Noema Nicolussi: Arakelov-Bergman Laplacians on degenerating Riemann surfaces and graphs

The Laplace-Beltrami operator for the (Arakelov-)Bergman metric is an interesting operator in spectral theory on Riemann surfaces. Its spectral theory is in particular relevant to arithmetic geometry. Since the early 90s, there has been great interest in understanding the behavior of the Laplacian and related objects when the underlying Riemann surface degenerates to a singular Riemann surface.

In this talk, we discuss a recent approach which explains the degeneration behavior of some of these objects using analogous objects on graphs.

Based on joint work with Omid Amini (École Polytechnique).