Mathematical Sciences: Partial Differential Equations & Analysis

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This award supports the work of three researchers investigating several areas of broad mathematical interest and importance. Some work will be conducted jointly while other will be addressed by single investigators. One can divide the studies into three main themes: nonlinear partial differential equations, wavelet analysis and complex function theory. In the first, efforts will be made to explore how the one-dimensional quantum version of the inverse scattering methods which replace the classical action-angle variables can be applied to understanding the corresponding higher dimensional systems. Work is also progressing on the investigation of inverse geophysical problems involving surface wave phenomena. Progress on model scalar cases suggests that analysis of the full system of equations can be obtained. Studies of adapted wave form analysis, a collection of FFT- like adapted transform algorithms, continues to discover and refine new orthonormal bases which decompose functions and operators into almost diagonal form. The interplay between these theoretical advances and the application to numerical signal processing has led to highly successful decomposition techniques using entropy based stopping time searches. The applications lead, in turn, to a host of new questions in harmonic analysis. One of these will be that of creating a new concept of theoretical dimension of a function designed to measure the number of parameters necessary to describe the function with wave forms taken from a given library. Studies on the fine structure of harmonic measure also continue. Here the goals are to find the asymptotic behavior of harmonic measure in a variety of settings. Work will also be done estimating the length of level lines of the Green's function on bounded simply connected domains. It is believed that level line length varies with the fourth root of the value of constancy. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions. Harmonic analysis focuses on the decomposition of functions into component parts which best reflect their oscillatory characteristics, while complex function theory seeks to describe the geometric and analytic properties of differentiable functions of a complex variable and their generalizations to higher dimensions.