Lie Groups

李群

• 批准号: 9705709
• 负责人: Joseph Wolf
• 金额: $ 8万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705709&HistoricalAwards=false
• 关键词: Lie; Groups

Abstract 9705709 Wolf Prof. Joseph Wolf will continue his research on Lie groups, harmonic analysis and representation, with some applications to complex analysis, Riemannian geometry, numerical analysis and control theory. This includes (1) his work on double fibration transforms and the structure of spaces of maximal compact subvarieties in flag domains, (2) his work on infinite dimensional Lie groups that are direct limits of finite dimensional groups, (3) his study of the Harish-Chandra--Schwartz space of a general semisimple Lie groups, (4) applications of his results with Milicic, Hecht and Schmid on equivalence of various constructions of representations of semisimple Lie groups, and (5) his line of research connecting semisimple representation theory with certain aspects of applied mathematics. Here (1) has already proved useful in automorphic cohomology, in indefinite metric quantization, and in geometric realization and understanding of certain singular representations of interest in physics. The new work proposed is to complete the precise description of the linear cycle space and apply the result to concrete analytic properties of the associated double fibration transforms. (2) is now at the point of understanding the direct limit analogues of the discrete series from both a geometric and an analytic viewpoint. (3) is a matter of completing a very beautiful topic in harmonic analysis. (4) is a question of direct reading of the character and growth properties of a representation from its defining basic datum. And (5) combines methods of control theory with the theory of Riemannian symmetric spaces. That line of research concentrates, at the moment, on the development of point placement techniques for observation of the heat equation on the sphere, along with various associated quadrature and approximation techniques. These research projects all are investigations of the role of symmetry in pure and applied mathematics. Symmetry simplifies calculations by eliminatin g variables or constraining their range, and the structure of the symmetry group and its representations imposes patterns that always lead to better understanding of complex situations. Thus indefinite metric quantization, which is tightly connected to project (1), has applications in particle physics. In fact it is essential for modern understanding of quantization of the photon and other particles that require a subquotient structure. The point placement and quadrature aspects of (5) are closely allied to the development of non- invasive methods of observation and methods of efficient computation. That sort of consideration comes up in many situations, such as medical applications of tomography and location of toxic spills. All five projects are closely related. In project (5), especially, methods developed for use in pure mathematics (specifically abstract harmonic analysis and differential geometry) turn out to have extremely interesting consequences for observability, controllability and practical computation.

摘要9705709狼 约瑟夫·沃尔夫教授将继续他的李群，调和分析和代表性的研究，与一些应用复杂的分析，黎曼几何，数值分析和控制理论。 这包括（1）他的工作双纤维化变换和结构的空间的最大紧凑subvarieties在旗域，（2）他的工作无限维李群是直接限制有限维群体，（3）他的研究的Harish-Chandra-Schwartz空间的一般半单李群，（4）应用他的结果与Milicic，Hecht和Schmid关于半单李群表示的各种构造的等价性，以及（5）他的 半单表示论与应用数学某些方面的联系。 在这里，（1）已经被证明在自守上同调、不定度量量子化、几何实现和理解物理学中感兴趣的某些奇异表示中是有用的。 提出的新工作是完成线性循环空间的精确描述，并将结果应用于相关的双纤维化变换的具体分析性质。 (2)现在的理解点的直接限制类似物的离散系列从几何和分析的观点。 (3)就是要完成一个关于谐波分析的很好的课题。 (4)是一个直接阅读的问题，从其定义的基本数据的字符和增长特性的表示。 （5）将控制论方法与黎曼对称空间理论相结合。 这条线的研究集中，目前，在发展的点放置技术的观察热方程的领域，沿着与各种相关的求积和近似技术。 这些研究项目都是关于对称性在纯数学和应用数学中的作用的调查。 对称性通过消除变量或限制它们的范围来简化计算，对称群的结构及其表示方法强加的模式总是能更好地理解复杂的情况。 因此，与方案（1）紧密相连的不定度规量子化在粒子物理学中有应用。 事实上，它对于现代理解光子和其他需要亚商结构的粒子的量子化至关重要。 （5）的点放置和求积方面与非侵入性观察方法和有效计算方法的发展密切相关。 这种考虑在许多情况下都会出现，比如断层扫描的医疗应用和有毒物质泄漏的位置。 这五个项目是密切相关的。 特别是在项目（5）中，为纯数学（特别是抽象的调和分析和微分几何）而开发的方法对可观测性、可控性和 实用计算