Work on this project focuses on questions arising in the mathematical theory of several complex variables. The research concerns the nature of mappings between domains in spaces of several complex variables - possibly of different dimension. A specific problem of interest is that of constructing maps from balls of different dimension which are proper and holomorphic. Proper means effectively that the mappings look finite-to-one locally. The maps are to be in certain Lipschitz classes up to the boundary. This degree of smoothness has yet to be obtained; the best result to date assumes differentiability up to the boundary. In such a case, the only mappings turn out to be rational. If the first domain is not a ball, then the target domain in general cannot be a ball. A more likely target is the unit sphere of a Hilbert space. In such cases, pseudoconvex domains with analytic boundaries can be mapped by proper maps. This project will go further in lowering the condition on the boundary to demonstrate that differentiable proper maps can be obtained.
