Matthew Gursky

University of Notre Dame

March 19, 2019

In this talk I want to discuss 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. I will present some recent work (joint with Kelleher and Streets) that attempts to address each question when the base manifold has positive scalar curvature. I will also mention some other geometric applications in 4-d.