- It is the province of knowledge to speak and it is the privilege of wisdom to listen. (Oliver Wendell Holmes, “The Poet at the Breakfast Table”)
Your advisor is one of the best sources of guidance you have; not only in directly assisting you with your research topic, but in directing you (both explicitly and implicitly) to relevant researchers, conferences, publications, open problems, folklore, or other pieces of good mathematics. Your advisor also knows your situation well and can give career advice which is tailored to your specific strengths and weaknesses (unlike the generic advice in these pages).
If things get to the point that you are actively avoiding your advisor (or vice versa), that is a very bad sign. In particular, you should be aware of your advisor’s schedule, and conversely your advisor should be aware of when you will be available in the department, and what you are currently working on.
For similar reasons, you should give your advisor some advance warning if you want to take a long period of time away from your studies.
If your advisor is unavailable, you should regularly discuss mathematical issues with at least one other mathematician instead, preferably an experienced one. [Also, it is not uncommon for a student to have both a formal advisor, who handles all the official paperwork, and an informal advisor, with which you discuss research and career issues.]
Of course, you should not rely purely on your advisor; you also need to take the initiative when it comes to your mathematical career.