Extremal Eigenvalues and Minimal Surfaces

I'll describe recent progress on the existence theory for metrics maximizing Laplace and Steklov eigenvalues on surfaces, and discuss applications to the existence of embedded free boundary minimal surfaces in the 3-ball and nonorientable minimal surfaces in the 4-sphere with any prescribed topology. Based on joint work with Misha Karpukhin, Rob Kusner, Peter McGrath, and Romain Petrides.