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.
