||In this talk, we will address on several types of trace formulas in scattering theory and some possible applications in inverse problems. The asymptotic behavior of the PDE solution at infinity or on the boundary justifies the existence of scattering matrix.
Inverse scattering problem asks for the information on the scatterer/perturbation given
partial/complete information on the scattering matrix. Actually, the singular structure of
this scattering matrix has a deep relation with the geometric information of the scattering scatter. This relation is best represented via geometric trace formulas.
We will start with the scattering resonance theory and follow up by wave/heat invariant
theory. The geometric trace formula would be any functional correspondence between the resonances and the wave/heat propagator. As the finale, we would consider some progress on the application of these trace formulas.