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A Breakthrough In Sphere Packing: The Search For Magic Functions

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D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016. Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015. For c = 4 and c = 12, these functionals exactly repro- duce the “magic functions” used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. Notices d’autorité : VIAF GND Blog de Gil Kalai sur les travaux de Viazovska [archive] David de Laat, Frank Vallentin: A Breakthrough in Sphere Packing: The Search for Magic Functions, Nieuw Archief voor Wiskunde, Septembre 2016, Arxiv [archive] Henry Cohn, A conceptual breakthrough in sphere packing, Notices AMS, fév. 2017, [1

These methods are based on finding a „magic function“ and exploiting the properties of this magic function to get a sphere packing. Viazovska’s breakthrough in [Via17] boils down to finding a magic function that yields a sharp bound in 8 (and later 24 dimensions). You can also have a look at this article with title A breakthrough in sphere packing: the search for magic functions written by David de Laat and Frank Vallentin for the Dutch mathematics magazine Nieuw Archief voor Wiskunde in 2016. In this article, the work of Maryna Viazovska is presented. From sphere packing to Fourier interpolation Henry Cohn Abstract Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll

HENRY COHN arXiv:1611.01685v1 [math.MG] 5 Nov 2016

Sphere packing | Hui Wang

Nieuw Archief voor Wiskunde: September 2016 In a periodic packing, spheres are not restricted to just the corners of a fundamental cell. No reason to believe densest packing must be periodic, but periodic packings come arbitrarily close to the maximum density.

A breakthrough in sphere packing: the search for magic functions. Zbl 1381.52027 de Laat, David; Vallentin, Frank 8 2016 “A breakthrough in sphere packing: the search for magic functions”. In: Nieuw Arch. Wiskd. (5) 17.3 (2016). Includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska, pp. 184–192. arXiv: 1607.02111. [Oko23] Andrei Okounkov. “The magic of 8 and 24”. In: ICM—International Congress of Search for a magic function To prove that E8 gives the densest packing in dimension 8, we need to find a radial Schwartz class function f on R8 satisfying the following:

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.

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 In March 2016, Ukrainian mathematician Maryna Viazovska provided the final solution for the Sphere Packing Problem in dimension 8 and later on – in collaboration – in dimension 24. Both papers Sphere Packing Problem in dimension 8, and 24, put an end to almost four centuries of studies of Kepler’s Conjecture. Mathematicians used “magic functions” to prove that two highly symmetric lattices solve a myriad of problems in eight- and 24-dimensional space.

  • Nieuw Archief voor Wiskunde: September 2016
  • AMS :: Proceedings of the American Mathematical Society
  • 8-dimensionale Kugelpackungen

A Breakthrough in Sphere Packing: The Search for Magic Functions Article Jul 2016 David de Laat Frank Vallentin

AMS :: Bulletin of the American Mathematical Society

Download Citation | Some properties of optimal functions for sphere packing in dimensions 8 and 24 | We study some sequences of functions of one real variable and conjecture that they converge

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Abstract. Viazovska’s solution of the sphere packing problem in eight di-mensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new Google matrix has billions of dimensions (a googol is 10100); Megapixel as points in R 3000000, etc. Sphere packings in communication theory/coding theory: Figure: From: Tsukerman, Communication and ball packing, Plus Mag. Search for a magic function To prove that E8 gives the densest packing in dimension 8, we need to find a radial Schwartz class function f on R8 satisfying the following:

A breakthrough in sphere packing: the search for magic functions. Nieuw Arch. Wiskd., 5/17:184–192, 2016. [21] B. de Smit, D. Dunham, H.W. Lenstra Jr., and R. Sarhangi. Artful mathematics: the heritage of M. C. Escher. Not. Am. Math. Soc., 50:446–457, 2003. [22] M. Dewar and M. Ram Murty. An asymptotic formula for the coeficients of j(z

David de Laat and Frank Vallentin, A breakthrough in sphere packing: the search for magic functions, Nieuw Arch. Wiskd. (5) 17 (2016), no. 3, 184–192. Includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska. MR 3643686 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 Without realizing how profoundly dificult the remaining step was, I imagined that we had almost solved the sphere packing problem in eight and twenty-four dimensions, and our inability to find the magic functions was extremely frustrating.

By Erica Klarreich In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest

FROM SPHERE PACKING TO FOURIER INTERPOLATION

Blog von Gil Kalai zum Beweis von Viazovska (englisch) David de Laat, Frank Vallentin: A Breakthrough in Sphere Packing: The Search for Magic Functions. Nieuw Archief voor Wiskunde, September 2016, Arxiv. Henry Cohn: A conceptual breakthrough in sphere packing. Notices AMS, Februar 2017, Online (PDF; 23,7 MB). Clay Research Award 2017 We establish a precise relation between the modular bootstrap, used to con- strain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra U(1)c maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in d = 2c dimensions. We also show that the analytic Henry Cohn – A conceptual breakthrough in sphere packing David de Laat, Frank Vallentin – A breakthrough in sphere packing: the search for magic functions 1. Vorbesprechung: 20.1-24.1.2020 2. Vorbesprechung: 27.2.-28.2.2020 Themen: Die Gitter E8 und Lambda24 Die lineare Programmierungsschranke Anwendungen der Interpolationsformel

With her homeland mired in war, the sphere-packing number theorist Maryna Viazovska has become the second woman to win a Fields Medal in the award’s 86-year history. To squeeze together three-dimensional cubes, you stack them like moving boxes. Mathematicians can easily extend these arrangements, packing cubes in higher-dimensional space to perfectly fill it. Packing spheres is much harder. Mathematicians know how to pack circles or soccer balls together in a way that minimizes the empty space David de Laat, Frank Vallentin: A Breakthrough in Sphere Packing: The Search for Magic Functions , Nieuw Danemark voor Wiskunde, September 2016, Arxiv Henry Cohn, Ein konzeptioneller Durchbruch in der Kugelpackung , Notices AMS, Feb. 2017, pdf