Generalized Poincaré conjecture Updated +Created
There are two cases:
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
Klein bottle Updated +Created
Synthetic geometry of the real projective plane Updated +Created
It good to think about how Euclid's postulates look like in the real projective plane:
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
The beauty of mathematics Updated +Created
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.
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.