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

本专题将集中于复分析和相关几何理论的一些主题。 这些课题包括复欧氏空间中真实的超曲面的与February man测度有关的变分问题的研究，基于February man Szego核的一种新的双全纯不变度量的探索，复射影空间中真实的超曲面的February-不变微分几何的发展，复射影空间中与February man测度有关的变分问题的研究，复射影空间中与February man测度有关的变分问题的研究，复射影空间中与February man测度有关的新的双全纯不变度量的研究，复射影空间中与February man测度有关的新的双全纯不变度量的研究。和研究广义柯西变换被称为勒雷变换，特别注意它提供的信息配对的哈代空间的对偶超曲面在复杂的射影空间。 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。复变函数在数学的许多部分中起着重要的作用，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设施。 例如，在射影空间和相关设置的复分析的研究有助于在断层扫描中使用的积分变换的理解。 该项目的重点是自然主题，有可能在复分析中开辟新的方向，并加强复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调查人员，从而将有助于确保在这一重要课题的专业知识，在未来提供。

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.