Holomorphic and CR mappings in Several Complex Variables

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The proposed research has interplay with many other fields of mathematics, theoretical physics, and applied science. The PI will study geometric objects (such as CR manifolds) and develop techniques (such as microlocal analysis) that arise from several complex variables. They are deeply connected with various areas in physics and engineering such as quantum field theory and control theory, and have extensive applications to topics including magnetic hydrodynamics and electrical networks. Hermitian symmetric spaces studied in the project play a fundamental role in mathematical physics. For instance, the classical domain of type IV is biholomorphically equivalent to the future tube. The future tube in four-dimensional complex Euclidean space is a basic object in modern cosmology as it is the natural defining domain for holomorphic relativistic fields. The PI's research will deepen the understanding of the geometric structure of such objects. The principal investigator proposes to study mapping problems in complex analysis and Cauchy Riemann geometry. More precisely, the project focuses on studying the geometric, analytic and algebraic aspects of holomorphic and CR mappings by employing techniques from partial differential equations, algebra, and differential geometry, in addition to complex analysis. In particular, the PI would like to investigate rigidity, existence and regularity problems for mappings between complex or CR manifolds,as well as related questions that arise in arithmetic algebraic geometry and complex geometry. The PI will also study the intrinsic connections between the Kahler geometry of a domain in a complex space and the CR geometry of its boundary. The techniques that are needed for this study come from several complex variables (CR invariant theory, asymptotic behavior of Bergman kernel and Chern-Moser theory, etc), Kahler geometry and geometric analysis. The objective of the research project is to further the present understanding of geometric function theory in several complex variables, as well as its profound connections with aspects of algebraic geometry, complex geometry, dynamical systems, and number theory. The PI also expects the project to develop substantially new methods and ideas that will influence these areas, and provide interesting research topics for graduate students and postdocs as well.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.