Group and Individual Project III 2026-27

Matrix Groups

Martin Kerin

A visualisation of SO(4) by rotating 4-space in two orthogonal 2-planes. Source: Patrick R. Nicolas, A Journey into the Lie Group SO(4).

Project Research Area Pure Mathematics Description A matrix group is a group of invertible matrices. This description sounds simple enough and seems purely algebraic. In fact, you've already encountered several examples of such groups in Linear Algebra I and Algebra II. However, as we know from linear algebra, invertible matrices represent geometric motions (i.e. linear transformations) of vector spaces, so maybe it’s not so surprising that matrix groups are useful within geometry. It turns out that matrix groups are amazingly ubiquitous and pop up not only throughout mathematics, but also in many situations involving objects with symmetries, such as molecules in chemistry, particles in physics, computer graphics, general relativity, quantum computing, neural networks and data analysis. Matrix groups are special cases of Lie groups, which are simultaneously algebraic and geometric objects. The fact that they lie at the interface of the algebraic and geometric worlds means that Lie groups have been studied from many different viewpoints and provide a rich source of concrete examples of various mathematical phenomena. The goal of this project is to focus on matrix groups from a geometric viewpoint, thus acquiring hands-on experience and intuition in a setting that is less abstract than that of general Lie groups. Group Project The group project will revolve around learning basic concepts and results in the study of matrix groups. By the end of the project, you will have learned the definition of a matrix group; the definition and properties of the quaternions; many examples of matrix groups; the connection between matrix groups and symmetries; the definition and properties of the Lie algebra associated to a matrix group; the basic topology of matrix groups; what it means for two matrix groups to be considered equivalent. Mode of Operation and Evidence of Learning The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats. Individual Project The individual project will build on the knowledge we have gained in the group project and will explore additional advanced topics. A few examples of topics you will be able to investigate are: Homogeneous spaces Transitive actions on spheres Lie group actions on manifolds The geometry of matrix groups The octonions and the exceptional Lie group G2 Homotopy types of skateboard tricks Mode of Operation and Evidence of Learning The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats. Prerequisites Complex Analysis II Mathematical Methods II Algebra II Recommended Corequisites Analysis & Topology III Differential Geometry III Some Resources Kristopher Tapp, Matrix Groups for Undergraduates: Second Edition, American Mathematical Society, 2016. Harriet Pollatsek, Lie groups: a problem-oriented introduction via matrix groups, Mathematical Association of America, 2009. Andreas Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, American Mathematical Society, 2003. Morton L. Curtis, Matrix Groups, Springer, 1979. Marcos M. Alexandrino and Renato G. Bettiol, Lie groups and Geometric Aspects of Isometric Actions, Springer, 2015. John C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), no. 2, 145–205. Ruben Arenas, Constructing a matrix representation of the Lie group G2, HMC Senior Theses 166 (2005). Ádám Gyenge, On the topology of the exceptional Lie group G2, MSc thesis 2011, Budapest University of Technology and Economics. Justus Carlisle, Kyle Hammer, Robert Hingtgen and Gabriel Martins, Skateboard tricks and topological flips, preprint 2021 Additional Information If you would like more information about this project, please contact me at martin.p.kerin@durham.ac.uk