Thoughts on graduate school

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.

Build narratives
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.

32 thoughts on “Thoughts on graduate school

  1. I think this is very good advice. In response, I will content myself with one agree and one disagree:

    Prioritize reading readable sources: wholeheartedly agree.

    As a graduate student, it’s often hard to know whether something is hard to read or whether you yourself just need to worker harder (or be smarter, etc.) to read it. But if you’re really trying to understand something and having trouble, it’s almost certain that other people have been in the same position, and there’s a very good shot that there is a better exposition out there.

    Here’s a tip: make sure you are not missing out on an earlier paper that really explains something which is only summarized very briefly in the paper you want to read. My favorite example of this (and, not coincidentally, one of my all-time favorite papers) is the 1958 paper of Lang and Tate, on Galois cohomology of abelian varieties. Because “Galois cohomology” was not a standard subject at the time, L&T write NOT ASSUMING that the reader has seen it before, and their exposition is accordingly both simpler and more insightful than almost any other I have found. Reading an early paper can also be very revealing on understanding the (hi)story of the subject. And there is a good chance that a really good early paper will contain some ideas that have been forgotten, or leads that were never followed up upon.

    By the same token, sometimes a later paper gives a much better exposition than the original paper, but I think your advisor is much more likely to tell you about that.

    “The easiest way to be unhappy in graduate school is to be sure early on exactly what math you want to work on.”

    I disagree. The easiest way to be unhappy is to be dimly aware that you’re not doing what you should be doing and that, though you seem to be working as hard as you can, you’re not making any progress.

    You should start thinking about an advisor as soon as you get to grad school, if not before. This should not stop you from learning about as many disparate subjects as you want (most advisors approve of this). I think it’s a terrible mistake to just bop around in grad school for a couple of years, taking classes and passing general exams, without any larger plan. A key point that many students don’t seem to appreciate is that expressing an interest in some subject or some person doesn’t commit you to do it: choosing an advisor is not like signing a lease. It is much more helpful to approach someone, do some reading in a certain subject for a semester or so and then realize that what you really want to study is something else entirely than not to look for a research topic at all.

  2. Here’s one more piece of advice:

    Understand that “academic mathematician” is a profession and that as a graduate student you are being trained for this profession. There is more to professional training than just learning mathematics and trying to prove theorems!

    In my experience, the best schools can be very bad at imparting professional knowledge and instilling a sense of professionalism. As a student at Harvard, I had the vaguely Calvinist sense that my job was to carefully nurture my (possibly) already extant inner brilliance. If indeed I was one of the elect, then that would become apparent at the appropriate time.

    If you want a job at Princeton straight out of grad school, then the above is the perfect strategy: i.e., what you were doing anyway. For the rest of us, it is extremely important to learn about the profession and learn to be PROFESSIONAL, not flaky and vague as so many graduate students can be.

    I would like to follow up with detailed advice on professionalism, but that’s beyond the scope of this message.

  3. I don’t get your argument about staying in grad school for a long time. Yes, if I take 6 years then I’ll be at a better place 5 years out of grad school then if I take 4 years, but that’s comparing apples and oranges; to compare apples to apples, you should compare where I could be 5 years after a 6-year Ph.D. and where I could be 7 years after a 4-year Ph.D.

  4. One of the reasons to stay in grad school for an appropriate length of time (not necessarily a long time) is precisely that (in many cases) you have some control over this length of time: you can graduate when you are ready, when your projects have come to fruition, and when you can enter the job market on your own terms. By contrast, most first postdoc jobs are between 1 and 3 years, and typically for a fixed time period. Whatever this time period is, you had better hope that you emerge from it at the “peak” of your cycle (if you want to get a decent job). Having a decent publication list before leaving graduate school is a very useful cushion to reduce the risk associated with the uncertainty in your output during a (relatively short) postdoc period.

  5. Regarding length of time spent in graduate school, it may be that at some schools there is flexibility about this, but at others there is not. My own experience at Harvard was that the department decided when it was time for you to leave, not you, and that one had only four or five years to finish, in any case.

