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Recent aspects of sphere packings - Part I
Org: Karoly Bezdek (University of Calgary, Canada) et Oleg Musin (The University of Texas Rio Grande Valley, USA)
[PDF]
Org: Karoly Bezdek (University of Calgary, Canada) et Oleg Musin (The University of Texas Rio Grande Valley, USA)
[PDF]
- MARIA DOSTERT, Royal Institute of Technology (KTH), Stockholm, Sweden
Kissing number of the hemisphere in dimension 8 [PDF]
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The kissing number of spherical caps asks for the maximal number of pairwise non-overlapping unit spheres that can simultaneously touch a central spherical cap in n-dimensional Euclidean space. Bachoc and Vallentin proved using semidefinite optimization that the kissing number of the hemisphere in dimension 8 is 183. In this talk I will explain our rounding procedure to determine an exact rational solution of the semidefinite program from an approximate solution in floating point given by the solver. Furthermore, I will show that the lattice E8 is the unique solution for the kissing number problem on the hemisphere in dimension 8.
- ALEXEY GLAZYRIN, The University of Texas Rio Grande Valley, USA
Linear programming bounds revisited [PDF]
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This talk will be devoted to linear programming methods for sphere packing bounds. I will describe a new approach to classic bounds and, at the end, present a new short solution of the kissing number problem in dimension three.
- ALEXANDER KOLPAKOV, University of Neuchatel, Neuchatel, Switzerland
Kissing number in non-Euclidean spaces of constant sectional curvature [PDF]
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We obtain upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $\mathbb{H}^n$ and spherical $\mathbb{S}^n$ spaces, for $n\geq 2$, and show that $\kappa_H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac{n-1}{2}, \frac{1}{2}) \cdot e^{(n-1) r}$ for large $n$. Here $d_n$ is the sphere packing density in $\mathbb{R}^n$, and $B$ is the beta-function. We also produce numeric bounds by using semidefinite programs and spherical codes. A similar approach locates the values of $\kappa_S(n, r)$, for $n= 3,\, 4$, over subintervals in $[0, \pi]$ with relatively high accuracy. Joint work with Maria Dostert (KTH Stockholm, Sweden).
- OLEG MUSIN, The University of Texas Rio Grande Valley, USA
The SDP bound for spherical codes using their distance distribution [PDF]
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In this talk we present a new extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result for the distance distribution of a spherical code and discuss reasonable approaches for solutions of two long standing open problems: the uniqueness of maximum kissing arrangements in
$4$ dimensions and the $24$-cell conjecture.
- SERGE VLADUT, Aix-Marseille University, France
Lattices with exponentially large kissing numbers [PDF]
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The quality of a lattice $L \subset R^n$, considered as a sphere packing can be measured by its density and/or its kissing number. For $n \to \infty$ the classical Minkowski theorem implies the existence of lattice families with density behaving as $O(2^{-n})$. However, that classical method does not permit to construct lattices with exponentially large (in n) kissing numbers, and their existence was not known until very recently. I will explain how to construct such lattice families using rather roundabout way through coding theory and algebraic geometry.