Mathematical Sciences: Harmonic Analysis, Representation Theory and Partial Differential Equations

数学科学：调和分析、表示论和偏微分方程

• 批准号: 8907766
• 负责人: Frederick Greenleaf
• 金额: $ 8.12万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8907766&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Analysis; Representation

Professor Greenleaf will apply the theory of group representations to problems in mathematical analysis on nilpotent Lie groups and their homogeneous spaces. Particular emphasis will be placed on studying the invariant differential operators on homogeneous spaces, describing algebras of such operators in terms of Kirillov's orbit picture, and developing solvability criteria for large classes of differential operators on these spaces. Lie groups arise naturally in mathematics and physics as groups of symmetries. An interesting example that lies readily at hand consists of all rotations of a sphere, made into a group by applying rotations successively. Understanding this group is important in any situation in which spherical symmetry is present. The simplest noncommutative example of a nilpotent Lie group (the class of most immediate interest here) is the Heisenberg group of upper triangular three-by-three real matrices with ones on the diagonal. The Heisenberg group plays a fundamental role in the quantum mechanical treatment of the harmonic oscillator, as the symmetry group of a certain differential equation. The strong relationship between differential equations and the representation theory of the underlying symmetry group is at the heart of Professor Greenleaf's project.

格林立夫教授将运用群论 幂零数学分析中问题的表示 李群及其齐性空间。将特别强调 上的不变微分算子 齐性空间，描述这种算子的代数， Kirillov的轨道图像，并发展可解性 这些上的大类微分算子的准则 空间. 李群在数学和物理中自然出现， 对称群一个有趣的例子， 由一个球体的所有旋转组成， 通过连续地应用旋转。了解这个群体是 重要的是在任何情况下，球对称是 礼物幂零李的最简单的非交换例子 组（这里最直接感兴趣的类）是 上三角三乘三真实的矩阵海森堡群 对角线上有一个。 海森堡群扮演着一个 在量子力学治疗的基本作用 谐振子，作为某个 微分方程之间的密切关系 微分方程的表示理论 基础对称群是教授 格林立夫的项目