Lie Groups

Prof. Joseph Wolf将继续他在李群、谐波分析和表示方面的研究，并在复分析、黎曼几何、数值分析和控制理论方面有一定的应用。这包括(1)他关于双纤维变换和最大紧子变在flag域上的空间结构的工作，(2)他关于有限维群的直接极限的无限维李群的工作，(3)他对一般半单李群的Harish-Chandra—Schwartz空间的研究，(4)他的成果与Milicic， Hecht和Schmid在半单李群表示的各种构造的等价性上的应用，(5)他将半简单表示理论与应用数学的某些方面联系起来的研究方向。这里(1)已经被证明在自同态上同调，不定度量量化，以及物理中某些奇异表示的几何实现和理解中是有用的。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双纤维化变换的具体解析性质。(2)现在正处于从几何和解析两种观点来理解离散级数的直接极限类似物的阶段。(3)是在谐波分析中完成一个非常漂亮的题目。(4)是一个从定义的基本数据中直接解读表征的特征和生长特性的问题。(5)将控制论方法与黎曼对称空间理论相结合。目前，这方面的研究主要集中在发展用于观测球面热方程的点放置技术，以及各种相关的正交和近似技术。这些研究项目都是研究对称在纯数学和应用数学中的作用。对称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所施加的模式总是导致对复杂情况的更好理解。因此，与项目(1)密切相关的不定度量量子化在粒子物理中具有应用。事实上，它对于现代理解光子和其他需要子商结构的粒子的量子化是必不可少的。(5)的点放置和正交方面与非侵入性观测方法和高效计算方法的发展密切相关。这种考虑在很多情况下都会出现，比如断层扫描的医学应用和有毒物质泄漏的定位。这五个项目都是密切相关的。特别是在项目(5)中，为纯数学（特别是抽象谐波分析和微分几何）开发的方法在可观察性、可控性和实际计算方面产生了非常有趣的结果。

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.