Spectra and geometry of graphs and networks (MAGIC096)
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Combinatorial graphs and networks appear in many theoretical and practical contexts. It is beneficial for every working mathematician and theoretical computer scientist to have a good familiarity with these concepts. A very active research area is spectral graph theory, where graphs and their properties are studied via the eigenvalues of their associated adjacency matrices. This topic is particularly appealing since it needs very little background knowledge and leads efficiently to many beautiful and deep observations and results. We will cover various useful aspects of general interest with a geometric viewpoint, amongst them: variational characterisation of eigenvalues, Cheeger isoperimetric constants or expansion rates and inequalities between them and eigenvalues, leading to the timely topics of expander graphs, spectral clustering, construction of codes, Ramanujan graphs, and Cayley graphs as geometric representations of discrete groups. We will discuss fundamental connections between eigenvalues and dynamics like mixing properties of random walks and electrical networks. We will study eigenvalue relations under graph constructions like coverings, zigzag products, and line graphs, and we will also investigate the interplay between eigenvalues and specific discrete curvature notions (like one based on optimal transport). In summary, the course provides a variety of interesting tools which can be used to study graphs and networks both from a geometric and an algebraic viewpoint via eigenvalues.
Autumn 2017 (Monday, October 9 to Friday, December 15)
The course will be self contained. Only basic knowledge in linear algebra (linear maps, symmetric matrices, eigenvalues), graph theory (vertices and edges of graphs, graph automorphisms), algebra (discrete groups and generators, symmetry groups), and probability theory (discrete probability spaces) is needed.
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