Abstract Algebra: Beyond and Higher?

[alvin51015]

I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.

 

[jedishrfu]

It's a pretty big field

http://en.m.wikipedia.org/wiki/Abstract_algebra

Like algebraic topology or maybe algebraic number theory?

 

[HallsofIvy]

There are many fields of abstract mathematics. They are generally not "ordered" so that we cannot think of one as being "beyond and higher" than another. But I think perhaps "Category Theory" (http://en.wikipedia.org/wiki/Category_theory) might qualify here.

 

[pwsnafu]

Representation theory probably counts.

 

[micromass]

Algebraic geometry should count

 

[Luis Gomez]

I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.

I would say Set Theory. Set theorists keep adding axioms to their Set theory and expect us to do math(such as category theory, mathematical analysis and pretty much all of the known math) withing their formal Axiomatic Set Theory.