The sphere packing problem in dimension $24$

Cohn, Henry; Kumar, Abhinav; Miller, Stephen D.; Radchenko, Danylo; Viazovska, Maryna

doi:10.4007/annals.2017.185.3.8

Cited by 246 publications

(263 citation statements)

References 9 publications

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“…This breakthrough result was itself made possible by the seminal works of Viazovska [Via] and the same authors [CKMRV1] on the solution of the sphere packing problem in the same dimensions (see also the expository papers [Coh, dLV]). As one of very few proofs of crystallization in dimensions larger than one, it represents major progress on the topic.…”

Section: Introductionmentioning

confidence: 98%

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

Petrache¹,

Serfaty

2020

Proc. Amer. Math. Soc.

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The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the E8 lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.Conjecture 1 (Cohn-Kumar [CK]). In dimension d = 2, 8, resp. 24, the lattice Λ 0 is universally minimizing in the sense that it minimizes E p among all possible point configurations of density 1 for all p's that are completely monotone functions of the squared distance.Coulangeon and Schürmann proved in [CS] a local version of this conjecture with the result that Λ 0 is a local minimizer of E p . Then Conjecture 1 was recently proved in dimensions 8 and 24 -it remains open in dimension 2.

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Section: Introductionmentioning

confidence: 98%

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

Petrache¹,

Serfaty

2020

Proc. Amer. Math. Soc.

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“…Consider, for instance, ordered packings of hard spheres in R d . Asymptotic packing bounds for d → ∞ (18) have little to say about the singularly dense triangular lattice in d = 2, root lattice in d = 8 (19), or Leech lattice in d = 24 (20). Does something similar occur for amorphous materials?…”

Section: Introductionmentioning

confidence: 99%

Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions

Charbonneau

Kurchan

Parisi

et al. 2017

Annu. Rev. Condens. Matter Phys.

299

421

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Despite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution of a microscopic model in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing meanfield predictions with the finite-dimensional simulations, we identify robust aspects of the theory and uncover its more sensitive features. We conclude with a brief overview of ongoing research.

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“…One particularly attractive case is the 4 root lattice, which is surely the best sphere packing in ℝ 4 . This lattice shares some of the wonderful properties of 8 and the Leech lattice, but not enough for the four-dimensional linear programming bound to be sharp.…”

Section: Future Prospectsmentioning

confidence: 99%

“…By this standard, Viazovska's proof is remarkably simple. It was understood by a number of people within a few days of her arXiv posting, and within a week it led to further progress: Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and I worked with Viazovska to adapt her methods to prove that the Leech lattice is an optimal sphere packing in twenty-four dimensions [4]. This is the only other case above three dimensions in which the sphere packing problem has been solved.…”

mentioning

confidence: 99%

“…We won't cover all the details completely, but we'll see the main ideas and how they fit together. Readers who wish to read a complete proof will then be well prepared to study Viazovska's paper [15] and the follow-up work on the Leech lattice [4]. See also de Laat and Vallentin's survey article and interview [13] for a somewhat different perspective, as well as [1] and [7] for further background and references.…”

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confidence: 99%

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