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I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, elliptic curves); but I still don't know enough to make any definite statements.

I have done a good amount of research on the different graduate schools, but I am hoping for some advice from people more involved in the field. What are some universities with good graduate programs in mathematics, and in particular algebra and algebraic number theory? Where is the interesting research going on in those fields?

I have seen the US News list on best algebra/number theory/algebraic geometry grad schools. And have looked into the research in Princeton, MIT, UC San Diego, Stanford, Berkeley, Brown, University of Chicago. So I was hopping for answers of less known universities, or more information about the known ones which is not easily available.

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Even though this isn't directly your question, I feel like this may help. Berkeley is really good if you're interested in arithmetic geometry, with an emphasis on geometry. Martin Olsson, for example, is pretty fantastic, as well as his advisor Arthur Ogus, as well as Vojta, Ken Ribet, etc. Also, there are some people there who, while still probably strongly arithmetic geometers, do some more "pure" number theoretic things--people like Xinyi Yuan .There is a chance that they might also pick up some more number theorists in the next year or two, but that is still speculative. –  Alex Youcis Nov 10 '13 at 0:26
My point being, if you're looking for good-old-classic algebraic number theory, you might want to inspect Berkeley more closely. I would suggest, since it's not on your list, looking at UPenn for good number theorists. They have Ted Chinburg, David Harbater, Florian Pop, etc. They are also, well at least David Harbater (or so it seems to me!), doing things that are more "pure number theory". Also, I'm sure you know, but Harvard, obviously, has one of the best number theory groups in the country (the Taylor group is really influential). –  Alex Youcis Nov 10 '13 at 0:29
Also, Northwestern has a fantastic program--even if just because Frank Calegari is there. –  Alex Youcis Nov 10 '13 at 1:34
Also, if that's in your scope, Europe has some great researches doing number theory. I would mention Bonn, Heidelberg, Paris-Sud XI, Regensburg, etc. –  BIS HD Nov 11 '13 at 15:51
@Zlatan: I am very curious how you arrived to the conclusion that Bonn is a "very bad university"? Did you get offended by Faltings? (Do you even know who he is?) –  studiosus Jun 28 '14 at 11:20

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