I am an undergraduate at a university here in the U.S., and I am hoping to apply overseas to graduate school. Now, the general process for applying to graduate mathematics programs in the US seems to be to come out of undergrad and apply to a Ph.D program. However, (as a top choice) if I am looking at Cambridge, it seems that it would be more appropriate for me to apply to a Master's of Advanced Study program, and then apply to a Ph.D program (possibly not necessarily at Cambridge).
I am wondering what math.SE has to give me in terms of advice? I would dearly like to study an algebraic topic, and so far Commutative Algebra or Algebraic Geometry seem to be the most interesting. I have seen the first 14 chapters of Dummit and Foote (up through Galois Theory) either in classes or in self study excepting the end of chapter 6 and chapters 10-12 (well, I've looked through 10-12, and worked some exercises in the first three sections of both 10 and 11, but I would definitely have a lot to learn from a class involving modules over PIDs and tensor algebra).
So I am wondering which level of overseas program would be appropriate for my application? Thanks for any and all advice.
Edit As Clayton pointed out, I've given no background on my other maths education. By the time I graduate, I will have taken:
- Analysis up through seeing Brouwer's fixed point theorem as well as integration on differential forms
- Topology, and a first class on Algebraic Topology (out of Hatcher).
- Misc: some stochastic processes (Poisson processes and queueing and martingale basics i.e., optional stopping theorem, as well as basic probability.