Geometry, Measures, and Integral Operators for Boundaries of Complex Domains

复杂域边界的几何、度量和积分算子

• 批准号: 0901205
• 负责人: David Barrett
• 金额: $ 28.6万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901205&HistoricalAwards=false
• 关键词: Geometry; Measures; Integral; Operators; Boundaries

This project will focus on a variety of topics in complex analysis and related geometric theories, including the following interconnected subjects: i) the study of invariant geometric properties of real hypersurfaces in complex projective space; ii) the study of the Leray transform, a multivariate generalization of the Cauchy transform, with special attention to the information it provides about pairing of Hardy spaces on dual hypersurfaces in complex projective space; iii) continuation of a project with Mike Bolt to investigate geometric and analytic topics relating to a variety of invariant arc lengths including that of Laguerre; iv) extension and application of the theory of Fefferman's equi-holomorphically invariant boundary measure with particular attention to an associated variational problem. Functions of complex variables arise naturally and play an important role in many parts of mathematics, physics and engineering. This project is focused on issues of boundary geometry and analysis which are fundamental in nature. In particular, the study of the Leray transform and related matters has substantial connections with the Laplace transform, an important tool in the study of partial differential equations. The project borrows from (and, it is hoped, will contribute to) a variety of geometric theories of independent interest. Finally, the project will involve work with younger investigators and thus will help to ensure that expertise in this area is available in the future.

本专题将集中于复分析及相关几何理论的多个课题，包括以下相互关联的课题：i）复射影空间中真实的超曲面的不变几何性质的研究; ii）Leray变换的研究，柯西变换的多元推广，特别注意它提供的关于复射影空间中对偶超曲面上的哈代空间配对的信息; iii）继续与Mike Bolt合作的一个项目，研究与各种不变弧长（包括拉盖尔弧长）有关的几何和分析主题;（iv）推广和应用February man的等全纯不变边界测度理论，特别注意一个相关的变分问题。 复变函数是自然产生的，在数学、物理和工程学的许多方面都起着重要的作用。 这个项目的重点是边界几何和分析的问题，这是基本的性质。 特别是Leray变换及其相关问题的研究与偏微分方程研究中的重要工具拉普拉斯变换有着实质性的联系。 该项目借用（并希望，将有助于）各种几何理论的独立利益。 最后，该项目将涉及与较年轻的调查员合作，从而有助于确保今后能够获得这一领域的专门知识。