Mathematical Sciences: Group Representations and Partial Differential Equations

数学科学：群表示和偏微分方程

• 批准号: 9208180
• 负责人: Frederick Greenleaf
• 金额: $ 6.27万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9208180&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Group; Representations; Partial

Greenleaf will apply the theory of group representations to problems in mathematical analysis on nilpotent Lie groups and their quotient spaces. These investigations will seek a description of the algebra of invariant differential operators in terms of the coadjoint orbits in the dual of the Lie algebra of the group. Initial efforts will address a basic conjecture: The invariant differential operators on the quotient form a commutative algebra precisely when the quasi-regular representation has finite multiplicities. Success in these investigations will lead to understanding of solvability, and existence of fundamental solutions, for invariant differential operators. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.

Greenleaf将把群表示理论应用于 幂零李群的数学分析问题， 商空间。 这些调查将寻求 不变微分算子代数的描述 的李代数的对偶中的余伴随轨道的项 本集团 最初的努力将解决一个基本的猜想： 商形式A上的不变微分算子 交换代数恰当拟正则 表示具有有限的多重性。 成功在这些 调查将导致对可解决性的理解， 基本解的存在性，不变微分 运营商 李群理论，以挪威人的荣誉命名 数学家Sophus Lie，一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性， 李群的表示理论对 数学本身，特别是在分析和数字方面， 理论和理论物理学，特别是量子 力学和基本粒子物理学。