Microlocal Analysis (MA), which is roughly speaking local analysis in phase space, arose as a natural development of the methods of geometrical optics pioneered by Hamilton and others. Hörmander introduced in the early 70's the concept of wave front set of a distribution and developed a calculus of Fourier integral operators (FIO). These fundamental developments have led to numerous results in the study of singularities of solutions of partial differential equations (PDEs). They have also had important applications in other fields.
In this one quarter course, we will develop the foundations of MA: The wave front set of a distribution, applications of the wave front set to define restriction and multiplication of distributions, and the theory of pseudodifferential operators. We will prove Hörmander's propagation of singularities theorem for hyperbolic equations. MA has been used in inverse problems to determin the singularities of medium parameters. In this course, we will apply MA to to reflection seismology, which arises in oil exploration. In this inverse method, one attempts to find the index of refraction (sound speed) of the subsurface of the earth by measuring at the surface the reflections of sound waves sent into the earth. The techniques used in reflection seismology are relevant to imaging using ultrasound.
Prerequisites. Knowledge of the theory of distributions and the Fourier transform as in the third quarter of the Linear Analysis course (Math 556). No previous knowledge of PDE is required but it is suggested that students without this prerequisite take Math 557 concurrently.
Recommended Text. Microlocal Analysis for Differential
Operators, by A. Grigis and J. Sjöstrand.