Analysis on Singular Spaces

奇异空间分析

• 批准号: 9971981
• 负责人: Victor Nistor
• 金额: $ 7.74万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971981&HistoricalAwards=false
• 关键词: Analysis; Singular; Spaces

AbstractNistorThe project is devoted to application of certain areas of Analysis, especially Index theory, K-theory, and Hochschild and cyclic homology, to a broad range of problems that have implications in several other areas of mathematics and physics. A common issue in these problems is to identify the relevant algebras that model specific situation of interest. Sometimes these algebras already exist, sometimes they have to be constructed. These algebras will then be studied with the indicated homological tools, and the results will be interpreted as providing information on the specific situations that are studied. Some of the applications include Index theory on singular spaces, the spectral and Index theory of Dirac operators coupled with vector potentials, and determinants of elliptic operators. A different but closely related type of application is a cohomological study of the representation space of a p-adic group.The index of an operator in its simplest form is a number. Many quantities from mathematics and even from physics and chemistry can be identified with the index of a suitable operator - for example, the number of electrons occupying a certain energy level in an atom is the index of a suitable operator. The Dirac equation (or operator) is one of the fundamental equations in physics; the number of solution of the Dirac equation is very closely related and can often be identified with the index of that operator.

该项目致力于将分析的某些领域，特别是指数理论、k理论、Hochschild和循环同调应用于广泛的问题，这些问题对数学和物理的其他几个领域有影响。在这些问题中，一个共同的问题是确定相关的代数来模拟感兴趣的特定情况。有时这些代数已经存在，有时它们必须被构造。然后，这些代数将用指示的同调工具进行研究，结果将被解释为提供所研究的特定情况的信息。一些应用包括奇异空间上的指标理论，Dirac算子与向量势耦合的谱和指标理论，以及椭圆算子的行列式。一个不同但密切相关的应用类型是对p进群的表示空间的上同调研究。操作符的最简单形式的索引是一个数字。数学，甚至物理和化学中的许多量都可以用一个合适的算符的指标来识别——例如，在一个原子中占据某一能级的电子数就是一个合适的算符的指标。狄拉克方程（或算子）是物理学中的基本方程之一；狄拉克方程的解的个数是密切相关的，通常可以用该算子的指标来识别。