- Bhargav Bhatt:
- On finding an advisor: I would like to reiterate (Karen’s) general advice above, especially points (4), (6) and (10). Adding to what was written in point (4), I’d recommend the following exercise: for each of the faculty members in your broad area of interest, skim through some of their recent papers (usually on the arXiv) to get a sense of the problems they work on, the objects they end up thinking a lot about, and the techniques they use. Doing this exercise in my case (in May 2019) would tell you that I enjoy thinking about problems in characteristic p or p-adic algebraic geometry, and that there’s quite a bit of theory involved.
- On transitioning from being a student to a researcher, especially in background heavy subjects like algebraic geometry: It is very easy to spend a lot of time early in graduate school learning various theories or background materials that you think might be useful in your future research (or simply because they sound cool). While this is a reasonable approach for the core material in the subject (e.g., Hartshorne’s book in algebraic geometry), it is not the quickest or the most effective way to learn advanced mathematics. Instead, it’s faster, more effective (and definitely more fun) to first find a problem (even a “toy” problem), and then learn whatever is necessary to solve it. If in the process, you find yourself rediscovering things people already knew, do not be disheartened: this indicates you’re on a good track, and is a phenomenon that essentially all research mathematicians encounter regularly. [A word of warning about this approach: when on the quest to solve a specific problem, it is often convenient to blackbox various background results in order to put them to immediate use. This is of course reasonable (as long as the results are trustworthy and you use them correctly). But one should also keep track of such blackboxes; at the end, it is important to know what you don’t know.]
- Alex Wright: I picked my advisor without having a great idea of what he did, since I hadn’t ever taken a course on Teichmuller theory or dynamics. I learned a little about the subject, but I picked the advisor more than the subject. I was looking for someone who worked on problems rather than big theories, but nonetheless used and developed theory related to the rest of mathematics.I picked my advisor at the end of my first year. I understand that’s early for U of M, but my advice is to pick earlier rather than later. If you’re interested in working with me, I suggest you come talk to me at the beginning of your first summer, even if you are still extremely uncertain.If you want to find out what I think about, check out my surveys “From rational billiards to dynamics on moduli spaces” and “A tour through Mirzakhani’s work“. You can also talk to my current students (including Francisco Arana-Herrera, currently still at Stanford). For students seriously considering working with me, I can share a five page document of advice and expectations for you, that I wrote specifically for new PhD students working with me. I hope to take an average of one student a year for the time being.
- Jenny Wilson:
- Your PhD advisor does not need to fill all mentorship roles. Be prepared to seek out different mentors to advise you on different aspects of your research and professional development, for emotional support and encouragement, as an advocate within the department, and so forth.
- If you’re interested in a particular prospective advisor, talk to their recent students about their experiences. Look at their webpage and their publications. Talk to them, and other faculty.
- Some factors to consider:
- What is their advising style? Are they hands-on and involved in their students’ work, or more laissez-faire? How often do they meet their students? Are they travelling or off-campus often? Do they expect their students to find their own problems and research directions?
- Were their recent students happy with their experiences? Do they continue to support their students past graduation?
- What is their mathematical style? Are their strengths in big-picture thinking or technical details? Do they work abstractly or concretely? Do they work with a computer? What are their philosophies around the best way to conduct research?
- What is their recent track-record in advising students? Have their recent students graduated? What sorts of jobs have their students or post-docs recently found?
- Do you feel comfortable talking to them? Would you be comfortable telling them, for example, if you were stuck and frustrated with your research, and having difficulty making progress? Would you be comfortable telling them if needed to take a hiatus for health or personal reasons?
- Do you feel that you can communicate with them effectively, either discussing math or non-mathematical topics?
- What are their expectations for their students? Do they have specific expectations around (for example) coursework and seminar attendance, how you spend your summer, amount of time devoted to research, how you prepare for your meetings, the structure of your meetings, how and what you communicate between meetings, etc.
- Be direct when you talk to your candidate advisor about their expectations, and be direct when you discuss whether they will take you as a student. Serious problems can arise when a student and faculty member have different understandings of whether they’ve committed to working together!
- Wei Ho:
- You should try to get to know potential advisors in a lot of venues, not just by taking classes (only a handful of faculty will be teaching the relevant grad classes each year). Go to learning seminars, research seminars, seminar lunches/dinners, tea, etc, because you’ll get to know the faculty in many different environments. Don’t make a decision based on one lukewarm interaction with a faculty member.
- Look at potential advisors’ papers, or see if you can attend (or find online) some of their research talks, whether in the junior colloquium or elsewhere.
- Anonymous: These points might be too negative or impolitic, but having seen friends and classmates left in bad advisory situations, I will state them. I’m not exactly sure what advice to give to avoid these situations, and they may be less important for students planning to go into non-academic careers.
- Choose a mainstream field of math. Is there a larger active group and regular conferences? If so, where in the world are the fields’ strongest research hubs?
- Avoid controversy if possible. Does the field (or advisor) have conflicts or political divisions?
- Choose an advisor who has an active research program and who is up-to-date in their field. Some signs of this include recent publications, federal funding, invited talks at conferences, etc.
- Consider how successfully the potential advisor has integrated their recent students into the field. Have their recent students had opportunities to attend summer schools or conferences, or give talks? Have their recent students succeeded at obtaining postdoctoral positions (if desired)?