The principal investigator will conduct research in an area of mathematics which involves differential geometry and several complex variables. The problems to be investigated concern the intrinsic geometry of convex domains based on rescaling and distortion theorems, compact complex manifolds and harmonic maps, and canonical structures on the boundary of hyperbolic space. The research to be conducted by the principal investigator involves differential geometry, convexity, and the theory of several complex variables. The first is an outgrowth of Newton's calculus and involves rates of change on surfaces. The second is the study of shapes which have no indentations. And the third is an area of analysis where functions have more than one independent complex variable instead of several real variables, as is the usual.