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。这些主题包括复欧几里德空间中与Fefferman测度相关的变分问题的研究；基于Fefferman Szego核的生物全纯不变度量的探索；复射影空间中实超曲面的莫比乌斯不变微分几何的发展；以及广义柯西变换（即Leray变换）的研究，特别关注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。最后一个主题对常系数全纯偏微分方程的研究具有潜在的意义。复变量函数在数学的许多领域都扮演着重要的角色，包括偏微分方程、傅立叶理论和积分几何。这些主题反过来又为物理学和工程学提供了重要的基础。例如，研究射影空间和相关设置中的复分析有助于理解断层扫描中使用的积分变换。拟议的项目侧重于自然主题，有可能在复杂分析中开辟新的方向，并加强复杂分析与数学其他部分之间的联系。该项目将涉及学生和年轻的研究人员，因此将有助于确保在未来获得这一重要主题的专门知识。

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.