Mathematical Sciences: Analysis: Partial Differential Equations, Fourier & Complex Analysis and Group Project in Mathematical Wavelets

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Modern techniques of mathematical analysis will be applied to fundamental problems in differential equations, complex analysis, wavelet theory, and dynamical systems in this group research project. Work on inverse scattering continues efforts to study higher dimensional problems by means of a new approach derived from methods of several complex variables: the d-bar method. Although this approach has had considerable formal success in two space dimensions and some success in higher dimensions, very few problems have been analyzed rigorously. In particular, work will be done on problems with large potentials or with general singular scattering data. Recent analyses of the d-bar-Neumann problem associated to domains in a complex manifold have shown that the corresponding trace of the heat kernel (for the heat equation) has a small time asymptotic, expansion whose coefficients are integrals of local geometric invariants, including the logarithmic terms. Very little is known about these invariants beyond some general results about their form and degree of homogeneity. Work will continue in efforts to elicit more concrete information. New proofs of the boundedness of the Cauchy integral and other singular integral using special bases has opened up areas of exploration both in operator theory and numerical analysis. Using basic Littlewood-Paley theory, one can translate the T(1) theorem of David and Journe into an effective numerical tool. This powerful result establishes the boundedness of a singular operator solely through its action on the constant functions. Work will be done in representing integral operators in wavelet bases as matrices which decay rapidly off the diagonal. These techniques can be shown to produce considerable reduction in complexity in the discrete approximation to the operators. A wide range of applicability of these ideas to signal and image processing is expected as a consequence of this research. Other work will focus on relationships between the Cauchy integral, analytic capacity and rectifiable sets. Work directed at characterizing rectifiable curves has led to new insights into the traveling salesman problem. Further applications will be considered relative to the problem of estimating the integral of the derivative of a conformal mapping and on determining the relationship between analytic capacity and Favard length of compact planar sets. Although they are not equivalent quantities, it is likely that they do agree on sets of finite one-dimensional Hausdorff measure. Work will be done to resolve this issue. //