Application of Harmonic Analysis and Geometry of Sets

建议：DMS-9971311主要研究人员：彼得·W·琼斯、罗纳德·R·科夫曼摘要：科夫曼和琼斯教授将继续他们在欧几里得空间中集合的分析和几何方面的工作。研究的中心是对集合或函数进行分类并提供对这些对象的有效描述的方法。在集合的水平上，这涉及到，例如，根据Klein群的Hausdorff维度或相关的几何概念来描述与有理映射相关联的Julia集或Kleian群的极限集。另一个例子涉及集合的潜在理论方面，包括允许快速计算的算法。对于函数，相关的目标是找到具有规定错误的简短描述。集合和函数分析的一个共同方面是寻找降维方法(带有规定的误差)，以便将问题带入更经典方法适用的范围。琼斯和科夫曼打算解决的问题是现在面临的数学家们被迫处理大量数据的问题。虽然现代计算机在处理许多计算问题时非常有用，但当数据量变得太大时，就会出现严重的限制。因此，人们寻求显著减少处理的信息量，同时确保在处理过程中只发生小错误，而不是处理全部数据。在进行了这样的简化之后，可以使用更经典的方法来分析简化的数据。调和分析是在过去的一个世纪里发展起来的一个领域，目的是精确地完成这种归约，尽管是针对纯数学领域的问题。计划中的研究集中在利用这些旧方法和开发新方法，以获得处理大量具体数据集的有效工具。这种技术在实际的日常情况中可能是多么有用的一个例子出现在电话行业，在该行业中，非常希望只发送一个人产生的声音的一小部分，但仍然让在线路另一端收听的人理解该消息。在这种情况下，开发一种用于压缩数据的有效方法将允许在一条电话线上发送的呼叫数量大幅增加。

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.