Since Lagrange’s work in the 18th century, the theory of minimal surfaces has become a central topic in differential geometry, geometric analysis and mathematical physics. While minimal surfaces in Euclidean space are plentiful, those in spheres remain far more elusive. In this colloquium, after providing a general overview of the theory, I will discuss a recent joint work with Michele Ancona, Anna Roig Sanchis, and François Labourie, in which we answer a question of Yau by proving the existence of negatively curved minimal surfaces in any sphere of large dimension.
