Classification of closed surfaces Updated 2025-07-16
So simple!! You can either:
A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.
You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!
Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.
Classification of finite fields Updated 2025-07-16
There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as .
Every element of a finite field satisfies .
It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!
Classification of finite rings Updated 2025-07-16
accounts for them all, which we know how to do due to the classification of finite fields.
So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)
Ciro Santilli is very fond of this result: the beauty of mathematics.
How can so much complexity come out from so few rules?
How can the proof be so long (thousands of papers)?? Surprise!!
And to top if all off, the awesomely named monster group could have a relationship with string theory via the monstrous moonshine?
The classification contains:
Video 1.
Simple Groups - Abstract Algebra by Socratica (2018)
Source. Good quick overview.
Classification of regular polytopes Updated 2025-07-16
The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts per dimension are:
DimensionCount
2Infinite
35
46
>43
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.
Generalized Poincaré conjecture Updated 2025-07-16
There are two cases:
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?