Becoming a great mathematician, or computer scientist, is not a matter of following career recipes. And yes, it is amazing how many people manage to listen to advice, and then go on to make the same mistakes for themselves. That caveat about the utility of advice (in particular advice with hindsight) aside, HLF laureates did have some useful hints about what to do and what not to do.
Look for existing structure!
Andrew Wiles’s advice in his lecture on Tuesday was particularly poignant because it runs counter to the fairy tale version of his proof of Fermat’s Last Theorem – you know, the version where a young Andrew is fascinated by the theorem even as a child, and the rest is fate. On the contrary, Wiles insisted that he wouldn’t have invested significant time looking for a proof if there hadn’t been more modern developments, linking the ancient theorem to concepts like elliptical curves, modular forms, and automorphic representations. It was only because of this embedding in a more complex system that Wiles was sure that what he was doing wasn’t going to be a complete waste of time. Even if he didn’t manage to prove Fermat’s last theorem, he would surely manage to produce some worthwhile proofs on the way.
Thus his advice to the young researchers: “Pick a problem that really appeals to you, but pick one that has some structure, so that even if you fail, you will have proved some things.”
For more advice, read this
Finally, here is a recommendation if you are interested in more advice – including some from HLF Laureate Sir Michael Atiyah. It is part of the “Princeton Companion to Mathematics,” a chapter entitled “Advice to a Young Mathematician,” with contributions from Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak.
I’ll just quote one particular bit concerning self-doubts, which Atiyah proclaims quite common:
I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage. Only the mediocre are supremely confident of their ability.
…and encourage you to read the rest: Princeton University Press has kindly put the PDF of this particular chapter online, available for download.