Advice is what we ask for when we already know the answer but wish we didn’t. (Erica Jong)

Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages).

Disclaimer: The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommenddiscussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice.  Also, it should be clear that most of this advice is targeted towards academic careers in mathematics; of course, there are many other career options available besides this, but I have no particularly informed advice to offer for such alternatives.

• Primary school level
  • Advice on gifted education.
• High school level
  • Advice on mathematics competitions.
  • What universities should one apply to?
• Undergraduate level
  • How can one become better at solving mathematical problems? Note that there is more to maths than grades and exams and methods; there is also more to maths than rigour and proofs.  It is also important to value partial progress, as a crucial stepping stone to a complete solution of a problem.
  • Don’t base career decisions on glamour or fame. But you should study at different places.
  • Does one have to be a genius to succeed at maths?
• Graduate level
  • It is important to work hard, and work professionally. But it is also important to enjoy your work.
  • Think ahead to understand the way forward; ask yourself dumb questions to understand the way before.
  • Attend talks and conferences, even those not directly related to your own work.
  • Talk to your advisor, but also take the initiative.
  • Don’t prematurely obsess on a single “big problem” or “big theory”.
  • Write down what you’ve done, and make your work available. In this regard, I have some advice on how to write and submit papers.
• Postdoctoral level
  • Learn and relearn your field, but don’t be afraid to learn things outside your field.
  • Learn the limitations of your tools, but also learn the power of other mathematician’s tools. In particular, you should continually aim just beyond your current range.
  • In your research, be both flexible and patient.
  • You should definitely travel and present your research if given the opportunity. But be considerate of your audience; talks are not the same as papers.
  • Be sceptical of your own work, and don’t be afraid to use the wastebasket.

I am also (slowly) in the process of gathering my thoughts on time management from the perspective of a research mathematician.

• Here are some general thoughts on this topic.
• Batch low-intensity tasks together to take advantage of economies of scale and to reduce distraction.

More advice:

• John Baez’s page on career advice.
• Po Bronson’s article on the relative importance of innate intelligence versus effort.
• Fan Chung’s advice for graduate students.
• Lance Fortnow’s “Graduate Student Guide“.
• Oded Goldreich’s “On our duties as scientists“.
• Richard Hamming’s “A stroke of genius: striving for greatness in all you do“.
• Matt Might’s “Illustrated guide to a Ph.D.“
• Gian-Carlo Rota’s “Ten lessons I wish I had been taught”.
• J. Michael Steele’s “Advice for Graduate Students in Statistics.”
• Ian Stewart’s “Letters to a Young Mathematician“.
• Ravi Vakil’s “For potential students“.
• The Princeton Companion to Mathematics‘ section on advice to younger mathematicians, with contributions by Sir Michael Atiyah, Béla Bollobás,Alain Connes, Dusa McDuff, and Peter Sarnak.
• AMS advice page for new PhDs
• AMS graduate student blog