I read the article “Advice to a Young Mathematician” written by five authors Sir Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff and Peter Sarnak. In the following I summarize in eight clauses what I found useful and what was pointed out by at least two of the authors.
1. Are you passionate about mathematics?
Without strong internal motivation you cannot succeed, but if you enjoy mathematics, the satisfaction you can get from solving hard problems is immense.
2. Be patient and ready for frustration and disappointment.
Only mediocre are supremely confident of their ability.
Most of the time one is stuck, and if it is not the case for you, then either you are exceptionally talented or you are tackling problems that you knew how to solve before you started.
…bear the problem in mind all the time: it worked for Newton, and it has worked for many a mortal as well.
3. Work on all levels. Have always problems of various degree to work on.
Some very worthwhile problems that you feel you should have a good chance of solving, given enough time, effort and luck
Then you should have a range of more challenging problems, with the most difficult ones being central unsolved problems. One should attack these on and off over time, looking at them from different points of view.
4. Work only on interesting problems
Taste is above everything
…you should choose problems that you find difficult not to think about.
5. Read as much as you can.
…both in your general area and in mathematics as a whole.
…combine reading modern treatments with a study of the original papers, especially papers by the masters of our subject.
6. Sometimes do not read. Have a fresh view.
On the other hand, it is often useful not to read up everything about an open problem you are about to attack: once you have thought deeply about it and apparently got nowhere, you can (and should) read the failed attempts of others.
7. If your method does not solve the problem, change the problem. (a quote by Saharon Shelah).
I probably most needed to learn how to ask good questions. As a student, one’s job is not only to learn enough to be able to answer questions posed by others, but also to learn how to frame questions that might lead somwhere interesting.
8. Take advantage of sociality.
Many people these days work jointly, which, besides the obvious advantage of bringing different expertise to bear on a problem, allows one to share the frustration.
Go to your departament colloquium every week …, besides learning …, you can often have an idea stimulated in your mind when the speaker is talking about something quite different.