Spectral Theory of Graph Laplacians and its Applications

The notion of graphs is a fundamental concept in mathematics. Roughly speaking, 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). One of their fascinating aspects is that they play a role in several very different contexts and hence allow to connect different fields. On the one hand, they appear in applied questions and can model, for instance, street networks, social networks or blood circulation. On the other hand, they are important for several abstract mathematical areas, for instance group theory (the ``dots then represent abstract objects). In this project, we investigate the properties of Laplace operators on graphs. These play a crucial role in understanding differential equations on graphs, such as the heat equation (describing ``diffusion in networks, e.g. spreading of heat / information) and the Schrödinger equation (``description of a particle in the network in quantum mechanics). It has been discovered that there is a beautiful and fascinating interplay between the properties of Laplace operators and geometric properties of graphs (simple examples are their ``diameter or ``connectivity). The aim of this project is to understand this interplay with respect to several new questions. For instance, we would like to understand how different versions of the Schrödinger equation can be classified by the ``geometric properties of infinitely large graphs at infinity. Another project goal concerns the connection between the theories of Laplace operates on graphs and Riemann surfaces.

Graphs are a fundamental concept in mathematics. Very simply put, a graph is the mathematical term for a "network" and consists of "points connected by lines" (e.g., a subway map or family tree). A fascinating aspect of graphs is that they play a role in very different contexts and thus build a bridge between different topics. For example, they appear in applied questions such as the modeling of road networks, social networks or blood circulation systems, but also in abstract mathematical areas such as group theory (the "points" then correspond to abstract objects). In this Schrödinger project, we investigated the properties of Laplace operators on graphs. These operators play an essential role in understanding differential equations on graphs, such as the heat equation (describes "diffusion" in networks, e.g. propagation of information or heat) or the Schrödinger equation ("quantum mechanical description of a particle in a network"). Here, very beautiful and fascinating connections arise between the properties of Laplace operators and the "geometric properties" of graphs (simple examples of these are, for example, their "diameter" or "connectivity"). For example, we were able to link the theories of two different types of Laplacian operators on graphs (discrete Laplacian operators vs. quantum graph Laplacian operators) and obtain new results from this link. Furthermore, we investigated the connections between geometric properties of graphs and so-called Hardy inequalities. We were also able to establish links to differential equations on "degenerating surfaces", i.e. surfaces that deform into a singular geometric object.