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I find this question distasteful, since it's asked with nothing more than a vague insinuation of Viazovska potentially not being "that good" as motivation. Here
by andrelaszlo 4y ago
I find this question distasteful, since it's asked with nothing more than a vague insinuation of Viazovska potentially not being "that good" as motivation. Here's the long citation:
"A very long-standing problem in mathematics is to find the densest way to pack identical
spheres in a given dimension. It has been known for some time that the hexagonal packing of
circles is the densest packing in 2 dimensions, while in 1998 Hales gave a computer assisted
proof of the Kepler conjecture that the faced centered cubic lattice packing gives the densest packing in 3 dimensions. The densest packing wasn't known in any other dimension until in
2016 Viazovska proved that the E8 lattice gave the densest packing in 8 dimensions and, very shortly afterwards, together with Cohn, Kumar, Miller and Radchenko, proved that the Leech lattice gave the densest packing in 24 dimensions. Viazovska's approach built off work of Cohn and Elkies, who had used the Poisson summation formula to give upper bounds on the possible density of sphere packings in any dimension. Their work had suggested that in 8 and
24 dimensions there might exist a radial Schwartz function with very special properties (for
instance it and its Fourier transform should vanish at the lengths of vectors in the respective lattice packings) which would give an upper bound equal to the lower bound coming from the known lattice packings. Viazovska invented a completely new method to produce such
functions based on the theory of modular forms.
Viazovska has developed these ideas in other directions. With Radchenko she proved the
unexpected result that any even Schwartz function such that it and its Fourier transform
vanish at the square root of every non-negative integer must be identically zero. In fact they showed that any even Schwartz function can be written [...] for certain special functions a_n and b_n.
With Cohn, Kumar, Miller and Radchenko she showed that the E8 and Leech lattice not only
gave optimal sphere packings in dimensions 8 and 24, but that they minimize energy for every
potential function that is a completely monotonic function of squared distance."
- stainforth 4y agoSo what's so special about 8 and 24 dimensional space?
- ykonstant 4y agoThe short answer is that we don't really know; they are related to some of the most mysterious aspects of representation theory and number theory. What we know is that these dimensions support lattice structures with exceptional symmetries. The intricate geometry of these symmetries interacts with number theory to provide spectral data (automorphic forms) that have very rare properties; this is the starting point for Maryna's investigations. Okounkov's popular scientific exposition "The magic of 8 and 24", linked in the article, goes into considerable detail on the constructions.
- kryptiskt 4y agoThey contain extremely many symmetries. The Leech Lattice is certainly a uniquely symmetrical object.
- rowanG077 4y agoI agree that it is a distasteful question. But what is much more distasteful that it wouldn't be out of the realm of possibility for the answer to that question to be yes. Not that I am in any position to make that assessment.
- cycomanic 4y agoSo you admit you are in no position to make that assessment, but at the same time make the assessment that it is not out of the realm of possibility? Maybe if in no position to make an assessment it is better not to make one?!
- rowanG077 4y agoI'm not the person who made the statement. They asked the question. That's exactly the opposite of making an assessment. That person tries to find someone who CAN make that assessment to answer the question.
- cycomanic 4y agoYou wrote: > I agree that it is a distasteful question. But what is much more distasteful that it wouldn't be out of the realm of possibility for the answer to that question to be yes. So you said it is not out of the realm of possibility for it to be the case, that is an assessment. By your own statement, you are in no position to judge, so how do you know it is possible?
- rowanG077 4y agoHow do I know it's possible? Because it's decided by people. People are sensitive for these kinds of things. What I meant with assessment in this case is that I cannot judge whether this is happened in this specific case. As in I don't know enough about mathematics to make an assessment. Not I don't know enough about committees to say it's possible or not. I thought that would be clear from the post but I can see how you misunderstood it.