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Register a new userDual linear programming bounds for sphere packing via modular forms
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We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.
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This expository paper describes Viazovskas 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.
We examine several currently used techniques for visualizing complex-valued functions applied to modular forms. We plot several examples and study the benefits and limitations of each technique. We then introduce a method of visualization that can take advantage of colormaps in Pythons matplotlib library, describe an implementation, and give more examples. Much of this discussion applies to general visualizations of complex-valued functions in the plane.
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $pi/3$ on the hemisphere in $mathbb R^8$, and we prove that the three-point bound for the $(3, 8, vartheta)$-spherical code, where $vartheta$ is such that $cos vartheta = (2sqrt{2}-1)/7$, is sharp by rounding to $mathbb Q[sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $f colon mathbb{R}^{12} to mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $widehat{f}$ by $widehat{f}(xi) = int_{mathbb{R}^d} f(x)e^{-2pi i langle x, xirangle}, dx$, and suppose $widehat{f}$ is real-valued and integrable. We show that if $f(0) le 0$, $widehat{f}(0) le 0$, $f(x) ge 0$ for $|x| ge r_1$, and $widehat{f}(xi) ge 0$ for $|xi| ge r_2$, then $r_1r_2 ge 2$, and this bound is sharp. The construction of a function attaining the bound is based on Viazovskas modular form techniques, and its optimality follows from the existence of the Eisenstein series $E_6$. No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of $f$ and $widehat{f}$ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.
For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.
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