- The best teacher is the one who suggests rather than dogmatizes, and inspires his listener with the wish to teach himself. (Edward Bulwer-Lytton)
While you should talk to your advisor, you should not be completely reliant on him or her; after all, you are going to have to do mathematics primarily on your own once you graduate!
If you feel like you want to learn something, do something, or write something, you don’t have to clear it with your advisor – just go ahead and do it (though in some cases other priorities, such as writing your thesis, may be temporarily more important, and you should of course keep your advisor updated as to what you’re doing mathematically).Research your library or the internet, talk with other graduate students or faculty, read papers and books on your own (both in your field and in nearby fields), attend conferences, and so forth. (See also “ask yourself dumb questions”.)
In a somewhat related spirit, while it is certainly acceptable to have mathematical role models, one should not try to mimic them too slavishly; you need to develop your own personal style, exploiting your own strengths and mitigating your own weaknesses, which will not be identical to those of your role models. Ultimately, it is better to follow the mathematics than to follow a mathematician.