the dichotomy between BigSchools and friendly/less competitive places. I’d say this is more of a weak correlation than any kind of causation. You can learn who’s Big by asking people or reading, but you can’t really get any kind of feel for the friendliness of a place except by visiting.

Second, size can be a good thing. If you really know you want to do CA, that’s great, and you should make a beeline for a place that’s got a great group in CA, irrespective of size. If not, then one of the advantages of a bigger place is that you’re less constrained if you do change your mind. Another advantage is that you may have opportunities to round out your view of mathematics that you wouldn’t at a smaller place — you can’t ever figure out that Lie Theory is really the love of your life if you never get to have a course in it. (Confession: I never had one, and I’m starting to think that Lie Theory may be the love of my life.)

Third, size (in terms of number of students, or number of faculty, or name recognition) doesn’t really correlate at all with strength in Commutative Algebra. You can get a decent idea of where the hotspots are in CA by scanning the list of people at commalg.org. Generally, the places with more than one or two people mentioned on that list are the best-regarded places. Here are a few particular places: (redacted, mostly, but here are the names: MI, NE, KS, Purdue, Mizzou, Illinois, Cornell, Berkeley, Syracuse, Washington. Also, but less so, UT, TX, A&M, KY.) You’ll note MIT, Princeton, Harvard, etc. aren’t on the list. They’d much rather do sexy things like mathematical physics, etc.

Generally, in my experience, “rankings” of graduate schools are almost worthless. For example, Princeton is generally ranked at the top of every list — but no one there does algebra, at least not the kind I’m interested in. So it would have been a huge mistake for me to go to Princeton, or at least wouldn’t have led to where I am now (which I quite like). I would have had an impressive name on my diploma, but wouldn’t have had anyone there to work with or learn from. And among people who know commutative algebra, Nebraska or Purdue is a much more impressive name than Princeton — they know where the active people are. (Note: here is a weighted-preference ranking engine, which looks interesting.

I think this is also true for other fields of pure mathematics: for each field, there is a generally accepted group of places that specialize in that area, and are known to be the best places for that area. They may or may not include the big-name schools (Princeton, Harvard, etc.). The problem is that not everyone is familiar with every field, and it’s easier for them to be impressed by a big name that they’ve heard of. This is definitely something to keep in mind. It does make a difference in getting a job — it’s very rare for someone to get an academic position at a “better” department than the one they graduated from (where “better” means “more well-known to non-specialists”).

In commutative algebra, there are at least two subfields: people either specialize in Noetherian rings, or non-Noetherian rings. For the Noetherian group (which is what I know most about) the generally accepted list would probably include Purdue U., U. of Nebraska, U. of Kansas, U. of Illinois at Urbana-Champaign, U. Utah, U.Minnesota, U. Missouri, U California-Berkeley, U. Michigan, U. Texas … those are just off the top of my head, so I may have forgotten one or two.