MATH 8366 Advanced Microlocal Analysis
This course provides an introduction to methods and applications of Microlocal Analysis. Microlocal Analysis is an instrument that was developed to apply Fourier transform to solve partial differential equations and carry out a qualitative analysis of the solutions. Topics include: basic concepts and computational technique of distributions, generalized functions, the local theory of distributions, the singular support of distributions, the convolutions of distributions, the structure of distributions, approximations by test functions, Schwartz space; Fourier transforms of test functions and distributions, Paley-Wiener theorem, Schwartz kernel theorem, Sobolev spaces, symbols, pseudo-differential operators (PDOs), the kernel of pseudo-local operators, PDOs and Sobolev spaces, amplitude functions and PDOs, transpose and adjoint of PDO operators, proper PDOs, product of PDOs, asymptotic series and expansions, product formula for PDOs, symbols of transposed and adjoint operators, symbol of composition and commutator of PDOs, elliptic operators, wave front set of distributions, propagation of singularities, Fourier integral operators, applications to elliptic and parabolic partial differential equations on manifolds, and propagation of singularities for hyperbolic partial differential equations.
Prerequisite
Consent of instructor.
Offered
As scheduled