Unprovability Results in Complexity Theory

Several of the central conjectures in complexity theory, including the celebrated P vs NP question, remain wide open despite several decades of effort. It has been speculated that the difficulties in their resolution might have connections to incompleteness phenomena in logic. I will describe the framework of bounded arithmetic, which studies fragments of Peano Arithmetic where attention is limited to statements and reasoning of bounded complexity. I will briefly survey unprovability results in this area, and explain some challenges in extending these results to questions like P vs NP.