Function Theory of Several Complex Variables

This project concerns the understanding of complex numbers and complex functions. Complex numbers and functions of complex variables are indispensable tools in many areas of mathematics and have deep applications to other areas of science and engineering. The solutions of many problems in applied sciences could ultimately depend on improvements in complex analytic tools. Results of the research to be carried out in this project may lead to the discovery of novel properties of complex-valued functions. This project also has significant educational and training aspects. Graduate students, undergraduate students, and junior researchers will be actively involved in the project. Also, the principal investigator will continue to organize international conferences on several complex variables and complex geometry, bringing together many mathematicians to discuss their research and teaching. This project involves work on a number of problems in the broad area of several complex variables and Cauchy-Riemann geometry. The problems under consideration also have connections to other mathematical fields, including differential geometry, complex singularity theory, algebraic geometry, and classical dynamics. More specifically, the PI will continue his investigation of rigidity problems in several complex variables, along with their applications and interactions with complex geometry and algebraic geometry. He will continue his research on the equivalence problem in several complex variables, pursue his ongoing study of the complex structure of the holomorphic hull of a real submanifold in a complex space, and further his investigations on the existence and regularity problem for Levi-flat submanifolds bounded by real submanifolds with CR singularities. In addition, he will continue his work on boundary invariants of weakly pseudo-convex domains, finite type conditions and the Bloom conjecture, and canonical metrics on Stein spaces with isolated normal singularities. Finally, he will continue his study on the boundary unique continuation problems for holomorphic functions and his work on transversality problems in the Cauchy-Riemann setting. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.