# Spectral Theory Seminar

### Fourier Integral Operators: the construction of parametrices for strictly hyperbolic Cauchy problems

After having seen how to construct approximate fundamental solutions for elliptic linear PDEs, in the first lecture of this series, we will now review the construction of similar approximate solutions, this time for the Cauchy problem in hyperbolic problems, using Fourier Integral Operators. These operators can be seen as a generalization of pseudo-differential operators, and bring into the picture a strong symplectic geometry component, related to the hamiltonian evolution in phase-space: the cotangent bundle of space points and Fourier frequencies.