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

摘要卡明克将研究涉及算子代数和指数理论几何应用的几个问题。Ruelle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。我们将研究乘性指数理论，并探讨其与赫兹布鲁赫比例原理和塞尔伯格迹公式的联系。此外，还将给出广义指数理论，并将其应用于量子化问题。这个项目涉及使用数学方法来研究在描述物理世界时发生的情况。一般来说，当使用数学对现象进行建模时，测量结果的微小变化可能会从根本上改变理论的预测。这是“混乱”概念的一个方面。对可能发生的“混乱”类型进行分类是本工作的目标之一。所使用的方法最早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。他们利用一个简单的想法，即一个人进行两次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。这进而导致将这些“广义数”与混沌系统相关联。此外，它们是可计算的，并在很大程度上提供了一份完整的可能性清单。有趣的是，这些数学思想最初并不被期望在这种情况下有用，只有在它们从纯数学角度密集发展后，才产生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，它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，它们也可以用来研究“混沌”，我们期待这种关系将导致在波波的一般理论的进步。

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.