4 ms·

Universal Math Solutions in Dimensions 8 and 24

45 points by eaguyhn 7y ago

Cogito 7y ago
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
- maxConfusion 7y ago
  There is a new lecture series here to add a resource: https://youtu.be/xALXm2XHDWc https://youtu.be/xALXm2XHDWc
fithisux 7y ago
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 7y ago
  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 7y ago
    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 7y ago
      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 7y ago
        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.

45 points by eaguyhn 7y ago

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

maxConfusion 7y ago
There is a new lecture series here to add a resource: https://youtu.be/xALXm2XHDWc https://youtu.be/xALXm2XHDWc

fithisux 7y ago

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 7y ago
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 7y ago
  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 7y ago
    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 7y ago
      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.