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Hi,

I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to study [insert your favorite hot topic here] as opposed to pinning down the fundamentals of the course materials I was studying. Probably the best class I took was a seminar where the prof had us read and discuss from classic texts in differential geometry and pdes. Anyways, now that I'm starting graduate school, I'd like to avoid other common pitfalls that graduate students make. So my question is:

What are common pitfalls, mistakes or misconceptions that you wished somebody had told you were wrong? I'm interested in pretty much anything from how to conduct research, to what courses to take, or anything else.

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    $\begingroup$ Don't spend too much time on MathOverflow. $\endgroup$
    – Jonas Meyer
    Commented Jun 7, 2010 at 2:35
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    $\begingroup$ Leave your work at the office. Or at least set aside specific times of day where you don't think about work and tend to your hobbies and relationships. This has much more to do with your mental well-being than any attempts at letting your unconscious have a go at math related problems. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jun 7, 2010 at 11:41
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    $\begingroup$ Allow me to put this highly controversial answer as a comment to avoid downvotes: learn category theory. I found that in my first year, this was by far my biggest weakness in graduate school. It came up in all of my classes, but was not well covered in any of them. $\endgroup$
    – B. Bischof
    Commented Jun 7, 2010 at 13:54
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    $\begingroup$ I find that it takes a couple of years of graduate school for the students to understand how important it is to talk to people about math. You can't learn just from reading (as is possible in college). The sooner you implement this, the better you will fare. Show humility and talk to people who know more than you, it will more than pay off. $\endgroup$
    – Makhalan Duff
    Commented Jun 7, 2010 at 14:05
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    $\begingroup$ Since this came back to the top: You can't start going to talks too early. I went to a couple that "looked interesting" to me throughout my first two years and never went consistently because I didn't understand anything. I only later realized that no one understood anything, but the important thing was that I was seeing concept and hearing terms that later I randomly found a use for. Go to talks. It will be useful even if it doesn't feel like it is. I wish I went consistently my first two years like I have since. $\endgroup$
    – Matt
    Commented May 30, 2012 at 16:34

7 Answers 7

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Marie desJardins has a nice article on Surviving Graduate School that is definitely worth reading.

The top two pieces of advice I would give are:

  1. The most important thing when choosing an advisor is to find someone who will go out of his or her way to help you succeed, not someone who is famous, and not even someone whose research is in the right area.

  2. You need to make the transition from being a mathematics student to being a mathematician. That means thinking of mathematics as an arena where you seek out unsolved problems and obsess over them until you solve them, not as a vast sea of material to be learned. Don't get sidetracked trying to learn everything; that's impossible. Focus on finding an open problem you can solve, and solve it.

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    $\begingroup$ Timothy's advice is particularly nice in that Gauss started to "follow" this maxim before he was 18. My advice: "Compete with yourself, i.e., what you plan to do and what you have actually done should be parallel. Then you are the best, 1st for yourself then for others.... " $\endgroup$
    – Unknown
    Commented Jun 7, 2010 at 12:20
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    $\begingroup$ Piece 2 here is something I didn't learn until too recently...and boy do I regret it. When you learn new techniques or areas of mathematics in order to solve specific problems, you simply learn things more thoroughly. You find out a technique's limitations, "historical" context, strengths and weaknesses. Problems have a mnemotechnic value in this regard, as well. I wager that you'll always remember a technique when you've tried (probably unsuccessfully) to use it on your own problem. Problems also highlight and organize what you need to know. $\endgroup$
    – Jon Bannon
    Commented Jun 7, 2010 at 17:21
  • $\begingroup$ Finding an advisor like in (1) is surely a matter of luck. $\endgroup$
    – user57432
    Commented Mar 12, 2020 at 5:52
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    $\begingroup$ @user170039 : Everything is a matter of luck to some degree. But for example when I was a prospective graduate student visiting various schools, it was high on my priority list to ask the graduate students about how various professors treated their current and past graduate students. So if you are intentional about it, you can gather considerable information and reduce the amount of luck involved. $\endgroup$
    – Timothy Chow
    Commented Mar 12, 2020 at 17:07
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Bob Thomason once told me that the reason Grothendieck was so uniquely successful was that, while everyone else was out to prove a theorem, Grothendieck was out to understand algebraic geometry. I wish I'd realized that sooner.