    At Northwestern, where I am currently, it is quite unusual, as far as I know, for a student to stay more than five years.

    In general, I don’t think that “staying as long as possible” is appropriate for everyone, in any event. If one’s research is going well, then at some point one changes from being a mathematician-in-training to being a working mathematician. At this stage, it is natural to progress to the next point in one’s career.

  6. Danny’s points about staying longer are mostly what I was getting at. Obviously you shouldn’t stay longer than you need to a) get the best postdoc you could get and b) be ready to publish stuff quickly as a postdoc.

    I totally disagree with Alon about what the relevant comparison is to make. The only question is whether you can get a good tenure track job, and how good of one you can get. How quickly you get to that point is just icing on the cake. If you have the option of sacrificing a year in order to increase your chances on the tenure-track market, it seems pretty clear to me that you should do it. Comparing where you are x number of years out is irrelevant.

    One apples-to-apples comparison is who is likely to get a better postdoc: someone who stays in grad school two years longer or someone who graduated earlier does a mediocre two year postdoc and is looking for a second postdoc? I’m pretty sure it’s the former if only because the latter can’t get an NSF.

  7. 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

    Speaking as a director of graduate admissions for a big Ph.D. program, this point is worth emphasizing. The number of students a department can fund is bounded. So if everyone takes six years instead of five, it means, very simply, that we can only educate 5/6 as many people as we’d like to. I guess this is not “bad for the field” if you think we’re currently overproducing math Ph.D.’s, but I don’t.

    This doesn’t apply, of course, if sticking around longer is funded by your advisor, or a side blogging business!

  8. Maybe I should have just emphasized the first sentence above; I like, Matt, am not so convinced that staying longer typically benefits the student. Of course, there are cases where the student has a really good theorem that needs a couple more months to bake, or, on the flip side, where the student discovers a mistake in the thesis a month before he or she was about to start applying for jobs. But I think most people ramp up their activity a lot after grad school — they’re expected to develop their own research program, are interacting with more people outside of their advisor’s main interest, etc. Whereas another year of grad school might be another year of polishing and maybe slightly strengthening the thesis.

    As for the students who already have a flourishing research program outside the thesis, and who are going to get two or three extra preprints in that sixth year of grad school — they are the ones who don’t need to worry about eking out competitive advantage on the job market!

  9. Several people I know have been on the job market consecutive years and all of them got much better postdocs the second year. It’s possible that this is a coincidence, and I’d be happy to defer to those of you who have more experience if you tell me that my sample size was too small.

    Certainly I’m not suggesting that anyone stay another year to polish their thesis. The whole point is getting a 6-month or more headstart on developing a research program which you can publish quickly as a postdoc.

    I’m a bit surprised that you don’t think everyone needs to be eking out competitive advantage on the job market. My impression is that the tenure track job market is extremely difficult. Especially if you want to have any choice in the sort of town/city you’re living in. I don’t think my odds at getting say a top 40ish tenure-track job in an amenable location are high enough that I don’t need to be eking out a competitive advantage at every step along the way. I’d be happy to be wrong about that, of course!

  10. The question of when to graduate is certainly a controversial one. Among the blog writers, we have quite the range — all the way from 3 to 7 years.

    Here is a relevant question for this issue:

    When people are hiring postdocs, how much attention do they pay to the number of years in grad school?

    I mean if there is one candidate who finished in 4 years with a reasonably good thesis and another who finished in 6 years with a slightly better thesis and also a published side project, who is likely to get the job. My instinct is that the number of years will be ignored and the guy with the stronger letters/CV will get the position, but I am curious to know what others think (especially those who have recently had to make such a decision).

  11. Regarding how long to stay, I think there is another aspect to consider. Not everybody is going to end up with a tenured position. I think for a person who eventually leaves academia, that person may eventually regret “losing” an extra year of their life in grad school to increase their chances at a good postdoc.

  12. In regard to Joel’s question in 14 above, for myself, when reviewing post-doc applicants, I don’t pay attention to the number of years in grad school; much more than anything else, I pay attention to the letters. I would like to emphasize that I also don’t pay that much attention to publications, unless they are really first-rate.

