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