Executive Summary : | This project focuses on geometric function theory in several complex variables, specifically on univalent maps (biholomorphic maps onto their range). A biholomorphic map from C^n onto C^n is called an automorphism of C^n. For n=1, the class of biholomorphisms consists of only nonconstant affine holomorphic maps. For n bigger than 1, the class of automorphisms is huge. Two simple classes of automorphisms are shears and overshears. The study of shears and overshears was pioneered by Rosay and Rudin in 1988. Andersen and Lempert later proved that the group generated by overshears are dense in all automorphisms, leading to the Andersen-Lempert theory. The project aims to generalize the Andersen-Lempert theorem to more general spirallike domains defined by Gurganus and generalize the notion of spirallikeness to complex manifolds with density property. The main open problem in the field is whether there exists a solution for Loewner PDE in a complete hyperbolic domain in C^n with values in C^n. The project aims to complete this investigation and start writing the first paper by the end of the first year. Preliminary calculations give hopes to solve these problems in certain classes of domains. The project also includes training Ph.D. students, Research Associate, and Project Assistant/Student in this field of research. |