    I (and I think many others) recognize that it takes time for papers to get written and published, and that papers can differ greatly in quality. For this reason, I would not pay much attention to relatively minor publications. The fact is, in some fields it is easier to produce papers than in others, and so a small paper or two produced in graduate school is not that indicative of future potential (which is really what one is looking for when evaluating post-doc candidates). Ultimately, I rely on the letter writers to evaluate and communicate the potential of the candidate. (This involves in part evaluating and communicating the quality of the thesis, and of any other work, but is more than just that.)

    To give some sense of where this attitude comes from, it might help to consider the situation with applications at the previous career step: grad school. Frequently one will have grad school applicants with publications, typically arising out of an REU. In my experience, these have almost no correlation with the students’ research potential. There are lots of ways for people to learn about, and get interested in, math, and REUs are one of them, which happen to lead to publications (sometimes, at least). But having produced a publication because of participating in an REU doesn’t really indicate much about the students’ future potential; it was just a function of a particular experience they had. In general, grad school applicants are not expected to have produced publications, and so they just don’t figure much in the evaluation process; other metrics are more important.

    Publications produced in grad school are usually a little different, of course, and of a different caliber. But again, one should bear in mind that it can be easier to produce relatively minor papers in certain fields, than say spending a long time coming to grips with deep and challenging problems. My preferred post-doc candidate would be someone who has done the latter; since this is more or less independent of producing minor papers, I won’t pay much attention to these, and will instead read the letters to find out whether the candidate has indeed grappled with difficult matters. Just as with grad student applicants (at least in my view), one simply doesn’t expect post-doc applicants to have produced papers, and so there are other metrics that I would focus on.

    Here I am talking about fairly minor papers. Major papers produced in grad school are a different matter. But if they are really major, than there is not so much point staying in grad school an extra year to produce them: if you produce them during your post-doc years instead, then they will simply serve to strengthen your tenure-track application.

    In my experience, even for candidates at the tenure-track level, the number of papers produced is not very important. Rather, one wants to see a couple of major papers (and typically, “decent length paper in a top journal” serves as the bench-mark for a major paper), together with evidence (and this is what the letters are for) that one can expect more of those in the future. The minor papers on the CV are regarded as just that: minor. They don’t play that large of a role in the evaluation process.

  13. The REU analogy is one I’d never thought of. I’ve always been an REU-hater (this is part of the Ross program culture), and so your analogy really makes me rethink my position on interesting but minor papers.

    That said the point you’re making about REUs and graduate admissions sounds like it’s running against the grain of what other people I’ve talked to say. We run into this at Mathcamp a lot where the JCs (who are undergrads) feel a lot of pressure to go to an REU. I have a friend whose prof. told her point blank that if she wanted to go to grad school she had to do an REU. Even at top-tier undergrad institutions there seems to be a lot of pressure to do an REU (which struck me as even weirder, if you’re getting a rec from an undergrad thesis advisor at Princeton isn’t that info going to swamp whatever your application says about an REU?).

    Whenever this comes up I try to argue that REUs are dumb and you shouldn’t need to do them, but everyone else I talk to seems to think I’m very much wrong empirically and so I’ve been convinced that REUs are stupid but matter a lot for grad school applications.

  14. Dear Noah,

    That’s interesting. I certainly agree with your analysis regarding, e.g., letters from a senior thesis advisor at a top tier school, but perhaps I am also in a minority regarding REUs.

    I should add that if a paper is sufficiently interesting (whatever exactly this means), this can compensate for it being minor (again, whatever exactly that means). (I am writing this in part because you used the expression “interesting but minor”.) Not everything any of us do will always be ground-breaking, and this will be true of any candidate. What I wanted was to put forward an argument against the pressure to write for writing’s sake, which I think some people might feel (and which I took — perhaps not completely fairly — to be a subtext of your advice), and which is perhaps related to the pressure to do REUs that you describe.

