ABSTRACT-DMS 9701307 Semmes This project is largely concerned with notions of "complexity", i.e., concrete measurements of how complicated a given object is. Finite and combinatorial notions of complexity play a basic role in computer science, but one can also see analogous ideas at play in traditional areas of mathematics, in connection with analysis, geometry, and topology, for instance. In the present project we take much of our inspiration from the ideas and methods of real-variable harmonic analysis, but the structures that we consider will be predominantly geometric in nature. In asking what is "complexity" one should also ask what is "simplicity". Traditional mathematics offers a rich variety of answers to both questions, and these answers are often connected to various forms of computational efficiency. In this proposal we shall mostly be concerned with notions of geometric or combinatorial structure in which there are some nontrivial rules or patterns but perhaps also moderate amounts of bending, breaking, or other disturbances. In particular, we would like to develop new methods for identifying, measuring, and comparing forms of structure in ways that are flexible enough to accommodate substantial distortion.
