Microlocal analysis

Microlocal analysis is a branch of mathematical analysis that studies functions, generalized functions and partial differential equations by localizing them both in position and in frequency. The main idea is to describe a singularity not only by the point at which it occurs, but also by the cotangent direction in which it occurs. The term microlocal refers to localization at a small scale in phase space: locally near a point ${\displaystyle x}$ of a manifold and simultaneously near a nonzero covector ${\displaystyle \xi }$ at that point. This is finer than ordinary local analysis, which only distinguishes whether a function or distribution is regular near ${\displaystyle x}$. The information of position and covector in which a singularity occurs is encoded by the wave front set of a distribution, a conic subset of the cotangent bundle with the zero section removed.

Microlocal analysis was developed from the 1950s through the 1970s in connection with linear partial differential equations, Fourier transform methods, hyperfunctions and pseudo-differential operators. It is concerned with elliptic regularity, propagation of singularities, Fourier integral operators, geometric optics, scattering theory, spectral theory, semiclassical analysis and inverse problems.