I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is given that I have knowledge in the above topics which is the next field that is appropriate to pick up to continue my studies in Algebra. This I feel is a relavent question because its always fun to pick up a field that is just out your reach and try to understand and master it, this is where I need help.
When you suggest a field keep in mind that it should just be out of reach in terms of abstractness, and please specify weather the suggested field actually applies the above topics or is a new topic in itself.
If possible post links to recommended book lists by famous mathematicians working in that field
For instance the book list for algebraic topology by Hatcher is a good example
http://www.math.cornell.edu/~hatcher/Other/topologybooks.pdf
Thank you