TITLE: Advice for pure-math Phd students
QUESTION [54 upvotes]: Pursuing a Phd in pure math can be a daunting task.  A number of students who begin a Phd in pure math don't complete it, and there are high-quality dissertations and those which are not so high quality.
My question is: What advice do you, or would you, give beginning or first-year Phd students early in their studies which will likely increase their possibility of successfully completing a high-quality Phd in pure math?  Alternatively: What advice do you wish you were given when you started your Phd?
Are there particular qualities or habits, or is there a particular way of approaching or attitude towards Phd studies, shown by Phd students who complete a high-quality Phd in pure math compared to those who don't?
Is there general principles or advice which can be given to "fit all"?  Or do those students who successfully complete a pure-math Phd have "wildly-varying" styles, attitudes and approaches to their studies?
I guess another perspective on this would be: What have you found to be the main reasons for Phd students dropping out or completing a poor-quality pure-math Phd, and what advice could have been given to them early in their studies to prevent these reasons from occuring?
I ask this question because I noticed in my department this year that some pure-math Phd students dropped out in their second or third year of study for various reasons, most of which seemed preventable if they had the right advice early on from their supervisors.  Also, the qualities required by pure-math Phd students seem in some ways to be unique compared to other fields.  Finally, is it fair to say that professors/supervisors are sometimes not particularly skilled at giving this kind of advice, so many (most?) Phd students are "going without" advice that could really benefit them?

REPLY [16 votes]: From my personal experience (I am currently in my third year of my PHD), I find that the most valuable thing an advisor can provide is simply insisting that you keep working on your problem. Solving mathematical problems is tough, and your first one even more so. After getting a "result" (it was wrong) fairly quickly after getting my problem and quickly finding out that it was incorrect, I worked essentially fruitlessly at my problem for over a year. This was after dredging through dozens of papers relating to the topic. Many times I felt like I should mention to my advisor that perhaps this problem is not suitable and that I should work on something else. He never waivered, and gently insisted that I keep working. Finally this past summer I was able to make non-trivial progress, albeit not quite in the direction originally anticipated. 
In turn, the advice I offer is as follows: there is no replacement for hardwork and perseverance. The return on investment (in terms of monetary value) of a pure math PHD is fairly poor; you can earn a lot more money doing a lot less in many other professions. If the intrinsic value and personal satisfaction of the work is not enough and you are doing it for the money, then start coming up with a plan B. However, if you would rather hammer away at your problem all day than do anything else, then be prepared to work hard often. 
Finally, it is important to remember that the role of your advisor is to advise, not to do the work for you. It is important to remember that the key transformation to be undertaken during your PHD is the transition from apprentice to master. After the completion of your PHD you will no longer be your advisor's disciple but your own master, where you will presumably start an academic career where you will primarily fill an instructor or mentorship role. Thus I feel you should expect to go beyond the boundaries naturally set out by your advisor; indeed, it is important for you to be better than your advisor at something (mathematically). If your advisor is a strict upper bound of you in every mathematical aspect, then why should you get a job since presumably your advisor can always do it better than you? So, it is important to work hard on your problem and get good results, but always be mindful of other subject areas, especially if your techniques can be applied to said subject areas.