The course attempts to prove the following simple-sounding conjecture: Let E be a subset of the plane, let X be the space of all functions on the plane with second derivatives in L^p, and let X(E) be the Banach space of all restrictions to E of functions in X. Then there exists a bounded linear map T from X(E) to X such that Tf=f on E for any f in X(E).That's known for p>=2, but not for p<2. After providing a bit of elementary background, the lectures study examples, with audience participation welcome, in hopes of eventually understanding the general case.