Lesson plan
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 x of a manifold and simultaneously near a nonzero covector \xi at that point. This is finer than ordinary local analysis, which only distinguishes whether a function or distribution is regular near 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.
Article summary
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 x of a manifold and simultaneously near a nonzero covector \xi at that point. This is finer than ordinary local analysis, which only distinguishes whether a function or distribution is regular near 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.