Minimal surfaces are fundamental geometric objects which have been studied intensively since the 1700’s. Classes of minimal surfaces of particular interest are the complete embedded ones in Euclidean space, closed (compact boundaryless) embedded in the round three-sphere, free boundary compact embedded ones in the unit Euclidean three-ball, and self-shrinkers of the mean curvature flow. Since the 1970’s great progress has been made in finding new examples and characterizing the simplest ones.

Partial classifications under low area, genus, or index assumptions, seem now possible in the long term. Gluing constructions by PDE gluing methods and in particular doubling constructions have been very successful.

In my talk I will review recent and ongoing work on these constructions and their impact on the field.

Nikolaos Kapouleas got his undergraduate degree from Cambridge University, Trinity College, and his Ph.D. from UC Berkeley in 1988 under R. Schoen. He is a tenured Professor at Brown University since 1991. He is best known for gluing constructions in Geometry by PDE methods. In his early work he provided the first counterexamples to the Hopf conjecture for any genus larger than one. He gave an invited lecture at the ICM in 1994 and received the Bodossaki prize in 1997. His recent work is on minimal surfaces and other geometric objects.