The sphere packing problem in dimension 24
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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.read more
Citations
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Book
Mordell-Weil Lattices
Matthias Schütt,Tetsuji Shioda +1 more
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Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions
Patrick Charbonneau,Jorge Kurchan,Giorgio Parisi,Giorgio Parisi,Pierfrancesco Urbani,Francesco Zamponi +5 more
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.
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Hyperuniform states of matter
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.
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Hyperuniform States of Matte
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.
Journal ArticleDOI
The sphere packing problem in dimension 8
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.
References
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Book
A First Course in Modular Forms
Fred Diamond,Jerry Shurman +1 more
TL;DR: Modular forms, elliptic curves, and modular curves as Riemann surfaces have been used to define the Eichler-Shimura Relation and L-functions.
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Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors
TL;DR: In this article, the authors introduce vector valued modular forms for the metaplectic group and the regularized theta lift for the Fourier theta lifts, as well as a lifting into cohomology.
Journal ArticleDOI
Universally optimal distribution of points on spheres
TL;DR: In this article, the authors studied configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points).
Book ChapterDOI
Elliptic modular forms and their applications.
TL;DR: These notes give a brief introduction to a number of topics in the classical theory of modular forms, based on various courses held at the College de France in the years 2000–2004.