Hi Everyone,

Famous anecdotes of G.H. Hardy relay that his work habits consisted of working no more than four hours a day in the morning and then reserving the rest of the day for cricket and tennis. Apparently his best ideas came to him when he wasn't "doing work." Poincare also said that he solved problems after working on them intensely, getting stuck and then letting his subconscious digest the problem. This is communicated in another anecdote where right as he stepped on a bus he had a profound insight in hyperbolic geometry.

I am less interested in hearing more of these anecdotes, but rather I am interested in what people consider an appropriate amount of time to spend on doing mathematics in a given day if one has career ambitions of eventually being a tenured mathematician at a university. 

I imagine everyone has different work habits, but I'd like to hear them and in particular I'd like to hear how the number of hours per day spent doing mathematics changes during different times in a person's career: undergrad, grad school, post doc and finally while climbing the faculty ladder. "Work" is meant to include working on problems, reading papers, math books, etcetera (I'll leave the question of whether or not answering questions on MO counts as work to you). Also, since teaching is considered an integral part of most mathematicians' careers, it might be good to track, but I am interested in primarily hours spent on learning the preliminaries for and directly doing research.

I ask this question in part because I have many colleagues and friends in computer science and physics, where pulling late nights or all-nighters is commonplace among grad students and even faculty. I wonder if the nature of mathematics is such that putting in such long hours is neither necessary nor sufficient for being "successful" or getting a post-doc/faculty job at a good university. In particular, does Malcom Gladwell's 10,000 hour rule apply to mathematicians?

Happy Holidays!

I agree that hard work and stubbornness are very important (I think we should all take after Wiles and Perelman as much as we can). But it is also important how you spend the many hours you dedicate to mathematics. For instance, choice of problems is quite important: it is important to make sure that when you work on something, you spend your time usefully, i.e. you not only make progress on this particular problem, but also learn something new about mathematics in general. It is also important not to get hyperfocused on a fruitless attempt to solve a problem; after some time and effort spent on it, it becomes addictive. In such a situation, it is sometimes better to stop and ask for help/read something or switch to another problem for a while. Often, you'll wake up one morning a month or a year later and see that the insurmountable obstacle has magically disappeared! Or maybe this "Aha!" moment will come during a discussion with another mathematician, or while listening to a talk. For many people it is also helpful to have many simultaneous projects, so that when you get stuck on one, you can work on another. To summarize, I think that not only the number of hours matters, but also how efficiently you spend them, not only in terms of publishable results, 
but also in terms of your personal growth as a mathematician.  