In this project the principal investigator will work on several problems in multivariate complex analysis that are related to research in nonlinear analysis, partial differential equations, and classical dynamics. More specifically, the principal investigator wishes to continue his research into 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 will study the equivalence problem for real submanifolds in complex spaces and further the present understanding of the local hull of holomorphy for a real submanifold in a complex manifold. He also intends to investigate the simultaneous embedding and filling problem for a CR family of embeddable compact, strongly pseudoconvex, three-dimensional CR manifolds and to explore the Grauert-Siu-Ling direct image theory through an approach that makes use of the d-bar equation. Various geometric and analytic properties for CR mappings will be pursued, as well.
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. The project has significant educational and training aspects: several graduate students will be actively involved in this project. Finally, the principal investigator is planning to coorganize conferences on several complex variables and partial differential equations, bringing together mathematicians (including young people and members of underrepresented groups) to discuss their research and teaching.
In the grant period of this project, the principal investigator worked on severa importantl problems in multivariate complex analysis that are related to research in nonlinear analysis, partial differential equations, and classical dynamics. More specifically, the principal investigator continued his research into 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 studied the equivalence problem for real submanifolds in complex spaces and further the present understanding of the local hull of holomorphy for a real submanifold in a complex manifold. He invetsigated the CR embedding problems by using the intrinsic invariant theory, such as the Chern-Moser-Weyl tensor. During the grant period, the PI has pubished more than 7 reserach papers. In particluar, he and his former student solved a long standing open question raised by the great later mathematician, Jurgen Moser in 1985. That 60 pages long paper appeared in the top journal Inventiones. 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 find applications and to discoveries of new properties of solutions of these equations. The PI also played an important role in the educational and training aspect: Three Ph D students wrote excellent Ph D thesis under his supervision during the grant period. They all have gotten very good positions after their graduation. One (graduated in the summer of 2008) is currently an Asscoiate Professor from Wuhan University, China and a French-based European Research Fellow; one (graduated in 2009) went to University of California as S. E. Warschawski Visiting Assistant Professor, and one (graduated in 2010) went to Johns Hopkins University as a J. J. Silverster Assistant Professor. He was also the organizer of a major international conference on Several Compex Variables and Complex Geometry held in Wuhan, 2009. The conference has more than 30 participants from USA and is jointly supported by the NSF of USA, France and China. He organized several other international workshops in Several Complex Variables which also had many participants from USA. The PI is a principal speaker of a RTG workshop in the field in 2011 at the University of Michigan.
