Nature and technology abound with fluid interfaces such as the surfaces of oil droplets in water or the membrane surfaces of living cells.  These interfaces are typically crowded with adsorbed particles, proteins or other large molecules, which are effectively confined to a two-dimensional fluid.  This two-dimensional system, though, has a twist: it can spontaneously change its curvature and thereby substantially alter the interactions among the bound particles or proteins.  In biology, there are many examples where proteins change the shape of a membrane – a key part of a cell’s ability to exchange materials with its exterior (via endocytosis).  Despite the many known examples, there remain quite basic questions about how proteins and membrane curvature work together.   In this talk, I will describe our experiments with a family of membrane-binding proteins known as BAR, which have a strong affinity for highly curved membranes.  BAR proteins are shaped like a banana, which suggests a geometric mechanism for altering membrane shape – but in fact the mechanism remains controversial.  By measuring the binding affinity of BAR as a function of mechanical tension applied to the membrane, we aim to derive new insights into how the BAR protein and its soft, two-dimensional substrate work together.