Choosing where to get your PhD is an important decision. If you continue onto academia, your PhD might be the longest time you spend at any one institution until you get a permanent position. The most obvious choice is apply to the high-ranking schools. However, you should consider far more than that. Here, we'll look at some of the important factors to consider, with the context of mathematics in mind. However, most of what I say applies to some other fields as well.
Represented research areas
Unlike choosing an undergraduate program, where the curriculum doesn't differ much around the world (though it certainly can vary greatly in strength or intensity), a PhD will be on a very specialised topic. So, if you go to a school where analysis and statistics are the main topics represented and you like algebra, you probably won't like it. This can be worse for those places where you don't have to choose an advisor until the second year. So I suggest you look at the represented research areas on departmental websites and see what catches your interest. Unfortunately, some math department websites look like they were coded on a Super Nintendo, if that were even possible. So:
If you're at the undergraduate level and not sure of your interests yet, it could be a good idea to consider a masters program first before starting a PhD. I enjoyed doing a masters degree first, even though in the long run it is more expensive.
Total school atmosphere
If you're lucky enough to be nearby some schools you're interested in, you should visit them, meet some professors, and even sit in on some classes and departmental seminars. Just walk around and see what it's like. Some schools have a much nicer atmosphere than others. You should also get a sense of the surrounding city. This is true especially if you are a very independent worker: having an enjoyable city will in fact make working much easier. Conversely, living in a place you dislike for several years is quite draining.
Sadly, living temporarily in cities you don't like is very probable in at least one stage of climbing the academic ladder.
The most important aspect of doing a PhD is by far the advisor. So you should try and meet potential advisors, either in person if possible or by Skype if you live too far away. Half an hour talking to this person will give you more information than hours of web surfing.
Some schools only require choosing an advisor in the second year of the program. If you are not sure who you want to work with yet, this is a big advantage for these types of programs.
What should you look for in an advisor? I think the most important quality is probably that you like them and can get along with them, and this is probably even more important than their mathematical achievements. After all, you will have to spend four years meeting with them dozens of times per year!
If you are meeting potential advisors in person, watch out for those that are dismissive and don't take your meeting seriously. Don't consider these potential advisors, even if they are famous.
The highest rated school is not always the best
There are downsides to some highly-rated schools. Some of places have a lot of pressure and competition, which can be worse at some top schools. Also, any school regardless of its ranking will not have every research area represented and thus may not be a good fit. However, highly-rated schools will often have stronger programs and stronger people there that may help advance your career.
However I suggest considering each school individually with a critical eye, no matter what the ranking.
The program itself
Just like schools, PhD programs vary. Some require you to take courses. I feel that courses are important, because it shows that the faculty is actually interested in teaching the graduate students specialised topics that are hard to learn on your own.
Some programs have funding, which may be in part covered by light teaching duties. For example, at McGill part of my funding came from being a "teaching assistant" for one semester per year. This job entailed holding one-hour problem sessions once a week, holding office hours, and marking exams. To contrast, some PhD programs in the United States require you to actually teach full courses, which takes way more time. On the other hand, teaching actual courses gives you more experience should you want to work at a more teaching-oriented school later on. Wherever it comes from, you will most likely need a program with full funding.
But where do you get all this information?
An important thing to watch out for is the expected length of the PhD. For math these days, and especially in technical fields like algebraic geometry, it actually takes a very long time to get up to the point where you have enough foundational material to know what's going on. Like I said, your PhD will probably be the longest peaceful time you have before you'll have to start frantically looking for jobs during 1-2 year postdocs. During a postdoc, while you have time to learn new things, the main emphasis is on publishing papers with the skills you already have. So don't expect to catch up then.
But what does this all mean?!
In my opinion, three years is way too short to do a PhD for most people. Is it possible to write a thesis in three years? Absolutely. That thesis could even be good, and on paper that's the bulk of the degree. However, keep in mind you are writing this thesis under the tutelage of an experienced mathematician who can possibly write a typical paper in a couple of months. What are you going to do when you need to show your own independence?
I'd like to emphasise that during a PhD you should:
- Take a couple years of courses or learn the equivalent of that
- Read papers and books, both in an outside your thesis topic area
- Talk about mathematics with other graduate students and professors
- Write a thesis and hopefully get a paper or two out the door
If you are restricted to three years, something on this list will surely suffer, and as a result the quality of your education and life will suffer as well. Not only that, but your life will be more hectic and rushed, and a PhD should be a time where you develop not only your mathematical abilities, but your other interests, hobbies, and life philosophy as well. Three years is too short to do that.
If you see a three year PhD program look elsewhere because it is just a money grab and evidence that the given school is trying to compete with other schools based on quantity, rather than quality of the PhDs they churn out.