This project concerns two different geometric objects: Riemann surfaces and graphs. Rie-
mann surfaces are surfaces which locally look like the plane of complex numbers. Graphs
provide the mathematical framework of studying ``networks and can be visualized as
``dots which are connected by lines (see, for instance, a metro network or a family tree).
In the recent years, it has become clear that the mathematical theories of Riemann sur-
faces and graphs are closely related.
In this project, we will investigate connections between certain differential equations on
Riemann surfaces and graphs (e.g. heat equation, Schrödinger equation and Helmholtz
equation). On the one hand, we aim to use graphs to understand the behavior of differen-
tial equations on degenerating Riemann surfaces, that is, Riemann surfaces which un-
dergo a deformation into a ``singular Riemann surface. On the other hand, we plan to es-
tablish graph analogs of classical results on differential equations on Riemann surfaces. A
third geometric object -- called hybrid curve -- which is a mixture of a graph and a Riemann
surface, plays a key role in our approach.