Many PhD supervisors have lists of open problems in their subject, often hidden somewhere on their web site. Here are some from my university:

Here are 27 research problems in group theory from Rob Wilson: http://www.maths.qmul.ac.uk/~raw/resprob.html

Here are many problems in combinatorics and group theory from Peter Cameron: http://www.maths.qmul.ac.uk/~pjc/oldprobidx.html

(And also his links to other lists of problems: http://www.maths.qmul.ac.uk/~pjc/bcc/links.html#prob)