Mathematical Sciences: K-Theory, C*-Algebras & Index Theory of Elliptic Operators

本文将研究几个涉及算子代数和指标理论的几何应用的问题。吕尔代数在双曲动力学和算子代数之间的联系中所起的主要作用将得到发展。我们将研究一个乘法指标理论，并探讨它与Hirzebruch比例原理和Selberg迹公式的联系。此外，本文还提出了一种泛化似双曲面的指标理论，并将其应用于量化问题。这个项目涉及使用数学方法来研究描述物理世界时发生的情况。一般来说，当用数学建模现象时，测量值的微小变化会从根本上改变理论的预测。这是“混沌”概念的一个方面。对可能发生的“混沌”的种类进行分类是本工作的目标之一。所使用的方法首先出现在量子力学的早期阶段，并已成为数学中发展得很好的一部分。它们利用了一个简单的概念，即一个人执行两个操作的顺序会影响结果。从数学上发展这一概念导致了对矩阵或“广义数”的研究。这反过来又导致将这些“广义数”与混沌系统联系起来。此外，它们是可计算的，并且在提供可能性的完整列表方面有很长的路要走。有趣的是，这些数学思想最初并没有被期望在这种情况下有用，只是在它们从纯数学的角度深入发展之后，才出现了以这种方式应用它们的想法。这项工作介绍了“小波”的使用，这是为以最具成本效益的方式发送电视图像的实际工作而开发的。令人惊讶的是，它们也可以用于“混沌”的研究，我们期望这种关系将导致广义理论的进步。

Abstract Kaminker Several problems involving geometric applications of operator algebras and index theory will be worked on. The major role that Ruelle algebras play in the connection between hyperbolic dynamics and operator algebras will be developed. A multiplicative index theory will be studied and its connections with the Hirzebruch proportionality principle and the Selberg trace formula will be pursued. Also, a general index theory for eroupoids will be worked out and applied to problems of quantization. This project deals with using mathematical methods to study situations that occur when describing the physical world. In general, when modeling phenomena using mathematics, slight changes in measurements can radically change the predictions of the theory. This is one aspect of the notion of ``chaos''. The classification of the kinds of ``chaos'' which can occur is one of the goals of the present work. The methods used arose first in the early stages of quantum mechanics and have become a well developed part of mathematics. They take advantage of the simple idea that the order in which one does two operations effects the outcome. Developing that notion mathematically leads to the study of matrices, or ``generalized numbers''. This, in turn, lead to associating these ``generalized numbers'' to chaotic systems. Moreover, they are computable and go a long way to providing a complete list of the possibilities. It is interesting that these mathematical ideas originally would not have been expected to be useful in this setting, and it was only after their intensive development from a purely mathematical point of view that the idea of applying them in this way occurred. This work introduces the use of ``wavelets'', which had been developed for practical work in sending televised images in the most cost effective way. It is very surprising that they can also be used in the study of ``chaos'' and we expect that this relation will lead to progress in the general theory of wa velets.