Project Details
Description
In this project, the principal investigator will continue his work on several basic problems in multivariate complex analysis that are closely related to research in differential geometry, nonlinear analysis, and classical dynamics. More specifically, the principal investigator would like to continue his research on the equivalence problem in several complex variables and carry further his work on the complex structure of the holomorphic hull of a real submanifold in a complex space through the use of the attached holomorphic disks. 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 the theory of complex singularities and complex geometry. He intends to investigate the simultaneous embedding and filling problem for a Cauchy-Riemann (CR) family of embeddable, compact, strongly pseudo-convex, three-dimensional CR-manifolds. Finally, he will continue his work to understand whether an algebraic, strongly pseudo-convex hypersurface can be locally holomorphically embedded into the Heisenberg hypersurface in a complex space of higher dimension.
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. Finally, 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.
Status | Finished |
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Effective start/end date | 6/1/11 → 5/31/15 |
Funding
- National Science Foundation: $375,000.00