MATH 8366 Micro-Local Analysis
Topics include: basic concepts and computational technique of distributions generalized functions, 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, transposed and adjoint to 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, propagation of singularities for hyperbolic partial differential equations.
Prerequisite
Departmental approval
Offered
As scheduled