Function Theory of Several Complex Variables

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This project concerns the understanding of complex numbers and complex functions. Complex numbers and functions of complex variables have become, since the nineteenth century, indispensable tools in many areas of mathematics and its application to other areas of science and engineering. The solutions of many problems in the applied sciences could ultimately depend on improvements in these complex analytic tools and a deeper understanding of their basic properties. For example, in materials science the standard method for treating multidirectional stresses in a uniform way is to represent them as complex numbers or, in more complicated situations, as complex functions. It then turns out that, among other things, 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. This project has significant educational and training aspects: several graduate students and junior mathematicians will be actively involved in this project. Also, the principal investigator will continue to organize international conferences on several complex variables and complex geometry, bringing together mathematicians to discuss their research and teaching.More precisely, the principal investigator will continue his work on several basic problems in complex analysis of several variables that are closely related to research in differential geometry, algebraic geometry and classical dynamics. More specifically, the principal investigator would like to continue his research on the equivalence problem in several complex variables, carry further his work on the complex structure of the holomorphic hull of a real submanifold in a complex space, and further his study on the existence and regularity problem for Levi-flat submanifolds bounded by real submanifolds with CR singularities. He will continue his investigation of various rigidity problems in several complex variables, as well as their applications and interactions with super-rigidity problems in complex geometry and algebraic geometry. He will investigate the simultaneous embedding and filling problem for a Cauchy-Riemann (CR) family of embeddable, compact, strongly pseudo-convex, three-dimensional CR-manifolds.