Subscribe to Events

Inverse boundary problems aim to determine the internal properties of a medium from measurements made at its boundary. Mathematically, this amounts to recovering the coefficients of a partial differential equation inside a domain from information about solutions on the boundary. Such problems arise in numerous applications, from medical imaging to exploration geophysics. A celebrated example is the Calderon problem, which asks whether one can recover the electrical conductivity of a medium from voltage and current measurements on the boundary. This foundational question has inspired decades of research in the field.

In this talk, we will offer an introduction to the field of inverse boundary problems, beginning with the Calderon problem and then turning to recent advances in inverse problems for both nonlinear and nonlocal elliptic operators. Perhaps surprisingly, the presence of nonlinearity or nonlocality can actually be beneficial, allowing one to solve inverse problems that remain open in their linear or local counterparts. In particular, we present a complete solution to the fractional anisotropic Calderon problem on smooth compact Riemannian manifolds without boundary: the source-to-solution map of the fractional Laplacian, known on an arbitrarily small open subset, determines the manifold up to isometry. This provides a nonlocal analog of the classical anisotropic Calderon problem, which remains one of the central open questions in dimensions three and higher.