Title: Hear the Shape at infinity

Abstract: In 1966, Marc Kac asked the famous question "Can one hear the shape of a Drum ?". In his article with this question as the title, he translates this question to eigenvalue problems of a domain in R^2. That is, can you tell the shape of a domain if eigenvalues are known. This question can be generalized from Euclidean spaces to curved spaces, especially non-compact curved spaces. In this talk, we will recall the history of this classic question at first. Then we will explain the generalized question in curved spaces and known results. In the end, we will discuss our recent results for non-compact Kahler manifolds in this direction.