Workshop

I will motivate a quick introduction to pseudo-differential operators by discussing the so-called Calderon's problem, that is, whether knowledge of the Dirichlet-to-Neumann map (or the Neumann-to-Dirichlet map) at the boundary  determines the conductivity of a body in the interior. This problem has applications in Electric Impedance Tomography (EIT). I will in particular discuss uniqueness at the boundary, interior uniqueness using Complex Geometrical Optics (CGO solutions), and if time permits, layer stripping reconstruction algorithms in the case of isotropic conductivities.

We provide a general introduction to the field of inverse boundary problems for elliptic PDE, with the celebrated Calderon problem serving as a prototypical example. The emphasis of the talk is on the techniques based on Carleman estimates and their role in the construction of complex geometric optics solutions. We also survey some of the more recent developments, including partial data problems, inverse boundary problems on manifolds, as well as inverse boundary problems for non-linear equations.

This talk will be a (very gentle) introduction of real blow ups with applications in singular geometry. We will start with two motivating toy problems, including an eigenvalue resolution of self-adjoint matrices and a simple Fredholm operator on a manifold with boundary. Then if time permits we will mention some applications in singular uniformization problems, where the resolution or compactification via blow ups plays an important role in understanding the moduli spaces of constant curvature metrics with cusp and conical singularities.