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Here's my generic answer (applies to almost anyone learning math at any level), which I am well aware probably is not suited to everyone or all areas of math:

1) I find it easier to learn new math, if I know, as early as possible, what questions it will help me answer. I view math as a tool, so if I don't know why I'm learning a new tool, I have a lot more difficulty.

2) I also find it helpful if, before I try to learn anything new, I try to answer some of the motivating questions with the tools I already know. I find this extremely helpful, whether I succeed or not. If I succeed, then the new material isn't so new anymore, and I can anticipate what's happening. If I fail, then I have at least been able to figure out what I already know and isolate the critical difficulties. So even then I can anticipate at least some of the new math and focus on what is really new to me.

3) In general, try to answer any problem or question using the least sophisticated techniques you can get away with, i.e. without any of the new stuff you just learned. Bring in more sophisticated stuff only as you need to. If you succeed in solving the problem without using anything new, think about how the new stuff might have simplified your effort and make your proof a lot cleaner and more elegant.

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There are a couple of posts on this at the Secret Blogging Seminar. There's also Terry Tao's advice page.

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    $\begingroup$ I very much agree with Noah Snyder's advice in the first link that one should "Do one early side project." It's hard to overstate how much this boost in confidence helps. $\endgroup$
    – David White
    Commented May 30, 2012 at 14:14
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Well, I realize this is a pretty old question, but since it's been bumped I'll add some links which helped me during my grad career. The first two are written by a guy with a computer science PhD, but I've found his advice useful for math as well:

3 Qualities of Successful PhD Students, 10 easy ways to fail a Ph.D., Productivity Tips

I keep the last two links on my favorites list so that any time I'm tempted to procrastinate I can see them and remind myself to get back to work.

Another nice collection of advice online is Dianne Prost O'Leary's Survival Manual. It covers everyday life as a grad student, finding an advisor, starting research, thesis writing, and even beginning your career. Bonus points because the section on Maintaining Sanity mentions Imposter Syndrome, which affects basically every grad student I know and is worth being aware of. Also, Chapter 8 on research makes several valuable points, most significantly how to become an active reader (which is similar to advice from Ravi Vakil about attending seminars).

Finally, the University of Indiana has a great collection of advice for grad students at all levels. For example, here is a paper on How to be a Good Graduate Student which has useful advice about the daily grind and about beginning research. Here is one they link to called Tips for Ph.D. Study which I've found useful (especially for writing).

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I will pass on a few tidbits from some of my teachers, and one of my own.

First, my own. Don't be shy! The profs like it when you ask questions, anytime. Talk to everyone, all the time.

Second, one from Marty Moskowitz, who taught me algebra - go to seminars, even if you don't understand anything. Eventually you will.

Third, from Isaac Chavel, after I went to him with concerns that I just wasn't getting what he was talking about in differential geometry - no one ever learns anything the first time they see it.

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    $\begingroup$ First time? Sometimes you have to try not just one or two times but many more times. Of course, sometimes it's good to step away for a while from whatever you're stuck on. Sometimes, just grinding through things step by step, filling in all the details, works. Sometimes, it's better just to take a long walk. $\endgroup$
    – Deane Yang
    Commented May 30, 2012 at 14:58
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I think you should take a look at "A Mathematician's Survival Guide: Graduate School and Early Career Development", by Steven G. Krantz. The customer reviews on the amazon page are all highly laudatory. I did not read the book myself (since I'm just a student in my first year), but based on the enthusiastic responses I suppose you will be rewarded if you read it.

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    $\begingroup$ Hm.. upon giving this answer I didn't realize you posted this question almost two years ago. How is graduate school working out for you? Did the answers help you? If so, how? $\endgroup$
    – Max Lonysa Muller
    Commented May 31, 2012 at 15:05

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