Advice for pure-math Phd students 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?
 A: I can say a little more than just "work hard and work a lot"! There is a motivational video about bodybuilders. I always watched is as if he was talking about mathematics. (In the middle of the video in my mind i replaced the names of the bodybuilders with the names of famous mathematicians, genetics with smartness etc. ) I read a book about Erdős this week, and noticed the fact that he worked so much, other mathematicians had a hard time to keep up with him (with the time he spent doing math). 
Here is the video: (Actually the only important part of the video is the sound.) http://www.youtube.com/watch?v=Sk56VxaeqEQ
A: Working with an advisor who produces ``high-quality'' research will increase the likelihood you do as well. And not just someone who used to produce good mathematics, but someone who is producing such now. Also, focus on research as early as possible, i.e. read research papers as early as possible. You can learn a lot of things from papers that you won't find in a book, and you can let the reading of papers guide your reading of the standard text-book references. In terms of how to work, just work hard. Everyone has their own style, so just work hard and eventually you will find yours.
A: The first piece of advice I give all of my students is to have a plan B if the math thing does not work out. There is nothing more pitiful than a person who is trying to hold on to a bad job in academia because they somehow have it in their head that this is the only honorable option out there. 
It is hard to generalize, but another important thing is to talk to many people. Yes, one needs to learn stuff from books and papers, but mathematics these days is not a solitary occupation and one will not get far by only talking to their advisor. It is, however, crucial to be able to have a good working relationship with the advisor, in fact this should be the criterion for the choice of the advisor.
Sometimes students are just not strong enough or get unlucky with the choice of thesis problem or advisor. Such is life.
A: 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.
A: There are a few things regarding that:
In the first year attend a wide variety of graduate courses in Math for two reasons: first, get acquainted with the wide variety of areas to figure out what area of Math speaks to you most, and second, gear which professors you would like to work with. IMHO professors' research abilities and advising abilities are less correlated that their graduate level teaching abilities and advising abilities.
Attend colloquia not only in Math but also in related areas, such as Physics, Computer Science, Engineering, etc. A huge part of successful Ph.D. is the problem statement. Interesting mathematical problems may arise not only in Math but also in applications. An additional benefit is some exposure to application of Math in case the career in pure Math wouldn't work out.
Pick the best advisor you can, based not only on their track record in Math, but also in their availability, ability to find interesting connections, communicate clearly, and foremost pose interesting problems. Make sure you start working with an advisor in your 2nd year the latest.
Make sure you can devote 3-4 years to your PhD study; in particular, you are comfortable enough with the finances not to seek a job outside the university before the completion of your program.
Have a plan B that is not too far away from plan A. For example, if you decide to do research in Stochastic Processes get familiar with Economics; if you pursue research in Discrete Math learn how to program and perhaps minor in Computer Science, etc. Then if life requirements would turn you away from academia you would still be valuable in an industry that requires substantial Math skills. Moreover, you may be able to do the Math you love in that industry, which would bring you twice as much money, but, alas, no peer recognition… If you are doing Math for the love of Math rather than recognition a career in an applicable industry is not bad at all, as long as what you do is Math-related. I know many very intelligent people who are content doing things like Computational Geometry or Numerical Optimization in industry for most of their lives.
