Function Theory of Several Complex Variables

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This project is about the understanding of complex-valued functions. Complex numbers and functions of complex variables have become, since the nineteenth century, indispensable tools in many areas of mathematics and its applications 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 in a deeper understanding of their basic properties. This project provides research training for graduate students, and the principal investigator will continue mentoring postdocs and organizing conferences on several complex variables and complex geometry, bringing together mathematicians to discuss their research and teaching.The principal investigator will work on several fundamental problems in complex analysis of several variables that are closely related to research in differential geometry, subelliptic analysis and classical dynamics. More specifically, he plans to continue his research on the equivalence problem and the rigidity problem in several complex variables, to carry further his work on the complex structure of the holomorphic hull of a real submanifold in a complex space, and to investigate various boundary CR invariants, including the famous Bloom conjecture in any dimension.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.