My friend has said that the Lenovo thinkpad tablet is excellent for math related note taking. For example writing notes and equations on new pages and as well as jotting down notes on existing documents (pdfs).
Perhaps you should read the reviews on amazon.com for it. I have provided the link.
http://www.amazon.co...28129874&sr=8-1
edit: I think you need to buy the tablet pen separately.
A good way to develop ideas is to read some up-to-date (maybe at most 10 years) Ph.D thesis in your field. They often give a survey of the topic or questions relating to your field. Your school libraries has access to databases on thesis.
Perhaps the tasks he gets you to work on serves as a kind of exploration so if a specific problem interests you, then you tell him "Hey, this looks fun. I'm investigating this...".
r/gradschool should be renamed r/gradschoolapplications because most links are about that.
r/frugal, r/AskAcademia/, r/askscience, r/funny, r/pics, and r/explainlikeimfive.