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Comment by mathgenius | Hacker News Reader

Somewhat randomly, I've been working with a mathematician that has convinced me to study the Langlands program with him. I feel ill-equipped for the journey, having boycotted number theory ever since they tried to get me to learn the multiplication table. But on the other hand, there is plenty of geometry involved! Who would have thought that the dodecahedron is involved with solving the quintic? It really does seem like this area (the Langlands program) is at a vast nexus of many different branches of mathematics. It's intense.

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Shapes just come about when you are talking about objects with certain types of symmetries. One of the main "attacks" on polynomials is to realize that their roots are related to eachother, in terms of total, product, difference, etc.

The simplest of these methods is just Vieta's formulas:

https://en.wikipedia.org/wiki/Vieta%27s_formulas

I highly recommend Edward Frenkel's book, Love and Math. It was a great read about the mathematics itself, and also his troubles as a Jewish academic in Soviet Russia.

Love and Math: https://amzn.to/2qcqUb4

Right, but the dodecahedron is a modular curve, and I'm guessing this is how Hermite came up with the solution to the quintic using modular functions in 1858. If there are any real number theorists here they might be able to clarify this some more. Modular functions, modular forms, automorphic forms, these are all key ingredients of the Langlands program.

I believe you're talking about the free algebras on groups of modular forms of a certain 'weight.' That's not exactly the modular forms themselves.

Here's some more information: https://johncarlosbaez.wordpress.com/2017/12/31/quantum-mech...

That john baez article is about something else.

Try this: https://arxiv.org/pdf/1308.0955.pdf

What do you mean when you say the dodecahedron is a modular curve? Is there some N for which X_1(N) can be naturally thought of as a dodecahedron or something?

Edit: Okay glanced at the arxiv paper linked below. The dodecahedron is not a modular curve per se, but the link is as follows:

The quotient X(5) -> X(1), which geometrically is a quotient from P^1 (the sphere) to itself, can also be thought of as quotienting the sphere by the group of symmetries of its inscribed dodecahedron. I believe this group can naturally be thought of as PSL_2(Z/5Z)

It's quite complicated (https://en.wikipedia.org/wiki/Bring_radical) ..amazing he derived it so long ago , long before the internet and modern research tools were invented. he had to scour for all that information and derive this solution.. even with the help of the internet I find it hard to understand

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