For example, if someone asked you whether any real number Viazovska’s magic function method was powerful — almost too powerful, in fact.
“Even in light of the earlier work, I would not have expected this [universal optimality proof] to be possible to do,” Hales said.It might seem strange that dimensions eight and 24 should behave differently from, say, dimension seven or 18 or 25.
In 2006, Cohn and Kumar The five authors of a new paper proving universal optimality in dimensions 8 and 24, shown clockwise from top left: Henry Cohn, Abhinav Kumar, Maryna Viazovska, Stephen Miller and Danylo Radchenko.Bryce Vickmark (Cohn); Courtesy of Abhinav Kumar; Daniil Yevtushynsky (Viazovska); Julia Semikina (Radchenko); Olena Shmahalo/Quanta Magazine (Miller)In most dimensions, the numerical bounds Cohn and Kumar found bore little resemblance to the energy of the best-known configurations.
Heisenberg’s uncertainty principle — which says that the more you know about a particle’s position, the less you can know about its momentum, and vice versa — is a special case of this general principle, since a particle’s momentum wave is the Fourier transform of its position wave.In the case of a repulsive force in dimension eight or 24, Viazovska made a daring conjecture: that the limitations the team wanted to place on their magic function and its Fourier transform lay precisely on the border between the possible and the impossible.
There’s a host of different such problems, and it’s not obvious why they should all have the same solution. Likewise, in dimension 24, the Leech lattice arises from fitting extra spheres into the gaps in another well-understood sphere packing.For reasons mathematicians don’t fully understand, these two lattices crop up in one area of mathematics after another, from number theory to analysis to mathematical physics.
Eugene Higgins Professor of Mathematics at Princeton University (not
In sphere packing, each sphere cares only about the location of its nearest neighbors, but for something like electrons scattered through space, every electron interacts with every other electron, no matter how far apart they are. 77 hours on the supercomputer No point configuration is optimal with respect to every single goal: For instance, when an attractive force acts on the points, the lowest-energy configuration will not be some lattice but just a massive pileup, with all the points at the same spot.Viazovska, Cohn and their collaborators restricted their attention to the universe of repulsive forces. This could have unforeseen implications in mathematics and physics which research in other areas." In the two-dimensional plane, there’s a candidate for universal optimality — the equilateral triangle lattice — but no proof. In most dimensions, mathematicians don’t believe this is remotely likely to be true.But dimensions eight and 24 each contain a special, highly symmetric point configuration that, we now know, simultaneously solves all these different problems.
If you wanted to ship an eight-dimensional soccer ball in the smallest possible box, the ball would fill less than 2 percent of the box’s volume — the rest would be wasted space.In each dimension higher than three, it’s possible to construct a configuration analogous to the pyramidal orange arrangement, and as the dimension increases, the gaps between the spheres grow. According to project leader Jeffrey Adams, "This is basic It is, Schwartz said, “one of the best things I’ll probably see in my lifetime.”Mathematicians used “magic functions” to prove that two highly symmetric lattices solve a myriad of problems in eight- and 24-dimensional space. If 200 E-5s leave the Army in a particular MOS, then 200 E-4s can be promoted to E-5.
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