Research Projects

There are several naturally-occuring notions of convexity and hulls in the study of mulitvariate holomorphic functions. These notions endow the underlying set with some geometric rigidity. We wish to study this in the setting where the underlying set is a real submanifold of complex Euclidean space. This brings into play the CR (complex-real) structure of the submanifold, which can impact the convexity of the set. There are three concrete problems outlined in our proposal. One is a minimal embedding dimension problem (with a constraint on the image). The other two are about holomorphic fillings of n-spheres in C^n, either embedded ones in dimensions greater than two, or immersed ones in two dimensions. The study of holomorphic fillings of embedded two-spheres and two-tori in C^2 has a rich history, which motivates this part of our project.