More advice for prospective graduate students

Isabel Lugo had a nice post recently on advice for prospective graduate students. Although not all of her advice rang true to me (the first year of graduate school was pretty fun for most people I know), she makes an excellent point in comment 2 that I wanted to respond to.

Your mathematical interests will change during the first year in graduate school, because a lot of subjects “feel” different at the undergraduate level than at the graduate level, and there are some things you just don’t see as an undergraduate at all.

Personally, my favorite subjects in mathematics are:

  1. The representation theory of finite groups from Frobenius through Brauer
  2. Algebraic and Analytic Number Theory from Gauss through sometime in the late 1800s
  3. Quantum Topology from the Jones polynomial through the present

On the one hand you can tell from this that I like algebra more than I like geometry or analysis. This was something I was quite aware of as an undergraduate and beginning graduate student. However, all of these have something else important in common: they are/were all young subjects. With the exception of Euler’s prescient work on zeta functions, there’s not a whole lot of precursors prior to the beginnings I’ve stated above. You don’t see on my list anything like modern homotopy theory, ell-adic cohomology, the classification of simple groups project, or 20th century number theory. I went into graduate school thinking I wanted to do number theory because I love number theory up until about 1920. The lesson I should have taken from that is that I like younger topics in algebra, not that I want to do number theory. Older subjects feel different from younger subjects. You have more tools, bigger machines, but the most natural questions have already been answered and the field has moved on to harder things. But there was no way for me to know this as an undergraduate, because undergraduates don’t know enough to have been exposed to any material in an older subject.

There are lots of other key differences between fields: Do people do more theory building or problem solving? Is the topic dominated by a few giants’ research programs with other people following their lead or is it more each researcher having their own smaller programs? Is it on the intersection of multiple fields or more isolated from other fields of mathematics?

Obviously a first-year graduate student isn’t going to know the answers to these questions, but these are the sorts of questions that you need to ask yourself rather than just “which subject did I like as an undergraduate.”

23 thoughts on “More advice for prospective graduate students

  1. The first year of grad school was pretty not fun for most people I’ve talked to about it. But then again, a lot of my sample is people that I asked “how was your first year of grad school?” during my first year of grad school, and perhaps the ones that had a good time then just changed the subject.

  2. It’s nice to hear this from someone else. I’ve been assuming that this is true throughout my whole choosing-a-grad-school process, and some people have been worried that I am passing up, say, Berkeley (which does everything I think I like) for Harvard (which does almost nothing I think I like).

    Another point which I have found in my experience is that mathematical tastes are very strongly affected by teachers. If you have a wonderful teacher for something, you will like the subject a lot more. Perhaps once you get to a certain point some of your tastes get ingrained, but I have no doubt that it was really just a coincidence that I came to really love pointset topology and functional analysis; if I had had different professors freshman year I would like different things now.

  3. Just one objection to the claim that all the natural questions have been answered regarding older topics. My experience (I admit, it’s a bit limited, though) has been that the most natural problems are often some of the hardest to solve. The subject I’m studying (algebraic geometry) is full of such examples. It seems like every natural question I ask is open and, worse, no one has a clue how to go about solving it. Though admittedly, the other points still hold, the naturalness of the questions that are left isn’t in complete correspondence with age.

  4. Eric: it’s a balancing act. A high-quality department that does a variety of things, some of which you think you’re interested in. Yes, your tastes may change, but what if you choose a slightly more glamorous department which doesn’t do what you’re interested in yet, and your tastes don’t change that much?

    Charles: I run into a related problem. I keep finding questions that nobody has even bothered to ask, and I’m constantly surprised to find out that they’ve remain unanswered so long.

  5. In reply to Charles, I too would agree that many natural questions belonging to old subjects are still open (in the area of Dynamical Systems there are plenty indeed). At the beginning I was naively thinking “why so many new definitions and long theorems when the end result is really small and obvious compared to the original question?”. But then what grad school has taught me in that respect really is that “elementary formulation” does not imply “elementary solution”. This doesn’t soothe the fact that our lack of knowledge in certain areas certainly feels shocking, but at least becoming aware of the kind of subtle things that cannot be bypassed does a bit.

