A conceptual breakthrough in sphere packing

This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska.

• Publication year

  2016

• Venue

  arXiv: Metric Geometry

• Publication date

  2016-11-05

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1090/noti1474arXiv 1611.01685
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Packing, coding, and ground states

  2016cited by this paper
• The sphere packing problem in dimension 8The sphere packing problem in dimension 8

  2016cited by this paper
• Some properties of optimal functions for sphere packing in dimensions 8 and 24

  2016influential reference
• A breakthrough in sphere packing : the search for magic functions

  2016cited by this paper
• A FORMAL PROOF OF THE KEPLER CONJECTURE

  2015cited by this paper
• A Note on Sphere Packings in High Dimension

  2013cited by this paper
• A semidefinite programming hierarchy for packing problems in discrete geometry

  2013cited by this paper
• A proof of the Kepler conjecture

  2005cited by this paper
• Optimality and uniqueness of the Leech lattice among lattices

  2004cited by this paper
• New upper bounds on sphere packings I

  2001influential reference
• Cannonballs and Honeycombs

  2000cited by this paper
• Sphere packings, I

  1998cited by this paper
• What are all the best sphere packings in low dimensions?

  1995cited by this paper

• Packings in classical Banach spaces

  2026cites this paper
• Monte Carlo model to determine eddy currents in 3D winding configurations with and without ferromagnetic materials

  2025cites this paper
• Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid

  2025cites this paper
• Sealless Production Ultracentrifuge and Its Magnetically Self-Bearing Openable Motors for Purification in Viral Nanotechnology

  2025cites this paper
• Graphics Processing Unit Simulation of Magnetic Domains Under Alternating Field Excitation Including Iron Enhancement

  2025cites this paper
• Probabilistic combinatorics at exponentially small scales

  2025cites this paper
• The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

  2025cites this paper
• A short survey on almost orthogonal vectors in a few specific large dimensions

  2025cites this paper
• Dynamic space filling

  2025cites this paper
• A survey of degree-boundedness

  2024cites this paper
• Multiplexed Multispectral Filter Array by 3D Sphere Packing Design

  2024cites this paper
• Dichteste Kugelpackungen in höheren Dimensionen – Die Arbeiten von Maryna Viazovska

  2024cites this paper
• Computation of the Schläfli Function

  2024cites this paper
• An efficient solution space exploring and descent method for packing equal spheres in a sphere

  2024cites this paper
• The Mathematical Foundation of Post-Quantum Cryptography

  2024cites this paper
• From sphere packing to Fourier interpolation

  2023cites this paper
• Unsupervised Feature Learning with Emergent Data-Driven Prototypicality

  2023cites this paper
• On the Pólya conjecture for the Neumann problem in planar convex domains

  2023cites this paper
• Multispectral Filter Array Design by Optimal Sphere Packing

  2023cites this paper
• Dynamic space packing

  2023cites this paper
• A new lower bound for sphere packing

  2023cites this paper
• Lattices: From Roots to String Compactifications

  2023cites this paper
• Identification and analysis of 3D pores in packed particulate materials

  2023cites this paper
• Birds on a wire

  2022cites this paper
• On the Best Lattice Quantizers

  2022cites this paper
• Packing and covering in higher dimensions

  2022cites this paper
• The magic of 8 and 24

  2022cites this paper
• The work of Maryna Viazovska

  2022cites this paper
• SPHERE PACKING DENSITIES OF SUBLATTICES OF THE MORDELL-WEIL LATTICES OF TWO FAMILIES OF ELLIPTIC CURVES

  2022cites this paper
• Golay codes and quantum contextuality

  2022cites this paper
• Performance of random template banks

  2022cites this paper
• Dynamical fluctuations in the Riesz gas.

  2022cites this paper
• Classical Density Functional Theory: Representability and Universal Bounds

  2022cites this paper
• Sphere Packing Densities of Sublattices of the Mordell-Weil Lattices of two Families of Elliptic Curves

  2022cites this paper
• Random sequential covering

  2022cites this paper
• Dense Packings via Lifts of Codes to Division Rings

  2021cites this paper
• Varieties of mathematical understanding

  2021cites this paper
• Scalable Optimal Transport in High Dimensions for Graph Distances, Embedding Alignment, and More

  2021cites this paper
• The Optimal Lattice Quantizer in Nine Dimensions

  2021cites this paper
• Thermodynamic stability of hard sphere crystals in dimensions 3 through 10

