Work will be done on problems arising in the linear theory of partial differential equations. The research involves continued efforts to analyze boundary problems for wave equations. Current investigations focus on classes of operators, known as Airy operators, which describe reflections of waves by boundaries in the presence of grazing or gliding rays. This work has numerous applications to problems in scattering theory. Studies have focused on a symbol calculus for the Airy operators which arise when hyperbolic equations reduce to boundary problems via a parametrix construction. The operators are considerably different if grazing rays or gliding rays are present. Despite these differences a good operator calculus and even a useful symbol calculus is developing which will be sufficient to treat natural boundary problems in the presence of rays. A second area of emphasis in this project concerns various ways in which noncommutative harmonic analysis applies to problems in partial differential equations. Two particular directions of research are the spectral theory of Schroedinger operators associated with gauge fields and noncommutative microlocal analysis. Specific applications to chaotic behavior of classical dynamical systems are possible through knowledge of the spectral behavior of Schroedinger operators. Tools necessary to treat these problems include geometric invariant theory and group representations. Other work will concentrate on the application of noncommutative microlocal analysis in the study of hyperbolic equations with double characteristics, a more difficult subject than the hypoelliptic case. A calculus of Fourier integral operators will be developed using noncommutative harmonic analysis to extend recent work on pseudodifferential operators. The Heisenberg group has been central to much of the current research in this area. There is a wealth of other Lie groups which must be tried out for versions of noncommutative microlocal analysis. This line of research will undoubtedly be developed in the coming years. Numerous applications of this work to diffraction of waves, hyperbolic systems with viscosity and propagation of singularities can be expected.
