
Universal Math Solutions in Dimensions 8 and 24 - eaguyhn
https://www.quantamagazine.org/universal-math-solutions-in-dimensions-8-and-24-20190513/
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Cogito
Seems like the crucial step here was being able to develop a systematic way to
address every possible type of 'repulsive force' in a given dimension, which
then allowed them to show that the given solution (for example E8 in dimension
8) optimally solved the packing problem for each possible force.

They talked about a few examples of the different repulsive forces that were
characterised, but I wonder what they mean by _all_ of those forces. It seems
like there are an infinite way to parameterise a force between two points
given a measure (by taking different functions of the distance between the
points) but also an infinite way of choosing a measure.

In the article they say

> _Viazovska, Cohn and their collaborators restricted their attention to the
> universe of repulsive forces. More specifically, they considered ones that
> are completely monotonic, meaning (among other things) that the repulsion is
> stronger when points are closer to each other. This broad family includes
> many of the forces most common in the physical world. It includes inverse
> power laws — such as Coulomb’s inverse square law for electrically charged
> particles — and Gaussians, the bell curves that capture the behavior of
> entities with many essentially independent repelling parts, such as long
> polymers. The sphere-packing problem sits at the outer edge of this
> universe: The requirement that the spheres not overlap translates into an
> infinitely strong repulsion when their center points are closer together
> than the diameter of the spheres._

So I guess that means that it doesn't really matter which kind of measure you
use as long as the repulsive force is monotonic. The auxiliary functions they
discuss probably incorporate however the measure and force are defined in a
nice way, so that the method for constructing the optimal auxiliary function
is agnostic towards them.

Reading more, in the linked paper's abstract [0], it seems they are only
dealing with euclidean distance, and the auxiliary function is points on a
sphere:

> _We study 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). Call a
> configuration sharp if there are m distances between distinct points in it
> and it is a spherical (2m-1)-design. We prove that every sharp configuration
> minimizes potential energy for all completely monotonic potential functions.
> Examples include the minimal vectors of the E_8 and Leech lattices. We also
> prove the same result for the vertices of the 600-cell, which do not form a
> sharp configuration. For most known cases, we prove that they are the unique
> global minima for energy, as long as the potential function is strictly
> completely monotonic. For certain potential functions, some of these
> configurations were previously analyzed by Yudin, Kolushov, and Andreev; we
> build on their techniques. We also generalize our results to other compact
> two-point homogeneous spaces, and we conclude with an extension to Euclidean
> space._

[0] [https://arxiv.org/abs/math/0607446](https://arxiv.org/abs/math/0607446)

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maxConfusion
There is a new lecture series here to add a resource:
[https://youtu.be/xALXm2XHDWc](https://youtu.be/xALXm2XHDWc)

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fithisux
I hate such articles. Please use theorems and proofs with pedagogy in mind,
other wise it is just a nother failed miracle cure for cancer.

~~~
jhanschoo
You can find the paper preprint here
[https://arxiv.org/pdf/1902.05438.pdf](https://arxiv.org/pdf/1902.05438.pdf)
linked to by the author.

What you're asking for is impossible. A paper is already written to
communicate to other mathematicians; teaching the prerequisites not mentioned
in the paper can't fit in anything but a graduate textbook.

The magazine article still serves a purpose. It sketches the shape of the
problem and the approach for the proof, and this is useful to math
practitioners of at least undergraduate level, in the sense that they may take
inspiration from the approach in their own problems, or may convinced enough
to have a look at the actual paper, or have to a first approximation a sense
of whether this paper can help them.

~~~
commandlinefan
I wish I could get a handle on what I needed to learn in order to understand
this stuff. I dug out my old undergraduate calculus textbook and started
working through all the problems; I'm about halfway through now, but terms
like "lattice", "zeta function" and "Laplace transforms" are still greek to me
(sometimes literally).

~~~
gtani
To bootstrap your understanding in general, pursue calculus, linear algebra
and probability in tandem, then go thru texts on real analysis and abstract
algebra. Also math summary books: Courant/Robbins, Ian Stewart has a couple,
Stillwell "Elements of"

~~~
commandlinefan
Thanks, much appreciated! I’m about half-way through my calculus textbook
right now, and I’m actually learning quite a bit since I’m going over some of
the sections my instructor skipped over (plus it’s been 30 years since I took
the class, so I’m re-learning a thing or two that I had forgotten…). Any
recommendations for real analysis and abstract algebra texts? I’m self-
studying, so I’m looking for books that include lots of practice problems with
answers (at least) in the back of the book so I can check to see if I’m on the
right track.

~~~
jhanschoo
I recommend heading over to reddit's r/math to ask for undergrad book
recommendations! They're friendly, and the quality of discussion there is
surprisingly high compared to other subreddits.

