Mathematical Sciences: K Theory and Harmonic Analysis

该项目包括三个与谐波分析和k理论相关的问题。第一个问题是研究局域上一般线性群的Kasparov定义的表示环的结构。我们发展了一种从极大抛物的表示环到满群的表示环的“归纳法”。这个问题的主要问题是一个解析问题，涉及在g -表示之间构造一个合适的算子，在卡斯帕罗夫的意义上，它几乎与g -作用相交换。第二个问题是关于紧群在横椭圆微分算子核和核中的不可约表示的多重性的显式计算。第三个问题是研究一类代数可解李群的拟正则表示的显式分解。主要的重点是将Anoussis的相当新的结果与Selberg迹公式结合应用于可解李群的轨道积分上。这个项目的总体重点是利用对称性来研究几个复杂的系统。对称的概念遍及各个科学领域，虽然很难给出精确的数学定义（但这样的定义确实存在），但我们每天都能看到对称。正方形的“正”或圆形的“圆”是对称的两个例子。对称性之所以如此重要，是因为只有系统具有内在的对称性，科学家才能进行分析。问题的对称性为我们提供了开发有效计算方法的有力工具。举个例子，这个提议所涉及的问题之一是找到有效的方法来非常明确地计算某些微分算子的“G-指标”：我们正在计算对称性存在的问题的某些类型的解。这导致了一种费恩玛n图的发展，它结合了微分算子和底层的对称性。费曼图技术已被证明在学术科学和工业应用科学中都是非常重要的。例如，底特律外的通用汽车研究实验室有专门研究费曼图技术的数学家，这种技术用于解决非常实际的问题背后的数学。

DMS-9623285 Jeffrey S. Fox University of Colorado The project consists of three problems related to harmonic analysis and K-theory. The first problem is concerned with studying the structure of the representation ring defined by Kasparov for the general linear group over a local field. We are developing a method of "induction" from the representation ring of a maximal parabolic to the representation ring of the full group. The main issue in this problem is an analytic one, involving the construction of an appropriate operator between G-representations that almost commutes , in the sense of Kasparov, with the G-action. The second problem concerns the explicit computation of multiplicity's of irreducible representations of compact groups in the kernel and cokernel of transversally elliptic differential operators. The third problem is to study the explicit decomposition of the quasi-regular representation for an algebraic solvable Lie group. The main focus is to apply the fairly new results of Anoussis on orbital integrals for solvable Lie groups in conjunction with the Selberg trace formula. The overall focus of this project is to use symmetry to study several complicated systems. The notion of symmetry pervades the sciences and while it is difficult to give a precise mathematical definition ( but such a definition does exist), we see symmetry every day. The "squareness" of a square or the "roundness" of a circle are two examples of symmetry. The reason symmetry is such an important idea is that it is systems that possess intrinsic symmetry that scientists can analyze. The symmetry of a problem provides us with powerful tools for developing effective means of computation. As an example, one of the problems that this proposal is concerned with is finding effective means of calculating, very explicitly, the "G- indices" of certain differential operators: we are counting certain types of solutions of a problem where symmetry is present. This leads to the development of a type of Feynma n diagram that incorporates both the differential operator as well as the underlying symmetries. Feynman diagram techniques have proved to be very important both in academic science as well as applied science in industry. For example, the General Motors Research lab outside of Detroit has mathematicians that specialize in Feynman diagram techniques, which are used to solve the mathematics behind very practical problems.