The past week or two while my thesis was out waiting for comments put me in somewhat of a retrospective mood. So I thought while things were fresh in my mind I’d try to pull together my advice for graduate students. I’m going to try to give advice which it is possible to disagree with, which will hopefully spark some discussion. (That way I can learn something too!) Necessarily all of this is going to be best suited for people in a similar situation to me and most of my friends (at a top school, planning to continue in research, etc.).
I’ll organize my thoughts around the following ideas.
- Prioritize reading readable sources
- Build narratives
- Study other mathematician’s taste
- Do one early side project
- Find a clump of other graduate students
- Cast a wide net when looking for an advisor
- Don’t just work on one thing
- Don’t graduate until you have to
Prioritize reading readable sources
When you are a beginning graduate student you face a problem: papers are written for experts and you are not an expert. However, the only way to become an expert is to read more math. How to beat this Catch 22?
Well the good side is that the key to becoming an expert isn’t knowing the main points of papers, it’s knowing the material that’s implicit to all the papers in the field. This means rather than choosing papers based on their main theorems you should instead choose what to read based on what is easiest for you to read! If you are struggling with a paper give up and read something else. Don’t spend a semester struggling through one paper. Read good textbooks, read expository sources whenever possible, when you find an author you find easy to read then read all their papers. (I wish I’d thought of this last bit of advice sooner.)
The you of the future will be much much better at reading all math, so the you of the present only has a comparative advantage reading things that are easy to read.
As a corollary, choose classes based on the teacher, not based on the material.
Mathematicians are people and people love stories. People don’t want to hear your theorems, they want to hear the story you’re telling with them. Similarly if you want to understand a field you need to distill it into stories.
So do this at every opportunity. When you’re learning about other people’s work guess what the narrative is that it fits into and see if you can make it work. Give a talk on that story. It doesn’t have to be what the author actually had in mind, but it needs to be a good interesting story. Here are some examples of the sort of stories I have in mind:
- In the 80s knot polynomials went through the following transition. First people thought about towers of algebras, then they replaced those with skein theory, then they related those to quantum groups. Attempts at categorification has gone backwards through this progression. First Frankel and Crane wanted to categorify quantum groups, when that proved difficult Khovanov instead categorified the skein theory. Finally, in Khovanov’s HOMFLY homology paper he went all the way back to categorifying towers of algebras and replacing them with towers of categories.
- Any time you have a combinatorial description of 3-manifolds you get a corresponding algebraic source of TQFTs. Triangulations->spherical tensor categories (Turaev-Viro-Ocneanu), surgery on links -> ribbon tensor categories (Reshetikhin-Turaev), and Heegaard splittings -> involutory Hopf algebras (Kuperberg). If you had another kind of combinatorial description you would get a new theory.
- Ocneanu’s TQFT is essentially the same as Turaev-Viro, so it stands to reason that you can reconstruct all of Ocneanu’s approach to the theory of subfactors (paragroups, biunitary connections, flatness, 4-partite fusion graphs) by just thinking about 6-j symbols.
- Deligne’s work tells you that it’s easy to detect supergroups among all tensor categories, but much harder to detect the difference between groups and supergroups.
Attending and speaking in seminars is invaluable for developing and learning narratives. The opportunity to develop narratives is also one of the pluses of blogging.
Study other mathematician’s taste
This is closely related to the last point. Just as you build narratives about math you should build narratives about mathematicians. Learn how to predict what kind of math makes your advisor happy. Which talks do they like? Which questions do they ask? Which results do they think are stupid? For example, my advisor hates negative results (“if you just change the assumptions then maybe you can do it”), and loves asymptotic questions (“but what if you send N to the infinity?”). Go to other professors seminars and when you’re thinking about your research think “How could I phrase this so that the representation theory profs would like it?” Jones? The topologists? etc.
It’s important as a graduate student to start to develop your own taste. But you don’t yet have enough experience to evaluate whether the math you like is also going to turn into good math. So you need to learn about other people’s taste and use that to inform your own taste. You don’t have to copy your advisor’s taste, because there are other people with good taste who will disagree with your advisor. But on the other hand if I think of something and I can’t think how to interest Kolya, or Vaughan, or Peter Teichner in it, then probably it’s not such a great idea.
The point here is to figure out why it doesn’t interest them, and internalize some of those reasons into your own taste in the future.
Do one early side project
The process of starting and finishing a project has certain similarities independent of the content or scope of the project. Having an early experience with going through the full process will let you realize that you don’t hate your thesis because you hate your mathematics or don’t want to be a mathematician, you hate it because you’re at that stage of research (for me it’s when the paper is about 3/4-written) where you hate the paper. Having had the experience of pushing through that and completing a small project is incredibly valuable. The experience of choosing a journal, submitting a paper, reading a referee report, etc. is also good to go through with a smaller project before you have to do it with something more crucial. Finally having something already published when you’re applying for postdocs is very valuable.
I had two of these side projects, and although each of them were very valuable individually, I think in some ways it was redundant to do two side projects. I probably would have gotten as much out of just doing one of them and then focusing more on my main research direction.
Find a clump of other graduate students
It’s much easier to learn and do mathematics when you have fellow travelers. A group of half a dozen people with somewhat similar interests who are the same age can make a huge difference. It also lets you start a student seminar. At Berkeley there was a big representation theory clump around our age and it was great. If I had been starting three years later it wouldn’t have been as appealing to work in representation theory. (Conversely, for people starting a few years after me there’s a great algebraic geometry clump at Berkeley.) The math you’ll be working on 5 years in will be enough different from your first glimpse of a field that it makes sense to adjust what you want to work on based on contingent social surroundings.
Cast a wide net when looking for an advisor
I don’t have a whole lot to say about the crucial issue of choosing an advisor, because it’s very difficult and totally mysterious. But here’s one observation:
The easiest way to be unhappy in graduate school is to be sure early on exactly what math you want to work on.
This often results in a hard time finding an advisor, or it results in a bad match in terms of personality and style of advising. It seems to be much better to look at a larger number of options, find someone who has some overlap with you mathematically, but who more importantly wants a student and is a good stylistic match. Top mathematicians are interested a wide range of topics, you don’t need your interests to align exactly in order to find a good project in the overlap of your interests.
Don’t just work on one thing
As I graduate student it seems to me you should be practicing behaving the way you’d behave as a real grownup. In particular, if you expect to be balancing several projects as a postdoc (and you sort of have to if you want to get several publications in the first two years of a postdoc before you have to apply for permanent jobs) then shouldn’t you also be balancing several projects as a graduate student?
How can you do more projects in the same amount of time? Collaborate, collaborate, collaborate. You’ll learn a lot from your collaborators, you’ll get projects done faster, and probably you’ll have more fun doing it.
Don’t graduate until you have to
The longer you stay in graduate school the stronger you will be on the job market at every point down the road. If you have the luxury to wait then do so. You’ll have more publications, you’ll be better prepared to publish a lot as a postdoc, and you will get a better job. As my advisor told me once “It is like a fine wine, you want to go on the market more mature.”
This is obviously highly dependent on your financial situation and your schools policies, so not everyone has the luxury of waiting. Furthermore it’s probably bad for the field as a whole if everyone stays longer. Nonetheless it’s good for you if you can do it.
Update: see lots of good discussion on this point in the comments. In particular I think Danny Calegari makes the point I was trying to make here a bit better. In particular, more important than “staying longer” is timing things so that you’ll have a productive postdoc.