I’ve been busy visiting grad schools, trying to decide where to spend my next 4-5 years (for my Ph.D., after one year at Cambridge).
Talking to professors and grad students, I feel like I’m going through a kind of initiation. The whole mathematical atmosphere feels different from undergrad. “Doing math” takes on different meaning. In undergrad, I take classes, try to learn a lot of stuff, and worry about grades; when I do research, I’m given a problem, sometimes one the professor already has an idea how to solve. In grad school I’ll learn more by independent study and by talking to people; for research I’ll eventually have to find my own problems. When I just got into undergrad, the question seemed to be, How will I be great? What will I do in math? Undergrad felt like a time of zooming out. I catch a glimpse of how large the area of math is, and now the question seems to be, Where, in this vast picture, will I fit in?
Grad students invite us to dinner and exude a quiet enthusiasm, a sense that math is a state of being. They feel like a community. They give me confidence: I’m on the right path.
What’s on my mind as I choose? The community of grad students. The professors: what are they interested in, and how they advise. The environment around the school. (Follow your gut, Gross says.) Princeton has a huge department, lots of professors in each subfield, so that the students in each subfield tend to form their own group. It’s in a small town, more isolated–but isolation can also mean focus. Harvard has a small department so that students in different areas interface more. It’s in Boston.
When I talk to professors, some of them sit me down and then give a mind-blowing introduction to their work (like Conrad, telling me about p-adic Hodge Theory, or Zhang, telling me about a Gross-Zagier formula that can be applied to the congruent number problem). Sometimes they tell me stuff to read. I ask them a standard set of questions, what kind of research do you do? Do your students do? What’s your advising style? Why grad school here? Similar with grad students: how’s your advisor? What are you working on? How’s the community? What made you come here?
Sometimes they give me insights of how people do math, how the math community works.
Math researchers come in many different types. And we need all of them. Some people are good at digesting lots of structure–barreling through Hartshorne, EGA–and work on developing abstract machinery; they have faith in the machinery. Others need to be motivated, they need concrete/classical problems to think about, and are experimentalists, understanding something by lots of examples. (There are 3 reasons why computation is important. One, is to give evidence for conjectures, or find them; the patterns tell us what to look for. Two, theoretical results tell how things exist, but how do we find them? Third, like Euclid’s algorithm, an algorithm can tell us something new.) Some are very specialized and others are scavengers who find a curious thing and think about it hard enough until they understand it, and something is proved.
How do you figure out which type you are? It’s important to figure how your mind works.
When the top places hire professors, they don’t see hire based on how smart and talented the professors are, they hire profs for being leaders.
We have different ways of thinking. Me, I just know, intuitively, an algebraic geometry result is true: it must have been done before. I go to Brian and he points out the exact place in EGA, knows how to fix the proof when the constraints are relaxed.
Professors have different advising styles. When I ask about advising styles, mostly their response is, it depends on the students. Some students like to meet regularly, and some like to drop in whenever. Some professors just give their students a thesis problem to work on. Sometimes they suggest a huge problem that can’t be solved, and a student picks out the part of it that can be solved. They might go through lots of different problems before the students finds something they want to work on. Sometimes students pick their own problems.
Their students usually work in different areas.
To give them space.
I can’t do what my students do.
What did you mean, a potential advisee has to know how to solve everything in Hartshorne?
[Don’t take it literally.] I expect you to know stuff, and not be afraid to do the exercises. You can’t know everything, but you have to be afraid of nothing. You have to be willing to be asked anything.
You must be willing to ask dumb questions, look at baby examples.
On your thesis, and research
There is almost universal agreement of what the role of grad school is.
You can work on your first project with others, but then you need to learn to do research on your own, blaze your own trail.
The goal of an advisor is not giving you a problem and guiding you to solve it. It’s like giving a fish versus teaching someone to fish. An advisor teaches you how to find the next problem. There’s no recipe for that.
The point of grad school is not writing a thesis. Of course, that’s important. The real goal is to be able to formulate your own questions, be an independent operator.
It has to be a good problem, that you can write a thesis on. But it should also not be a dead end.
It’s less about writing a thesis, and more about being in a position, when you’re a postdoc, to find your own problems.
[Saying that you should pick just pick a problem that interests you] is simplistic. You have to also like the techniques. The problems and techniques inform each other.
On talking to people
Take classes and read books and papers together with your classmates.
There are a lot of student-run seminars. A lot of learning comes from talking to people. Because the stuff hasn’t been written down yet. Two posts ago I wrote about how distributed mathematical knowledge is now, so communication becomes all the more important.
You have to talk to other people, you can’t expect professors to come to talk to you. You have to be interested in them too, go up and say hey, can we talk about…?
The great thing about Harvard is very interactive. Even the architecture is very open, you can’t avoid the common room. By talking to other people, you assemble a large list of questions and a large list of tools. And you might just find the tool you need for a certain question. (Interestingly, Feynman also said something similar.)
On hard work
Some people spend as much time on math as a job with a clear schedule. And some devote their lives to it.
(If you’d asked me, at the beginning of undergrad, why work hard in math? I would have said, because more time spent leads to more progress. And now? I think, the more time I spend on it, the more I’ll get a better big picture, and see how things connect. Because there’s just so much out there.)
Some other things grad students said
When you mature, the good news is spread out more. You got in a lot of places, you’re visiting many different grad schools. Enjoy your time now.
The hardest part about becoming a grad student is learning to manage time on your own.
Some grad students have crazy hobbies too, like silk dancing. The transition from resting state to resting state is the hardest.
There’s probably a lot more I don’t remember. But this is enough stuff to think about…