Gap and index estimates for Yang-Mills and harmonic map theory

In this talk we want to discuss joint work with Streets and Gursky on two related questions about the variational structure of the Yang-Mills functional in dimension four. The first is the question of 'gap' estimates; i.e., determining an energy threshold below which any solution must be an instanton, hence a minimizer for the Y-M energy. The second question is about non-minimal solutions, and in this case the problem is to estimate the index of a solution. Time permitting, we will also discuss related joint work with Gursky on a gap estimate in harmonic map theory.