On starting graduate school and common pitfalls... 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.

 A: 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).
A: 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:


*

*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.

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