DescriptionThis project will explore relations between analysis and geometry in the setting of discrete graphs. For example, discrete graphs can model connections between people in social networks. The geometry of these graphs, such as their diameter, connectivity and girth, captures key aspects of these networks. One can associate an adjacency matrix to any graph whose entries describe the number of connections between any two vertices. Spectral graph theory studies how properties of this matrix relate to the geometry of these graphs.The first aim of this project will be to link properties of the spectrum (eigenvalues) of the adjacency matrix to the graph geometry. A key concept for this is the Benjamini-Schramm topology which allows us to compare the local geometry of two graphs using probability theory. We also extend this study to random constructions of graphs. There are some good notes on these topics [3,4] that we can refer to. In the second part of the project, the student will be able to explore further areas of their choosing. One fascinating example is the application of graphs to quantum chaos through their spectral properties. Eigenvectors of the adjacency matrix give rise to probability distributions that model the location of a quantum particle on the graph. Quantum ergodicity shows that as the size of the graphs gets large, these probabilities become uniformly distributed on the graph for most of the eigenvectors. This means that we are equally likely to find the quantum particle at any vertex of the graph, i.e. its behaviour is in some sense chaotic [1]. The nature of the values of the eigenvectors at randomly sampled vertices in randomly constructed graphs also inform us about the probability distributions associated to the eigenvectors. These random entry values are seen to follow a Gaussian distribution [2] in line with the so-called Random Wave Hypothesis in physics in the graph setting. PrerequisitesNone specifically but the first part of the project will use ideas from probability and topology, thus some familiarity with these concepts will be beneficial.Resources
email: Joe Thomas |