Spectral theory of the complex Laplacian

In this expository talk, we will discuss aspects of spectral theory of the complex Laplacian.
Our emphasis is on the interplay between spectral behavior of the complex Laplacian and the underlying
geometry. In particular, we are interested in geometric properties that characterize positivity and
spectral discreteness of the complex Laplacian. We will also discuss recent results on spectral
stability of the complex Laplacian.