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:

- simple to state but hard to prove:
- Fermat's Last Theorem
- transcendental number conjectures, e.g. is $e+π$ transcendental?
- basically any conjecture involving prime numbers:
- many combinatorial game questions, e.g.:

- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:Lists:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes

- unpredictable magic constants:
- why is the lowest dimension for an exotic sphere 7?
- why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem

- undecidable problems, especially simple to state ones:
- mortal matrix problem
- sharp frontiers between solvable and unsolvable are also cool:
- attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
- related to Diophantine equations:

- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson

Good lists of such problems Lists of mathematical problems.

Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:
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.

Am I achieving insight, or am I just memorizing definitions?

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.

Bibliography:

- mathoverflow.net/questions/75698/examples-of-seemingly-elementary-problems-that-are-hard-to-solve
- www.reddit.com/r/mathematics/comments/klev7b/whats_your_favorite_easy_to_state_and_understand/
- mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts this one is for proofs for which simpler proofs exist
- math.stackexchange.com/questions/415365/it-looks-straightforward-but-actually-it-isnt this one is for "there is some reason it looks easy", whatever that means

Randomly reproduced at: web.archive.org/web/20080105074243/http://personal.stevens.edu/~nkahl/Top100Theorems.html

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".

Examples:

- classification of finite simple groups
- classification of regular polytopes
- classification of closed surfaces, and more generalized generalized Poincaré conjectures
- classification of associative real division algebras
- classification of finite fields
- classification of simple Lie groups
- classification of the wallpaper groups and the space groups

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.Cool examples:

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:

- Navier-Stokes existence and smoothness is basically the only problem that is really easy to understand the statement and motivation :-)
- p versus NP problem

Hard to understand the motivation!

- Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptographyOf 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 valueThis 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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