If you’ve decided to go to grad school but aren’t sure which one is right for you (or where to apply), it can be really useful to ask people who are in those programs (or were recently) about their experiences. Start with people you know (friends or professors, for example), but it’s also okay to reach out to fellow alumni from your undergrad, or even cold email grad students from the math department website at schools you’re interested in — just keep your initial email short (“Dear so and so, I’m so and so, interested in getting a math PhD at such-and-such, would it be okay if I ask you some questions about your experience?”) and be prepared to take no or silence for an answer. A few people will be happy to help out and you can learn a lot from them.
Here are some things you can ask people about their grad school experiences:
- What is the quals/prelims/etc. process like? (Usually there’s some kind of exam after one or two years, which you have to pass in order to continue, but that’s very different from place to place.)
- How much training is there for grad students, both in the subject matter (e.g. are there classes to help grad students get up to speed before their exams?) and in other skills required in academia (teaching, networking, travel, publishing, mentoring, applying for grants, etc.)?
- How do students and faculty feel about careers outside of academia? Academic jobs are incredibly competitive compared to applying for undergrad and grad school — usually hundreds of qualified candidates applying for a single spot — but at many places there’s an attitude that not staying in academia is failure.
- Is the atmosphere among grad students mostly cooperative or mostly competitive? Do grad students notify each other of opportunities, work together on papers, support each other through the dissertation process, etc.?
- How does choosing an advisor work? Are there faculty members who are known to be good/bad advisors? Is anyone on the faculty planning to retire/leave soon?
You also might want to think about what matters to you personally in grad school: there are the academic and professional things I’ve already mentioned, but there’s also: how much would they be paying you (it’s okay that this matters — grad school is much more like a job than undergrad!), is it in a location you want to live in, how easy would it be to travel (to conferences or to visit loved ones), are there amenities nearby that are important to you (and would you have a way to get to them), and so on? These will matter different amounts to different people, but it’s important to think of grad school both as a stepping stone to future possibilities AND as years of your life that you want to be a good experience in itself.