Geometric Harmonic Analysis

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Harmonic analysis on sets is an area that has been under intensive developmentsince the 1980's. The first results concerned the behavior of singular integrals on sets in terms of the geometry of the sets. The model operator, and impetus to the theory, was the Cauchy integral with the model geometric setting being a Lipschitz curve. This theory has witnessed an explosive growth in terms of understanding the relation between L2 estimates in terms of the multiscale geometry of the underlying set. One of the realizations of the 1990's was that there is an L2 theory of geometry in terms of so-called Beta numbers, and that there is a "dictionary" that translates theorems on geometry of sets into theorems on wavelets, and vice versa. While the theory of Beta numbers also gave a good understanding of the multiscale structure of e.g. a data set, it only provides a certain framework for the understanding of the geometry, and does not encompass a theory analogous to Fourier series or the study of heat flow. It is the development of such a theory, along with its relations to the already understood multiscale aspects, that is now required to provide a deeper understanding of harmonic analysis on sets. For example the problem of building local coordinates that capture most of the statistical behavior of (mostly) lower dimensional subsets has not been well developed mathematically, though many proposed methods have been studied. We propose to relate the top down methods (e.g. Beta Numbers and corresponding geometry) to bottom up methods of diffusion geometries. This new method of studying harmonic analysis on sets is based on the use of certain eigenfunctions related to the set. These eigenfunctions, coming from naturally defined matrices, allow the introduction of "local coordinates" on the set by picking the n largest eigenvalues, and using the corresponding n eigenfunctions as coordinates. The method proposed has a close relation to the theory of so-called prolate functions, as the resulting eigenfunctions have similar properties. This is in sharp contrast to the method of Coifman, Jones, and Semmes for defining Haar type L2 frames on sets resembling Lipschitz curves. The method of the proposal gives different functions with which one can naturally define local coordinates and study (approximate and correctly defined) heat flow on sets. Professors Jones and Coifman propose to study these methods and develop a theory that can be combined with previous results to relate top down behavior to bottom up behavior. This is done from the point of view of computational efficiency and the development of fast algorithms. They also propose to study the various geometrical descriptions to provide new methods of attacking older problems in harmonic analysis. In doing so, they aim to break new ground and broaden the applicability of other earlier methods.