Recently I've been more interested in puzzles.
Project Euler is the classic example of puzzles which require either higher math or computational skill (or both!).
Facebook has a collection of puzzles too, motivated from the engineering perspective.
But note: Facebook uses these puzzles for hiring people. Plus, the puzzles are not always mathematically oriented.
I suppose a good mathematician should always set up puzzles for themselves. As Socrates remarked:
SOCRATES: Indeed, Lysimachus, I should be very wrong in refusing to aid in the improvement of anybody. And if I had shown in this conversation that I had a knowledge which Nicias and Laches have not, then I admit that you would be right in inviting me to perform this duty; but as we are all in the same perplexity, why should one of us be preferred to another? I certainly think that no one should; and under these circumstances, let me offer you a piece of advice (and this need not go further than ourselves). I maintain, my friends, that every one of us should seek out the best teacher whom he can find, first for ourselves, who are greatly in need of one, and then for the youth, regardless of expense or anything. But I cannot advise that we remain as we are. And if any one laughs at us for going to school at our age, I would quote to them the authority of Homer, who says, that
'Modesty is not good for a needy man.'
Let us then, regardless of what may be said of us, make the education of the youths our own education. (Emphasis added, from Plato's Laches)
For example, I know a little bit about representations of Lie groups and Lie algebras (one can always learn more!)...but what about the representation of the quaternion group induced from the irreducible representations of SU(2)? How does it decompose into irreps? Etc.
Knuth remarked somewhere what helped him understand the representation theory for the symmetric group was writing a program which generated the permutation matrix representations.
I suspect writing a program which does these sorts of computations is a great puzzle for any mathematician that's savvy with programming.
And now, for something completely different.
A few papers I want to read:
When physics helps mathematics: calculation of the sophisticated multiple integral, 13 pages;
Some algebraic properties of differential operators, 15 pages;
Introduction to supergravity, 152 pages;
Fermionic impurities in Chern-Simons-matter theories, 31 pages;
Spinors and Twistors in Loop Gravity and Spin Foams, 16 pages;
Quaternionic Analysis, Representation Theory and Physics, 60 pages.