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

抽象Kaminker 涉及算子代数和指数理论的几何应用的几个问题将被工作。Ruelle代数在双曲动力学和算子代数之间的联系中发挥的主要作用将得到发展。 一个乘法指数理论将被研究，并将追求其与Hirzebruch比例原则和Selberg迹公式的联系。 此外，一个一般的指标理论eroupoids将制定和应用于问题的量化。 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，当使用数学建模现象时，测量的微小变化可以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良好的一部分。 它们利用了一个简单的想法，即一个人做两个操作的顺序 影响结果。 从数学上发展这个概念会导致对矩阵或“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，并且在提供一个完整的可能性列表方面有很长的路要走。 有趣的是，这些数学思想最初不会被期望在这种情况下有用，只有在它们从纯数学的角度进行了深入的发展之后， 以这种方式应用它们。 这项工作介绍了使用“小波”，这已经开发的实际工作中发送电视图像在最具成本效益的方式。 令人惊讶的是，它们也可以用于“混沌”的研究，我们希望这种关系将导致在wavelets的一般理论的进展。

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.