Geometric and Analytic Properties of Real Hypersurfaces in Complex Euclidean and Projective Spaces

这个项目将集中在复杂分析和相关几何理论的一些主题上。这些主题包括：研究复欧氏空间中实超曲面的Fefferman测度的变分问题；探索基于Fefferman Szego核的新的双全纯不变度量；发展复射影空间中实超曲面的Mobius不变微分几何；以及研究称为Leray变换的广义柯西变换，特别注意它提供的关于复射影空间中对偶超曲面上Hardy空间配对的信息。复变函数在包括偏微分方程组、傅立叶理论和积分几何在内的许多数学领域都有着重要的作用。这些主题反过来又为物理学和工程学提供了至关重要的基础设施。例如，研究射影空间和相关环境中的复分析有助于理解层析成像中使用的积分变换。拟议的项目侧重于自然主题，有可能在复杂分析中开辟新方向，并加强复杂分析与数学其他部分之间的联系。该项目将有学生和年轻的调查人员参与，从而有助于确保今后能够获得这一重要课题的专门知识。

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.