Researchers chip away at Smale's 7th unsolved problem in mathematics | Hacker News Reader

Researchers chip away at Smale's 7th unsolved problem in mathematics

phys.org

53 pointsdnetesn17 comments

Correction: The article states that Thomson's problem has only been solved for 2, 3, 4, 6, and 12 charges. However, in 2013 the 5-electron case was solved by Richard Schwartz. Here's the paper:

http://www.tandfonline.com/doi/abs/10.1080/10586458.2013.766...

Needless to say, there has been a lot of work on this subject from a number of different directions, and any particular new paper is unlikely a priori to be a revolutionary breakthrough. Look at Google Scholar or a similar citation graph and you can find thousands of papers about it. (Some key words: “spherical cap discrepancy”, “Fekete points”, “Riesz energy”, “logarithmic potential”, obviously also “Thomson’s problem”, etc.)

I actually was asked this problem in a programming interview. This is first I've read that it is an unsolved problem, which definitely makes me chuckle.

How did you answer?

I guess if you give someone a puzzle, there's a possibility they might solve the puzzle from memory. If you give an unsolved problem, you know they won't have seen the solution before and can see their thought process better(since you know thinking will be required).

The exact phrasing matters too, I guess. There are lots of ways to find a good packing, and even good bounds on how far your good packing is from the global optimizer. So a reasonable question to your interviewer would have been "does it need to be the exact global optimizer, or can it be a good approximation ?". If they wanted a good approximation it might not be a bad question, and the question may have been designed to get you to ask for the distinction? Or maybe they were just dicks, who knows.

Reminds me of reading about at least one case where a problem was solved in a situation like that. Anyone recall what it was?

Dantzig solved an open problem in statistics which he thought was being causally assigned as homework:

https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st...

I think you mean "casually", not "causally" (even though it's kind the interesting Freudian slip, IMHO).

but Wikipedia shows a huge table of values.it's just that we don't have a name or certain constructions for values above 12

Those are the best known configurations, we don't have proofs that they are the lowest possible.

Isn't this an optimization problem that can be solved with computer to solve a high degree equation?

Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.

And even if you can find a numerical solution to arbitrary precision, at best you only get arbitrarily close to the mathematically optimal solution (at best because a purely numerical approach cannot rule out that it misses a very small, but high peak in the function to be optimized)

As an extreme example, for n=2 and using binary floating point, you probably will find the mathematically correct solution, but there's no way to tell numerically whether your answer is off by a fraction of your floating point's epsilon value.

That's probably not important to physicists who want to know an answer, but it is for mathematicians.

What I find very surprising is (from https://en.m.wikipedia.org/wiki/Thomson_problem):

"Numerical solutions for N=8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively."

I would like to see the visualizations of the better solutions for n=20 (wikipedia links to the one for n=8) and/or hear a heuristic argument as to how that can happen.

For N=8 the solution listed on the Wikipedia page is a square antiprism (https://en.m.wikipedia.org/wiki/Square_antiprism) which seems intuitively reasonable. But, I don't see anything listed for N=20. Look at the table in S.5 of the article (https://en.m.wikipedia.org/wiki/Thomson_problem#Configuratio...) for more links to visualisations in the rightmost column, if one exists.

The function here has a simple continuity, so there is no risk of narrow peaks beyond some computable threshold. The challenge is the high dimensionality.

The unsolved problem referred to in the title is finding a configuration close to the global minimum. Most optimization algorithms need such a configuration as a starting point.

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