  2021cites this paper
• Atomic Liquids in Infinite Dimensions: Thermodynamics

  2020cites this paper
• Errata and Addenda to Mathematical Constants

  2020cites this paper
• Modular Forms

  2020cites this paper
• The Unity of Combinatorics

  2020cites this paper
• Packings, sausages and catastrophes

  2020cites this paper
• Frame Potential in CPn Some Numerical and Analytical Results

  2020cites this paper
• Germ order for one-dimensional packings

  2020cites this paper
• Polymer dynamics via cliques with applications to hard-sphere mixtures

  2020cites this paper
• Quantum stabilizer codes, lattices, and CFTs

  2020cites this paper
• Decentralized Attribution of Generative Models

  2020cites this paper
• Counting Glass States: The Complexity

  2020cites this paper
• The Gardner Transition

  2020cites this paper
• Atomic Liquids in Infinite Dimensions: Equilibrium Dynamics

  2020cites this paper
• Preface

  2020cites this paper
• Thermodynamics of Glass States

  2020cites this paper
• Infinite-Dimensional Models in Statistical Physics

  2020cites this paper
• Packing Spheres in Large Dimensions

  2020cites this paper
• Replica Symmetry Breaking and Hierarchical Free Energy Landscapes

  2020cites this paper
• Rheology of the Glass

  2020cites this paper
• Index

  2020cites this paper
• The Jamming Transition

  2020cites this paper
• Sphere packing and quantum gravity

  2019cites this paper
• Orthogonal polynomial expansions for the Riemann xi function

  2019influential citation
• Generalized Fourier Transforms Associated with the Oscillator Representation

  2019cites this paper
• Exponential improvements for superball packing upper bounds

  2019cites this paper
• Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

  2019cites this paper
• Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

  2019cites this paper
• Theory of Simple Glasses

  2019cites this paper
• On the Formation of Solid States Beyond Perfect Crystals: Quasicrystals, Geometrically-Frustrated Crystals and Glasses

  2019cites this paper
• Where will your next student find a job ?

  2019cites this paper
• Generalized Fourier Transforms Associated with the Oscillator Representation

  2019cites this paper
• Locally Optimal 2-Periodic Sphere Packings

  2019cites this paper
• Treatise On Solidification: A Comprehensive View on the Formation of Solid States Beyond Perfect Crystals

  2019cites this paper
• Eigenfunctions of the Fourier transform with specified zeros

  2019cites this paper
• Linear programming bounds for cliques in Paley graphs

  2019cites this paper
• The Jamming Transition

  2019cites this paper
• Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

  2018cites this paper
• Discrete Geometry 1137 Workshop : Discrete Geometry Table of Contents

  2018cites this paper
• Feature Systems of points with Coulomb interactions

  2018cites this paper
• Division Algebras, Clifford Algebras, Periodicity

  2018cites this paper
• Locally distributed spatial navigation in a scale-space model for grid cells

  2018cites this paper
• Division Algebras, Clifford Algebras, Periodicity

  2018cites this paper
• Alternative summation orders for the Eisenstein series $G_2$ and Weierstrass $\wp$-function

  2018cites this paper
• One-Dimensional Packing: Maximality and Rationality

  2018cites this paper
• Hyperuniform point sets on the sphere: probabilistic aspects

  2018cites this paper
• Energy Minimization in $CP^n$: Some Numerical and Analytical Results

  2018cites this paper
• The Theory of Modular Forms and the Sphere Packing Problem in Dimensions 8 and 24

  2018cites this paper
• ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

  2017cites this paper
• One-Dimensional Packing: Maximality Implies Rationality

  2017cites this paper
• Symplectic embedding problems, old and new

  2017cites this paper
• Extreme Points of the Vandermonde Determinant on Surfaces Implicitly Determined by a Univariate Polynomial

  2017cites this paper
• Locally Optimal 2-Periodic Sphere Packings

  2017cites this paper
• An optimal uncertainty principle in twelve dimensions via modular forms

  2017cites this paper
• On Minimal Covolume Hyperbolic Lattices

  2017cites this paper
• SYSTEMS OF POINTS WITH COULOMB INTERACTIONS

  2017cites this paper
• Towards a proof of the 24-cell conjecture

  2017cites this paper
• The sphere packing problem in dimension 24

  2016cites this paper
• A Brief Bibliography of Publications about the Kepler Conjecture

  2015cites this paper