The fundamental question underlying directly or indirectly most of my research 
is this:

* What is the nature of proof complexity?

That is, what are the intrinsic reasons that some formulas are hard to prove?
Can the proof complexity of some formulas be traced to the computational
complexity of associated computational tasks? Which mathematical structures relate
to the existence of feasible proofs?

These questions have many technical mathematical facets, the most famous being the P vs. NP problem and 
its variants. I approach these problems from the mathematical logic perspective. Most of this research is
covered in my Cambridge U. Press books.