{tag=The beauty of mathematics}

= The beauty of alebraic geometry
{synonym}

https://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates Interesting results in algebraic geometry accessible to 3rd year undergraduates


The beauty of algebraic geometry

<Ciro Santilli> intends to move his beauty list here little by little: https://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
  * <number of unknown rationality>, e.g. is $e + \pi$ <rational>?
  * <transcendental number conjectures>, e.g. is $e + \pi$ transcendental?
  * basically any conjecture involving <prime numbers>:
    * <twin prime conjecture>
  * many combinatorial game questions, e.g.:
    * <first-move advantage in chess>
* 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 (mathematics)> counter-examples. General surprise themes include:
  * <classification (mathematics)> of potentially <infinite> sets like: compact <manifolds>, etc.
    * <classification of finite simple groups>
    * <classification of regular polytopes>
    * <classification of closed surfaces>, and more <generalized Poincaré conjectures>
    * classification of <wallpaper groups> and <space groups>
  * 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>:
        * <decidability of Hilbert's tenth problem of a given degree and number of variables>
        * <Hilbert's tenth problem over other rings>
  Lists:
  * https://math.stackexchange.com/questions/139699/what-are-some-examples-of-a-mathematical-result-being-counterintuitive
  * https://math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite/2055458#2055458
* applications: make life easier and/or modeling some phenomena well, e.g. in <physics>. See also: <how to teach/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:
> 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.


Ciro Santilli @cirosantilli 37

 Tagged: The beauty of mathematics