New minimal surfaces in S^3 and B^3 via eigenvalue optimization

I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in S^3 and free boundary minimal surfaces in B^3 via constrained optimization problems for Laplace and Steklov eigenvalues on surfaces. Along the way, I'll highlight some new techniques for establishing existence of extremal metrics in more general situations, and point to some stubborn open problems.