering. | |
| Feb 13, 2020 at 18:38 | comment | added | Henry Cohn | Conformal field theory builds in modular invariance (which amounts to conformal invariance on tori), so it makes sense that modular forms should play an important role there, but then the question is why conformal field theory is connected with sphere packing. So far, it’s unclear whether there’s an even deeper connection, or it’s just that the LP bound for sphere packing is isomorphic to the spinless modular bootstrap. | |
| Feb 13, 2020 at 18:35 | comment | added | Henry Cohn | Regarding 3, 8, and 24, there are a lot of numerical relationships in this area, some of which seem deeply meaningful and others less so, and it is hard to say. I doubt 3 dimensions plays a significant role in LP or their generalizations (and Hales’s proof uses entirely different techniques), but maybe it will in the future. As for the sphere packing and quantum gravity paper, it’s really interesting and important, and raises as many questions as it answers. | |
| Feb 13, 2020 at 16:44 | comment | added | VS. | Regarding $3$, $8$ and $24$ it would be nice to know if there is a modular connection to $3$ as well. | |
| Feb 13, 2020 at 16:37 | comment | added | VS. | The linked paper in post on quantum gravity speculates some other deeper construction than modular forms of which modular forms emerge at $n\in\{8,24\}$ (I think). | |
| Feb 13, 2020 at 16:36 | history | edited | Henry Cohn | CC BY-SA 4.0 |
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| Feb 13, 2020 at 16:29 | comment | added | VS. | Would there be some connection between $3$, $8$ and $24$ or would it be wild to expect one? Since we know the best bounds in all three has it been predicted that some multiplicative behavior might be possible? Just wondering if there could be some underlying arithmetic to these optimality? | |
| Feb 13, 2020 at 16:27 | vote | accept | VS. | ||
| Feb 13, 2020 at 16:09 | history | answered | Henry Cohn | CC BY-SA 4.0 |