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?