OK, so before this accidentally turns into a philosophy blog and I forget about the fact that I was going to go through my interests, next up math.
My main mathematical interests are in algebraic topology and category theory. More specifically in the stuff that Stefan Stolz and Peter Teichner are doing in trying to examine QFT correlations with elliptic cohomology (as a beginning grad student I am nowhere near this level though). I’m also interested in the theoretical side to differential geometry (none of this calculate the curvature stuff), i.e. the Whitney embedding theorem etc. Also putting differential forms on manifolds and looking at cohomology.
Category theory interests me, but not all that strongly. I have major philosophical viewpoints on how language affects mathematics, and I agree with David Corfield’s opinion that category theory will eventually take over since it is a more compact and powerful language. In this sense it interests me. In a pure sense, it doesn’t (sort of like set theory interests me philosophically, but not mathematically).
On a weaker level, I’ve always found ring theory, point-set topology, complex analysis, and algebraic geometry fascinating. They aren’t the subjects for me as a career or anything, but very interesting none-the-less. Probably my least favorite subjects would be pure finite group theory, the excessive terminology in point-set topology, any analysis that requires excessive approximation, and numerical analysis.