  6. I think it is part of the advisor’s duty to communicate some perspective on what problems are open and feasible for graduate level work, and what problems should “wait until tenure” or be dropped altogether. Understanding these differences is often more difficult in older fields, because most of the easy, accessible problems have been solved already, and the techniques needed for solving more advanced problems have a nontrivial learning curve. In addition to the perspective, another reason why it is good to choose a research topic close to your advisor’s interests is that you’re more likely to get help if you get stuck.

    On the other hand, I think it is more important to find an advisor you can get along with than someone whose interests match yours. There are a few grad students who start off already knowing the topics of their theses and just need an advisor to sign papers, but for most people, an unfriendly advisor can really wreck their happiness for a long time.

    Random anecdote: I once told my advisor that I was learning about étale cohomology, and he replied that it was a beautiful theory, but that I had to be careful, because one could easily spend ten years learning about it and still not have a thesis.

  7. Isabel- It probably depends a lot on the school. I get the sense that people in Berkeley are pretty happy at the beginning, and then as the years go by, some of them get unhappier. Of course, the first year of grad school is also tough because you’ve just moved, you miss your friends from college, and so on. At least, my experience was that it was tough personally, but great mathematically.

    Eric- You didn’t pick a very good example of departments. Most people would argue that feeling unsure about your interests is a reason to go to a bigger, more diverse department, rather than a smaller, more focused one. Of course, it’s hard to argue against Harvard, but just as a general rule, the smaller the department, the more sure you should feel there are particular professors there you want to work with.

    Charles- I think you’re getting a little mislead by selection bias. When natural questions have easy answers, you tend not to even think of them as questions, whereas the ones you can’t solve stick out like sore thumbs.

    I also suspect you and Noah may have different ideas of what is natural. After all, algebraic geometry is the classic example of a field Noah hates, especially once cohomology is involved.

    John- Since when is running into questions no one has asked a problem? I mean, unless they’re questions people didn’t ask for a reason. The really good trick is when you can ask a question that people haven’t asked but will wish they had.

  8. I definitely mis-phrased what I meant by natural questions, I didn’t just mean that the phrasing was natural. For example, in algebraic geometry the result that most closely captures the type of thing I’m getting at is Bezout’s theorem, not say whether there are rational points on the Fermat curve (even though the latter is easier to state than the former). Similarly, Riemann-Roch for curves over C is pretty natural. I don’t love either R-R or Bezout, but that’s more that I don’t like *geometry*.

  9. Eric- I’m sure I’ve told you this before, but although I’m usually a Berkeley booster, for you I think living in Cambridge is going to be much better for you socially in terms of the culture of the undergraduates. Also if you’re willing to keep open the option of an MIT advisor (which you should) I find it hard to imagine you having trouble finding an advisor you like. Especially since your tastes are wide-ranging and abstract. For example, the whole Mike Hopkins crowd seems to do things that would be very much stuff you would like.

  10. About the “first year of grad school” point – I definitely had a lot of fun that first year, but I also was constantly struck by the fact that everyone around me seemed to know lots of stuff already that I didn’t know, and felt out of my depth at many points even just discussing things at tea with people. This was true of my interactions with the philosophy students as well as the math students. It certainly required an attitude adjustment even compared to the environment at at undergraduate school that supposedly gets many of the best and brightest students.

  11. Sure the question is old and over-hyped relative to other questions, but it is relevant. There are differences between subfields that roughly match this question. Say in number theory there’s a big difference between working on cryptography and working on the Langlands program. Some people would be much happier working on one or the other. Sure there’s theory building and problem solving in all subfields, but some have more of one or more of the other.

  12. Ben (#7): I have the opposite sense at Penn — people are miserable at the beginning, stick it out, and are happy at the end. (Or at least are happy at the middle — I have yet to see how I’ll feel about graduate school when I’m actually in thesis-writing mode.)

