DMS-9423746 PI: R. Beals, R. R. Coifman, P. W. Jones Abstract The principal investigators will continuing their work in a number of related areas of analysis: harmonic analysis, nonlinear Fourier analysis, and partial differential equations, with applications to complex analysis, physical problems, and numerical work. Beals and collaborators will make more use their methods from Hamiltonian mechanics to investigate fundamental solutions for subelliptic laplacians. Beals will also continue joint work on the quantized three-wave interaction and on geophysical inverse problems. Coifman and Meyer will continue to investigate nonlinear Fourier analysis and applications to homogenization and nonlinear dependence. Coifman will also continue to develop adapted wave form analysis to obtain geometric and numerical understanding of oscillatory operators. Jones will continue his work on the fine properties of harmonic measure and level lines of Green's function, as well as work on singular integrals and d-bar problems related to corona problems and analytic capacity. Jones also plans further investigations in quasiconformal mappings and problems relating Sobolev spaces and geometry of sets. Analysis is the area of mathematics that has grown directly from the invention of calculus and differential equations. It provides a framework for attacking problems that range from the purely mathematical and geometrical to the most concrete scientific and technical questions. In turn, these problems have provided the motivation for the continued refinement of the subject and, not incidentally, its continued utility. Thus there is no precise dividing line between pure theory and potential or actual application. For example, in little more than a decade sophisticated techniques of pure harmonic analysis have emerged in practice as wavelet analysis and have enormously extended the possibilities for analysis and storage of digital data. The sub jects to be investigated in this proposal reflect the full range of analysis as well as the actual and potential interplay between "pure" and "applied." Some have a history of a century or more and come from classical analysis and geometry, others have a history of decades and are motivated by physics and geophysics, others have arisen in the last decade and are connected with signal processing and the concrete use of computers in science and technology.
