Application of Harmonic Analysis and Geometry of Sets

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Proposal: DMS-9971311Principal Investigators: Peter W. Jones, Ronald R. CoifmanAbstract: Professors Coifman and Jones will continue their work on the analysis and geometry of sets in Euclidean space. The research is centered on methods to classify sets or functions and to provide efficient descriptions of these objects. On the level of sets, this involves descriptions of, for example, the Julia set associated with a rational mapping or the limit set of a Kleinian group in terms of its Hausdorff dimension or related geometric concepts. Another example relates to potential theoretic aspects of sets, including algorithms that allow for fast computations. For functions, the relevant objective is to find short descriptions with prescribed error. A common aspect of the analysis of both sets and functions is the search for methods of reducing dimensionality (with prescribed error) so as to bring the problem into a regime where more classical methods apply.The problems on which Jones and Coifman intend to work are those that now confront mathematicians who are forced to deal with massive quantities of data. While the modern computer has been very useful in treating many computational problems, severe limitations occur when the amount of data becomes too large. Instead of treating the full set of data, one therefore seeks to reduce significantly the amount of information being handled, at the same time ensuring that only small errors occur in the process. After such a reduction has been made, more classical methods can then be used to analyse the reduced data. Harmonic analysis is a field that has been developed over the past century to accomplish precisely this kind of reduction, albeit for problems in the realm of pure mathematics. The projected research centers on exploiting these older methods and developing new ones in order to acquire efficient tools for treating a wide array of concrete data sets. One example of how useful this technique might be in a practical, everyday situation arises in the telephone industry, where it would be highly desirable to send only a small percentage of the sounds produced by one person, yet still have the message understood by the person listening at the other end of the line. The development of an effective method for compressing data in this instance would allow a large increase in the number of calls that could be sent on a single telephone line.