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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.
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As far as I know (and I could very well be wrong and hopefully someone will correct me), at Cambridge Part III (which you named Master's of Advanced Study) is compulsory for everyone, even PhD students. – Git Gud Feb 21 at 20:30
I am really not very qualified to answer the question as a whole since I've only attended universities in the US, but it would seem to me that the level of program you apply to will be directly related to your preparation as an undergraduate. That you've read/seen Dummit and Foote is great, but that says nothing else about what exposure you've had to some of the core courses that graduate schools like to see (my own impression). Bottom line: you would have to provide more information on your preparation to get a really appropriate answer. – Clayton Feb 21 at 20:33
You might be interested in this program as well as another option, or you could apply directly to any number of masters programs at various European universities. Usually a masters is required for PhD enrollment I think, but why not try asking people in the departments you're interested in working in to see what they recommend? – vgty6h7uij Feb 21 at 20:33
Most PhD students in Cambridge take Part III or equivalent, but this year there's one who started straight after undergraduate studies in America. There is some cultural difference – for example, we don't do quals, and there are no compulsory classes. – Zhen Lin Feb 21 at 20:45
@ZhenLin From the sounds of things it is typical to do Part III then. Reading the "Unofficial Guide to Part III" by Tom K\"orner has helped, but I do have one question about number of different courses: I see that courses are either 2 or 3 units depending on how long they last, and then it is recommended you offer 17-19 credits for examination. Is this examination at the end of your year, or do you take an examination after each term? – JJR Feb 21 at 21:02
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