The investigator studies a range of central geometric and analytic problems in complex analysis by incorporating techniques from partial differential equations, algebra, and differential geometry, together with methods from complex analysis. The project will develop substantially new methods and deepen the current understanding of several important problems in several complex variables. Through research collaboration, mentoring, seminars, and other academic activities, the proposed work will provide accessible research opportunities and valuable training experiences for graduate students and postdoctoral researchers. The research focuses on three major directions: (1) geometric and algebraic aspects of the Bergman metric and kernel; (2) Bergman logarithmic flatness and obstruction flatness; and (3) rigidity problems in several complex variables. The proposed research also has significant interactions with other areas of mathematics, including algebraic geometry, complex geometry, dynamical systems, and mathematical physics. The project will develop innovative methods for studying both the interior geometry and boundary structure of complex manifolds through Bergman and Kahler-Einstein metrics. It will also deepen the understanding of rigidity phenomena for holomorphic and CR mappings, which are among the central objects of study in several complex variables. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the F