Not sure if this question is too general for math.stackexchange but I will try.
I find the classification of finite simple groups fascinating. That we have sorted all possible finite symmetries into neat little groups. On the way we learnt fun facts, like the existence of the monster group.
I also enjoy the idea that we have classified shapes. We have triangles, squares, general n-gons, and so on. We can extend, categorise, and classify these further into the 3rd/4th dimension, self-intersecting shapes, what have you. We've got some mathematical objects, and we've sorted them into boxes.
My question is, is there a list of all classifications out there? I just do mathematicals for fun, so would like to know where I can find a big list of all the classifications in mathematics. Is there a table/paper/mathexchange post that answers the following(for example?):
- Is the classification of knot theory complete?
- Is there a classification of fields/rings?
- What about abstract algebras in general?
- Topological shapes?
- Have all types of differential equation been classified?
- etc. etc.?
Where do I find this list/table, or does it not exist? Thanks
I don't think that there is something like such a list. We generally define certain properties to categorize what otherwise would be a wild collection of very different examples. Thus we defined properties which helped us to distinguish between different groups, vector spaces, rings, algebras, fields, etc. Some of them led to full characterizations as e.g. for semisimple Lie algebras, simple finite groups, finite abelian groups, finite fields or real division algebras. For others we still need more criteria. Finite groups in general are simply too arbitrary, although a valid category.
The problem with your question are those properties. How can we know an area is fully described? Simple finite groups is such a category, nilpotent groups are not. And they probably never will be, since they are just too many. So another criterion will likely be needed, e.g. a certain length of their composition series. See, unless we have a full description we do not know, whether our criteria are good ones or not.
However, I have a full categorization of all even prime integers ; -)