Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md
The most beautiful things in mathematics are results that are:
Good lists of such problems Lists of mathematical problems.
Specific examples:
Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:
Am I achieving insight, or am I just memorizing definitions?
Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.
One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for K-12, lower undergrad levels), but extremely hard to prove.
This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.
In mathematics, a "classification" means making a list of all possible objects of a given type.
Classification results are some of Ciro Santilli's favorite: Section "The beauty of mathematics".
Oh, and the dude who created the en.wikipedia.org/wiki/Exceptional_object Wikipedia page won an Oscar: www.youtube.com/watch?v=oF_FLN-TmCY, Dan Piponi, aka @sigfpe. Cool dude.
Good place to hunt for the beauty of mathematics.
He's a bit overly obsessed with polynomials for the taste of modern maths, but it's still fun.
Ciro Santilli would like to fully understand the statements and motivations of each the problems!
Easy to understand the motivation:
Hard to understand the motivation!
  • Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptography
    Of course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.
    As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.
  • Yang-Mills existence and mass gap: this one has to do with findind/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff value
    This is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.

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