    Kenny (#10): I think a large part of the “feeling out of your depth” arises from the fact that when you overhear things which make sense to you, you ignore them, but when you hear things which don’t make sense you think “oh, that person must be smarter than me”. I had a similar experience when I was studying for my oral exams last year. Some students in my department maintain the officious oral exam archive, where students write up the questions they were asked at the (end-of-second-year) oral exams. The oral exam is on two subjects of the students’ choice. I looked over that archive and thought “oh my god, I could never pass these exams!” But I didn’t have to pass other people’s exams; I had to pass my own. (Which I did.) The point I’m trying to make is that things we don’t understand can seem more difficult than things we do understand.

    (But there’s also a Dilbert comic in which the pointy-haired boss allows Dilbert six minutes to build some sort of gigantic computer network; the PHB thinks that everything he doesn’t know how to do is easy.)

  13. Greg,

    Definitely steering. Without power, I can get some friends to push it around as a large cart, but without steering, I’m constrained to a constant geodesic curvature trajectory.

  14. Scott,

    you must have better friends than most of us. If I had a powerless car and asked my friends to push it around, pretty soon I’d have no friends.

  15. Okay, “problem” may have been poorly chosen. But here’s where it can be tricky: if you solve a problem nobody thinks of as a problem yet, then you not only have to convince them that you’ve got a good solution, but that it was to a good problem in the first place.

  16. Regarding natural questions and algebraic geometry, the ones I’m talking about ARE of the nature of Bezout’s Theorem. Things about hypersurfaces of projective space and how they behave, in fact. I can follow not liking geometry, but these are still rather natural questions. Like “Is every cubic surface rational?” (Yes) “Is any cubic threefold rational?” (no) and “Is any cubic fourfold rational?” (open)

    On the first year at Penn, I have to say I’ve been enjoying mine, though weaseling out of the first year courses contributes to it.

  17. I have to disagree a bit with Ben’s assessment of happiness among first year students at Berkeley, but since I entered a year before him, I have a different sample. Most of the people I knew seemed to have a good time, but there were some sources of stress that caused some friends to leave, when in other conditions they might have been reasonably successful .

    The two obstacles that seemed to weed out the most students were the prelim and the search for advisors. Several of the students I knew did not pass the prelim exam on the first try, and this was a big source of pressure for them. Others who passed had trouble finding a compatible advisor. I suppose this sort of thing is common to many schools.

    In addition to the stress of standard requirements, there were a few unpleasantly competitive people who would, e.g., loudly declare that homework problems x,y,z were trivial, when other people, who were having difficulty with those problems, were in the same room. This sort of behavior can really sap someone’s confidence, and it doesn’t take many such people to turn an otherwise friendly environment into a caustic one.

    My ego was generally big enough to absorb such hits, but hanging out with my officemates and classmates was less fun when they had just had their parades rained on. In the end, this is a social problem rather than an administrative one, and I found that some combination of pointing out tactlessness and uninviting people from the office improved the atmosphere a lot.

  18. Hmm. My comment seems to suggest that I singlehandedly changed the environment at Berkeley, and that is pretty far from the truth. The general picture wasn’t as bad as I may have made it seem. I meant to say that I (among several others) tried to counter some nasty behavior and I think it made some of the people around me happier.

  19. Scott and Isabel- I was just throwing out my remembrance of things from half a decade ago. I would take it a grain of salt.

    What’s probably true is that the amount of self-doubt one experiences probably has relatively little to do with how qualified one is, and everything to do with one’s personality. Some people who are actually very talented and ready for grad school will feel like they’re the one who got in by mistake, and some people (such as myself) are cocky bastards to whom it wouldn’t occur.

  20. John- Of course, reasonable people differ about what are interesting problems to solve, but if convincing people that the problems you thought were worth solving are interesting ones to solve is giving you consistent trouble, it might be time to rethink things.

  21. Oh it’s no trouble at all when I’m face-to-face. Everyone likes the ideas when they actually see them, but I can’t sit down in front of every hiring committee (not all of us can afford to jet around the world quite so much).

    On the other hand, I’m sort of invested in this line of inquiry. “rethinking things” at this point amounts to retirement, since I don’t have the cushion to get on a whole new course.

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