Geometric Problems in Several Complex Variables

This project focuses on complex analysis, a branch of mathematics that investigates the theory of calculus over the complex. Complex analysis serves as an important tool in numerous applications: It plays a crucial role in physics (e.g., modeling airflow over airfoils and analyzing dispersion relations in optics), engineering (e.g., signal processing and control theory), and computer science (e.g., image processing and quantum computation). The theory of complex analysis in one variable is classical and well understood, but when additional variables are introduced, many mysteries remain. In this project, the investigator will further the theoretical understanding of complex analysis of several variables. The proposed activities also involve collaboration with and mentoring of junior researchers at the undergraduate, graduate, and postdoctoral level. The investigator will study the Gromov hyperbolicity of the Kobayashi metric; regularity properties of biholomorphic mappings between families of domains in complex Euclidean spaces; quantitative versions of the Hartogs’ extension theorem and analytic continuation; and proper holomorphic maps between unit balls. These topics involve a range of mathematical fields, including differential geometry, metric geometry, geometric group theory, Lie theory, and dynamical systems. Thus, the project will not only contribute to the field of several complex variables but also strengthen its ties with these other areas. 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.