  15. The only reason I can think of to stay in Graduate School would be to take more graduate courses… mostly because becoming formally, as has been said, an actual “professional” mathematicians instead of a student, tends to lead to other duties and things to do which may reduce drastically the possibility of attending carefully a good background-forming class in fields which are not directly related to one’s work, but which are likely to be of use and importance in the future, if only as a way to understand seminars and conversations (e.g., a solid algebraic geometry course for an analytic number theorist — and conversely –, a solid probability course for almost anyone not working directly in probability, etc).

    On the other hand, I can’t quite imagine life in graduate school being so good that you would want to stay too long — it might have changed, or be different in various places, but there was a definite “impecunious student” aspect that was an efficient motivation to try to get a PhD as fast as possible.

  16. On the other hand, I can’t quite imagine life in graduate school being so good that you would want to stay too long — it might have changed, or be different in various places, but there was a definite “impecunious student” aspect that was an efficient motivation to try to get a PhD as fast as possible.

    This is extremely far from universal. Obviously it depends on the person, but I think most of the people would not have found it a huge financial hardship to stay in graduate school a year longer. Of course, as JSE noted, it can be a financial hardship for the school.

    In part, I think Noah’s attitude is one born of going to graduate school in Berkeley, in a time when the department was actually short on graduate students to fill teaching positions, and there was an RTG grant in our research area. It was very easy in the years 2003-2008 to stay on for as long as you liked (at 5 years, I think I’m actually third fastest Ph.D. on this blog), and who wouldn’t? It was a very stimulating environment, and had a lot of other appealing aspects, even if you were impecunious.

    So, while I think there is an important point here that grad school is not a race, and you don’t get any special consideration for finishing sooner, you should take Noah’s view of the matter with a grain of salt. There are a lot of other situations where the costs of staying in grad school a year longer outweigh the benefits, just as there are a lot of situations where the reverse is true.

  17. On the pecuniary issue it’s also true that those of us who stayed longest are also the people whose partners are taking paycuts to move.

    Anyway the “stay longer” point was last in the list because it was the most questionable, so I’d love to hear if people have thoughts on the other points, or key things I’ve missed.

  18. I definitely would have made
    *go to as many conferences as is practically possible
    *talk to people as much as you can force yourself to
    *cultivate relationships with as many senior people as you can
    points on the list.

  19. I’ll accept Noah’s invitation to remark on another of his points. I think the suggestion “Don’t just work on one thing” is a good one. It has many interpretations, and probably all of them (at least that I can think of) make good sense.

    In Noah’s more detailed remarks, “thing” seems to mean “project”, and I think this is good advice. When you begin a post-doc, you will need several projects to work on (and when you apply for a post-doc, you will have to write a research statement describing these various future projects). If you have begun to seriously think about these in graduate school (especially projects that are not immediate extensions of your thesis), this will help a lot both for writing a research statement, and for actually carrying out research successfully as a post-doc.

    One could also interpret “thing” to mean “topic”, and I think this interpretation of “Do more than one thing” is also good advice. Learning advanced mathematics, and writing a thesis, tends to exert a pressure in the direction of specialization, and it is good to spend some time and effort in graduate school working against this pressure. Of course, the pressure to specialize exists for a good reason, and one shouldn’t fight too hard against it; you do have to write a thesis after all, and this normally means becoming an expert (perhaps the world expert) in some particular, usually highly specialized, topic. But one can fight against it in a reasonably gentle, productive way.

    For example, in my field (algebraic number theory) one normally has to learn some algebraic geometry no matter what one is going to do, and so in grad school it makes sense to try and learn a little more algebraic geometry than one might need just for standard arithmetic geometry purposes. This will help broaden your outlook in the long term, and also has a good chance of having a more immediate payoff, since algebro-geometric thinking often offers a powerful perspective on problems in number theory. (Representation theory is another subject that makes close contact with number theory, and similar remarks apply.)

  20. Tiny point, but given how prevalent ageism (still) is in mathematical culture, I can’t resist making it:

    Students in the early stages of graduate study should be described as “beginning graduate students,” not “young graduate students.”

  21. Could someone tell me.. where the picture in the (beginning of this page) was taken? I think I know the guy with the beard.. or maybe his doppelganger?? Does he work on quadratic forms (I didnt look at their blackboard..)?

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