Geometric inverse problems are an example of one of the most natural questions we can ask ourselves in Mathematics: can one recover the shape of an object using only some specific data measured from it? Perhaps the most famous of such questions in the last century is in the celebrated article "Can one hear the shape of a drum?" by Mark Kac, where he asks whether or not a domain in the plane is determined by the set of "pure tones" it produces. In this talk, we will explore inverse problems on surfaces and Riemannian manifolds in general, such as the problem of determining a Riemannian metric on a closed surface by knowing only the lengths of its closed geodesics.
