## Oliver Club

Since the 1700s, efforts to prove the existence of minimal submanifolds in various settings have given rise to a lot of beautiful mathematics, including the development of Morse theory and the Lebesgue integral. For minimal submanifolds of dimension one (geodesic curves) and codimension one (minimal hypersurfaces), the existence theory has matured dramatically in the last half-century (though deep open questions remain).

By contrast, the space of minimal submanifolds of higher dimension and codimension remains rather poorly understood, with few major advances since the 1960s. After surveying some of this history, I'll describe a new approach to the existence theory for minimal submanifolds of codimension two, based on connections to a well-studied family of geometric pdes with origins in the study of superconductivity. I'll highlight some initial successes of this approach, and point to some key open problems. (Based on joint work with Davide Parise and Alessandro Pigati.)