Topics in Fourier Analysis

提议：DMS-9970042首席研究员：Andreas Seeger摘要：拟开展的研究将涉及傅里叶分析中的一些主题，涉及奇异积分算子、具有退化正则关系的傅里叶积分算子、Radon变换、曲线和曲面上的平均以及相关的极大函数、波传播的正则性、非紧流形上的谱乘子以及这些主题之间的多种相互关系。P.I.建议进一步开发和完善其中的一些工具。这应该会导致对积分算子的更好的定性理解，这些积分算子出现在从数学物理的偏微分方程的研究中，以及在积分几何和层析成像的研究中。改善这种理解的潜在具体影响是巨大的。例如，层析成像不仅为医学成像中经常使用的重建方法提供了数学基础，还为解释地震数据提供了关键，赋予了它一种不能夸大的“现实世界”意义。

Proposal: DMS-9970042Principal Investigator: Andreas SeegerAbstract: The proposed research will be concerned with a number of topics in Fourier analysis related to singular integral operators, Fourier integral operators with degenerate canonical relations, Radon transforms, averages over curves and surfaces and associated maximal functions, regularity of wave propagation, spectral multipliers on noncompact manifolds, and the multitude of interrelations between these topics.The subject of Fourier analysis has provided powerful tools for various areas of mathematics and its applications. The P.I. proposes to develop further and refine some of these tools. This should lead to a better qualitative understanding of integral operators that arise in the study of partial differential equations from mathematical physics, as well as in the study of integral geometry and tomography. The potential concrete implications of improving this understanding are enormous. Tomography, for instance, not only provides the mathematical foundation for the reconstruction methods that are employed constantly in medical imaging but also furnishes the key to interpreting seismic data, lending it a "real world" significance that cannot be overstated.