Bill Meeks

Abstract:   In recent years the global theory of properly embedded minimal surfaces in three dimensional Euclidean space has enjoyed spectacular successes. First there was progress made in constructing rich moduli spaces of new and geometrically interesting examples. Shortly afterwards we obtained new insights into the global conformal, topological and geometric structures that arise in the examples. Accompaning these theoretical developments, there has been a return in interest in the classical examples studied by Riemann, Schwartz, Scherk and other geometers in the last three centuries and the extent to which these examples determine the geometry of all examples.

Part of my talk will cover the general theoretical developments, such as the maximum principle at infinity and curvature estimates, that have had a big impact on the theory in recent years. I also will describe how close we are to proving the uniqueness of many classical examples including the helicoid, the catenoid, Scherk's surfaces and an important family discovered by Riemann and defined in terms of elliptic functions. During my talk I will be showing computer graphics slides of many of these beautiful surfaces.