I am a (as of this summer) third year graduate student in a mathematics PhD program, and I was wondering if you also have a post on something like “Where to Look for New Problems.” Certainly faculty in any decent mathematics department are a great source of open problems, but I have also heard that it is good to go outside of the expertise of, say, your advisor, and to be working on multiple research projects simultaneously (the latter being advice from you in another blog post, if I recall correctly).

What might you suggest as an avenue for finding new problems for an aspiring mathematician?

]]>I learn mathematics as a hobby because I did not have a chance to study college-level mathematics at any good educational institutions.

Do one best learn mathematics which one does not know anything about , say symplectic geometry , best by starting from page 1 of a book on symplectic geometry , and work out all (or most) the proofs and problems as he read. Or is it much more efficient to start from a problem in , say , classical mechanics and work out the mathematical structure of hamiltonian systems on his own from scratch using symplectic topology books as a guide ? ]]>