Research in Geometric Analysis

几何分析研究

• 批准号: 9988854
• 负责人: Steven Krantz
• 金额: $ 14.5万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988854&HistoricalAwards=false
• 关键词: Research; Geometric; Analysis

ABSTRACT:Krantz proposes to study the groups of holomorphic self-mappings(automorphism groups) of domains in complex space using techniques of realand complex analysis, differential geometry, Lie theory, and partialdifferential equations. The goal is to identify invariants that will helpto classify domains up to biholomorphic equivalence. In anotherdirection, Krantz will study real-variable Hardy spaces on domains,pursuant to earlier collaborative work with Chang and Stein. Finally,Krantz will collaborate with Marco Peloso in his continuing study of theinhomogeneous Cauchy-Riemann equations in the Sobolev topology.Applications to mapping theory and the study of the worm domain are beingdeveloped.Professor Krantz will study aspects of complex analysis related tosymmetry. The notion of symmetry is measured by means of mappings ofdomains. The set of such mappings of a region in space to itself hasan algebraic structure which is called a ``group''. The properties ofthis group reveal geometric structure of the region under study andvice versa. This idea of attaching an algebraic invariant to ageometric object is one of the big thrusts in twentieth-centurymathematics, and uncovers new depth and texture in the subject.Krantz's study of these ``automorphism groups'' complements hiscontinuing study of certain partial differential equations on thesespatial regions. The partial differential equations can be used toconstruct mappings, and to calculate their regularity properties. Asan ancillary to these two projects, Krantz will study certain spacesof functions and mappings which are called Hardy spaces, and whoseexistence was inspired by the mapping questions. The geometric studyof complex function theory in several variables has applications incosmology (thanks to work of Penrose) and in geophysics.

摘要：Krantz提出利用实分析、复分析、微分几何、李理论和偏微分方程的方法来研究复空间中域的全纯自映射群（自同构群）。 我们的目标是确定不变量，这将有助于分类域的双全纯等价。 在另一个方向，“将军”将研究域上的实变量哈代空间，根据以前的合作工作与张和斯坦。最后，Krantz将与Marco Peloso合作，继续研究Sobolev拓扑中的非齐次Cauchy-Riemann方程。Krantz教授将研究与对称性相关的复分析方面。 对称性的概念是通过域的映射来度量的。 空间中一个区域到它自身的这种映射的集合具有一个代数结构，称为“群”。 这个群的性质揭示了研究区域的几何结构，反之亦然。 这种将代数不变量附加到几何对象上的思想是二十世纪数学的一大突破，它揭示了这一学科的新的深度和结构。Krantz对这些“自同构群”的研究补充了他对空间区域上某些偏微分方程的持续研究。 这些偏微分方程可以用来构造映射，计算映射的正则性。 作为这两个项目的辅助，Krantz将研究某些被称为哈代空间的函数和映射空间，这些空间的存在受到映射问题的启发。 几何研究复变函数理论在几个变量有应用在宇宙学（由于彭罗斯的工作）和天体物理学。