Timeline for Optimal sphere packings in dimensions different fom 8 and 24
Current License: CC BY-SA 4.0
12 events
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| Sep 12, 2022 at 18:43 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title
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| Sep 12, 2022 at 18:06 | answer | added | Timothy Chow | timeline score: 7 | |
| Sep 12, 2022 at 8:01 | comment | added | Roland Bacher | Another indication that optimal densities might not be achieved by lattice packings is the fact that already in $\mathbb R^3$ there are non-enumerably many packings with optimal density (also achieved by a lattice-packing). Optimal packings are therefore sometimes very far from unique or finite in number (perfect lattices yielding all locally optimal lattice packings are however in finite number). | |
| Sep 12, 2022 at 4:13 | comment | added | Dan Romik | @WillSawin is right, this has been done by Cohn and his collaborator Triantafillou. They proved the existence of obstructions to using the Cohn-Elkies linear programming bounds (which Viazovska’s method uses) to prove upper bounds for sphere packings that match the known lower bounds in dimensions 12, 16, 20, 28 and 32. See their paper “Dual linear programming bounds for sphere packings via modular forms”. | |
| Sep 11, 2022 at 23:58 | comment | added | Will Sawin | @TimothyChow I think it is possible to give an obstruction to the linear programming bound other than a dense packing by, basically, solving the dual linear programming problem. I think Henry Cohn mentioned an explicit example of such a thing in high dimensions in a talk of his I watched. I'm not sure if this can be made to work in low dimensions or whether one could get a lower bound better than the best upper bound for a lattice packing in a given dimension. | |
| Sep 11, 2022 at 20:50 | comment | added | Timothy Chow | @WillSawin But there's no known (or even heuristic) "obstruction" is there? It's just that the known bounds from linear programming are so far from tight that it seems too optimistic to hope for such a thing in dimensions other than 8 and 24? That's at least the impression I got from (for example) Henry Cohn's expository article. | |
| S Sep 11, 2022 at 16:35 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
Improved English
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| Sep 11, 2022 at 16:07 | review | Suggested edits | |||
| S Sep 11, 2022 at 16:35 | |||||
| Sep 11, 2022 at 14:52 | comment | added | Will Sawin | The structure that exists in dimensions 8 and 24 but is expected not to exist in other dimensions is not a lattice but a sharp Cohn-Elkies auxiliary function. | |
| Sep 11, 2022 at 11:59 | comment | added | Timothy Chow | Or, if you're asking whether (in dimensions 6 and 7, say) there is an obstruction to the existence of better "rigid algebraic" packings than $E_6$ and $E_7$, then yes, those are provably the densest lattice packings in their respective dimensions. (Though maybe your term "rigid algebraic structure" might include some non-lattice packings?) See also this related MO question. | |
| Sep 11, 2022 at 11:49 | comment | added | Timothy Chow | Lattice packings do exist in other dimensions, and in low dimensions they may well be the densest packings. For example, if you think of $E_8$ as a "rigid algebraic structure" then surely $E_7$ and $E_6$ are too, and they're probably the densest packings in dimensions 6 and 7. Are you asking whether there is some theoretical obstruction to mimicking the proof of the upper bound (on the packing density) in other dimensions? I think the main problem is that the linear programming bound is somehow "too loose" in other dimensions (but I'm not aware of any theoretical "obstruction" as such). | |
| Sep 11, 2022 at 8:41 | history | asked | Johnny Cage | CC BY-SA 4.0 |