This project will focus on a number of topics in complex analysis and related geometric theories. These topics include the study of variational problems connected with Fefferman's measure for real hypersurfaces in complex Euclidean space; the exploration of a new biholomorphically invariant metric based on Fefferman's Szego kernel; the development of the Mobius-invariant differential geometry of real hypersurfaces in complex projective space; and the study of a generalized Cauchy transform known as the Leray transform, with special attention to the information it provides about pairing of Hardy spaces on dual hypersurfaces in complex projective space. The last topic has potential implications for the study of constant coefficient holomorphic partial differential equations.
Functions of complex variables play an essential role in many parts of mathematics, including partial differential equations, Fourier theory, and integral geometry. These topics in turn provide vital infrastructure for physics and engineering. For example, the study of complex analysis in projective space and related settings contributes to the understanding of the integral transforms used in tomography. The proposed project is focused on natural topics with potential for opening up new directions within complex analysis and for strengthening the connections between complex analysis and other parts of mathematics. The project will involve students and younger investigators and thus will help to ensure that expertise in this important subject is available in the future.
