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

将应用现代数学分析技术 微分方程的基本问题，复 分析，小波理论和动力系统在这个组 研究项目。 逆散射的工作继续努力研究更高的 维问题的一种新的方法，来自 多复变量方法：d杆法。 虽然 这种方法在两个领域取得了相当大的正式成功， 在更高的维度上取得了一些成功， 对问题进行了严格的分析。 特别是，工作将 在具有较大潜力的问题上或在一般 奇异散射数据 关于D-bar-Neumann问题的最新分析 复流形中的域已经表明， 热核的迹（对于热方程）具有小的时间 渐近展开式，其系数是局部 几何不变量，包括对数项。 非常 除了一些一般的不变量外， 结果关于他们的形式和程度的均匀性。 工作将 继续努力获取更具体的信息。 Cauchy积分有界性的新证明 利用特殊基开拓了其他奇异积分领域 在算子理论和数值分析方面的探索。 使用基本的Littlewood-Paley理论，可以将T（1） 定理的大卫和柯恩到一个有效的数值工具。 这个强有力的结果建立了一个奇异 操作符仅通过其对常量函数的作用。 本文将研究积分算子的小波表示 基作为矩阵，在对角线上迅速衰减。 这些 技术可以显示出产生相当大的减少 离散近似算子的复杂性。 一 这些思想对信号和图像的广泛适用性 作为这项研究的结果，预计将进行处理。 其他工作将集中在柯西之间的关系 积分，解析容量和可求积集。 指导的工作 在描述可求长曲线方面， 旅行推销员问题 进一步的应用将是 考虑相对于估计积分的问题， 保角映射的导数和确定 解析容量与Favard长度的关系 紧凑的平面集。 虽然它们不等同于 数量，很可能他们同意有限的集合， 一维Hausdorff测度 将努力解决 这个问题 //

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. //