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.

该项目将重点关注复分析和相关几何理论中的各种主题，包括以下相互关联的主题：i）复射影空间中真实超曲面的不变几何性质的研究； ii) Leray 变换的研究，这是 Cauchy 变换的多元推广，特别关注它提供的关于复射影空间中双超曲面上的 Hardy 空间配对的信息； iii) 继续与 Mike Bolt 合作开展一个项目，研究与各种不变弧长（包括拉盖尔弧长）相关的几何和分析主题； iv) 费弗曼等全纯边界测度理论的扩展和应用，特别关注相关的变分问题。 复变量函数自然产生，并在数学、物理和工程学的许多领域发挥着重要作用。 该项目的重点是边界几何和分析问题，这些问题本质上是基本的。 特别是勒雷变换及相关问题的研究与偏微分方程研究的重要工具拉普拉斯变换有着密切的联系。 该项目借鉴了（并且希望能够贡献）各种独立感兴趣的几何理论。 最后，该项目将涉及与年轻研究人员的合作，从而有助于确保未来能够获得该领域的专业知识。