By foundations of mathematics I am referring to the mathematical, logical, and philosophical foundations of the subject. I'm interested in seeing which of these have active research going on within them, and what those active areas of research are, i.e. what questions they deal with. Another way to put it; what questions in the foundations of mathematics remain unanswered, and which of these unanswered questions remain active fields of research? Any information you can share on the subject would be greatly appreciated!