The lecture presents the interplay of analysis, geometry, probability and mathematical physics in the realm of graphs. We start with finite graphs and develop the connection of graphs and their corresponding quadratic forms, Laplace operators and Markov processes. With these notions fundamental phenomena of the mathematics of electrostatics and heat evolution can be studied. In the second part we consider infinite graphs. Here we take a look at further properties of the heat equation as well as the connection of spectral theory and geometry.
The lecture is based on the upcoming book "Graphs and Discrete Dirichlet Spaces". A preliminary version can be found here
More information can be found on the moodle page.