Course details
Semester
 Autumn 2024
 Monday, October 7th to Friday, December 13th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 0
 Total advised study hours
 40
Timetable
 Fridays
 11:05  11:55 (UK)
Course forum
Visit the https://mathsmagic.ac.uk/forums/magic096spectraandgeometryofgraphsandnetworks
Description
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.
Prerequisites
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.
Syllabus
Lecturer

Professor Norbert Peyerimhoff
 University
 Durham University
Bibliography
No bibliography has been specified for this course.
Assessment
The assessment for this course will be released on Monday 13th January 2025 at 00:00 and is due in before Friday 24th January 2025 at 11:00.
Assessment for all MAGIC courses is via takehome exam which will be made available at the release date (the start of the exam period).
You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).
If you have kept uptodate with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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