>>7799882
I prefer algebraic geometry and differential topology.

>>7799887
Algebraic Geometry scares me, seems like it's predominately related to ring theory, which I wasn't a fan of, Algebraic Topology is predominately homological algebra which is a cool as fuck field.

>>7799892
Rings are dual to affine schemes, so ring theory and algebraic geometry are basically the same thing. It's interesting though, because you can reinterpret properties of rings as geometric properties. For instance, product rings are the same thing as disconnected spaces.

>>7799887
Topological algebra, and geometrical difference

>>7799908
Either algebraic geography or geological differences

>>7799892
You need to learn commutative algebra (theory of commutative rings) properly to be considered a serious pure mathematician.

>>7799941
Would you recommend Eisenbud?

>>7799882
If by cool, you mean 'in', something that is hip, then anything topology related. But truly cool something would be algebraic geometry as other anon has mentioned

>>7799882
toporential algemetry and geobraic differology imo

>>7799924
>geography
Maxim overpleb

>>7799950
Algebraic memeology is best

>>7799944
It depends how advanced you are. When I learned it, the book Undergraduate Commutative Algebra by Miles Reid was good for me. It's a serious book, and it contains serious topics like primary decomposition, Nullstelensats, noether normalization, etc. and lots of good algebraic geometry exercises.

Commutative Algebra by Atiyah and Mcdonald is harder and more algebraic (not much geometry). It was too hard for me when I first looked at it, but after going through Reid's book I found it to be good.

>>7799944
Sorry, forgot to answer your question. It only works for me as a reference. I wouldn't learn things for the first time from it.

Differential Geometry,
and recently Syntetic Differential Geometry, which is very algebraic

differential topology is cool as shit

I'm going through a book called Functional Differential Geometry and it's pretty good. I recommend it.