Title: Inverse Problems and Geometry
Abstract: Geometric notions have played a significant role in understanding inverse problems for partial differential equations (PDE) with applications in mathematical physics, geophysics and medical imaging. Such inverse problems are as follows: For a Riemannian manifold with boundary, find its metric from knowledge of the boundary Cauchy data of all the solutions to a PDE defined on the manifold. Some of the PDEs of interest are Laplace, wave, diffusion, elastic, Maxwell, Schrodinger or Dirac equations. This problem cannot be solved in general; so it is the mathematician’s task to find reasonable conditions under which the problem becomes tractable.