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.