The Krein-de Branges theory of canonical Hamiltonian systems connects spectral problems for second order differential operators with problems of complex analysis. In the first part of the mini-course I plan to discuss recent results on the inverse problems for canonical systems obtained jointly with N. Makarov.

An important subclass of canonical systems consists of Dirac systems. Scattering transform for Dirac systems is generally viewed as a non-linear version of the Fourier transform. Many classical results of Fourier analysis have their analogs in non-linear settings. In the second part of the course

I will talk about an extension of Carleson's theorem on the pointwise convergence of Fourier series to the non-linear Fourier transform.