The sphere packing problem in dimension 24
01 May 2017-Annals of Mathematics (Princeton University and the Institute for Advanced Study)-Vol. 185, Iss: 3, pp 1017-1033
TL;DR: In this article, it was shown that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing.
Abstract: Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.
Citations
More filters
Book•
21 Oct 2019TL;DR: This chapter brings together the concepts from Chaps.
Abstract: In this chapter, we give the definition of Mordell–Weil lattice (in Sect. 6.5). First, we bring together the concepts from Chaps. 4 and 5 in order to gain a better understanding of the Neron–Severi lattice of an elliptic surface. This will lead to the announced notion of Mordell–Weil lattice which will be studied in detail in this chapter, but also throughout the remainder of this book.
359 citations
TL;DR: Comparing mean-field predictions with the finite-dimensional simulations, this work identifies robust aspects of the description of hard spheres around the dynamical, Gardner and jamming transitions and uncover its more sensitive features.
Abstract: 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 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 mean-field predictions with the finite-dimensional simulations, we identify robust aspects of the description and uncover its more sensitive features. We conclude with a brief overview of ongoing research.
262 citations
TL;DR: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids as mentioned in this paper.
254 citations
TL;DR: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids.
Abstract: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids. All perfect crystals, perfect quasicrystals and special disordered systems are hyperuniform. Thus, the hyperuniformity concept enables a unified framework to classify and structurally characterize crystals, quasicrystals and the exotic disordered varieties. While disordered hyperuniform systems were largely unknown in the scientific community over a decade ago, now there is a realization that such systems arise in a host of contexts across the physical, materials, chemical, mathematical, engineering, and biological sciences, including disordered ground states, glass formation, jamming, Coulomb systems, spin systems, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, number theory, stochastic point processes, integral and stochastic geometry, the immune system, and photoreceptor cells. Such unusual amorphous states can be obtained via equilibrium or nonequilibrium routes, and come in both quantum-mechanical and classical varieties. The connections of hyperuniform states of matter to many different areas of fundamental science appear to be profound and yet our theoretical understanding of these unusual systems is only in its infancy. The purpose of this review article is to introduce the reader to the theoretical foundations of hyperuniform ordered and disordered systems. Special focus will be placed on fundamental and practical aspects of the disordered kinds, including our current state of knowledge of these exotic amorphous systems as well as their formation and novel physical properties.
228 citations
Cites background from "The sphere packing problem in dimen..."
...(Note it was recently proved that the E8 and L24 lattices are the densest packings among all possible packings in dimensions 8 [154] and 24 [155], respectively....
[...]
TL;DR: In this article, it was shown that no packing of unit balls in Euclidean space with density greater than that of the lattice packing has density better than the E_8$-lattice packing.
Abstract: In this paper we prove that no packing of unit balls in Euclidean space $\mathbb{R}^8$ has density greater than that of the $E_8$-lattice packing.
152 citations
References
More filters
TL;DR: The Handbook of Mathematical Functions with Formulas (HOFF-formulas) as mentioned in this paper is the most widely used handbook for mathematical functions with formulas, which includes the following:
Abstract: (1965). Handbook of Mathematical Functions with Formulas. Technometrics: Vol. 7, No. 1, pp. 78-79.
7,538 citations
Book•
01 Jan 1982
TL;DR: In this paper, theta functions in one variable and motivation: motivation and theta function in several variables are compared. But the results are limited to one variable, and motivation is not considered.
Abstract: and motivation: theta functions in one variable.- Basic results on theta functions in several variables.
2,115 citations
Posted Content•
[...]
TL;DR: A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
Abstract: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
1,276 citations
"The sphere packing problem in dimen..." refers background in this paper
...See [5] for background and references on sphere packing and its applications....
[...]
TL;DR: In this paper, the authors provided an overview of spherical codes and designs, and derived bounds for the cardinality of spherical A-codes in terms of the Gegenbauer coefficients of polynomials compatible with A.
Abstract: Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension d ( X ) of the space spanned by X , its cardinality n = | X |, its degree s( X ), and its strength t ( X ).The chapter presents derivation of bounds for the cardinality of spherical A -codes in terms of the Gegenbauer coefficients of polynomials compatible with A . It also discusses spherical ( d , n , s , i)- configurations X . These are sets X of cardinality n on the unit sphere Ω d , which are spherical t -designs and spherical A -codes with I A I = s ; in other words, the strength t ( X ) is at least t and the degree s ( X ) is at most s . A condition is given for a spherical A -code to be a spherical t -design, in terms of the Gegenbauer coefficients of an annihilator of the set A . The chapter presents many examples of spherical ( d , n , s , t )-configurations; there exist tight spherical t-designs with t = 2, 3, 4, 5, 7, 11, and non-tight spherical ( 2s − 1)-designs. The constructions of these examples use sets of lines with few angles and association schemes, respectively.
1,009 citations
"The sphere packing problem in dimen..." refers methods in this paper
...ogramming bounds. This technique was successfully applied to obtain upper bounds in a wide range of discrete optimization problems such as error-correcting codes [7], equal weight quadrature formulas [8], spherical codes [13], [16]. In exceptional cases linear programming bounds are optimal [5]. However, in general situation linear programming bounds are not sharp and it is an open question how big t...
[...]
01 Jan 1983
TL;DR: The selberg trace formula (version A) as mentioned in this paper is a trace formula for the Poincare series and the spectral decomposition of L2(? \H,?).
Abstract: Development of the trace formula (version A).- Poincare series and the spectral decomposition of L2(? \H, ?).- Version B of the selberg trace formula.- Version C of the selberg trace formula.- Selected applications.- Some examples.
964 citations