Function Theory of Several Complex Variables

Abstract:

The principal Investigator proposes in this project to work on a number of problems in the area of several complex variables and Cauchy-Riemann Analysis. These problems are not just important from the point of view of complex analysis, but also from the perspective of non-linear analysis, partial differential equations, complex singularity theory and mechanical engineering. More specifically, the PI wishes to further the present understanding of various rigidity phenomena in complex analysis and complex singularity theory. The PI would like to continue his previous investigation on the holomorphic invariant theory for real submanifolds embedded in the complex Euclidean spaces. The investigation of the intrinsic connections of the holomorphic theory of real submanifolds in a complex manifold with many well-known problems in classical mechanics, non-linear partial differential equations and classical dynamics will also be pursued. The PI intends to carry out the simultaneous embedding problem for a CR family of embeddable compact strongly pseudoconvex three-dimensional CR manifolds. Various geometric and analytic properties for Cauchy-Riemann mappings will be studied, too.

Complex numbers and the theory for functions with complex variables have become, since the 19th century, indispensable tools in many areas of mathematics and in its application to other areas of science and engineering. Indeed, the solutions of many problems in the applied

sciences could ultimately depend on improvements in these complex analytic tools and a better understanding of their basic properties. For instance, in material science, the standard method for treating multi-directional stresses in a uniform way is to represent them as complex numbers or, in more complicated situations, as complex functions. It then turns out, for instance, that the direction of the propagation of cracks in materials is related to the properties of certain equations associated with these complex numbers or functions. Results of the research to be carried out in this project may lead to the discovery of new properties of solutions of these equations, which would then translate to a deeper understanding of the related mechanical properties of materials.