DescriptionCoxeter groups appear unexpectedly in many very distant domains of mathematics, as well as in physics, biology and even geology. Some of them can be viewed as symmetry groups of regular polytopes (e.g. Platonic solids). In general, any Coxeter group can be presented as a group generated by reflections in an appropriate vector space.The group part of the project will be devoted to understanding the classification of finite and affine Coxeter groups in all details. After that, you will be able to look at several different aspects of Coxeter groups, namely algebraic, combinatorial and geometric, and to play with some related objects (Coxeter diagrams, Davis complex, etc). Another possible (challenging) direction is to consider geometric Coxeter groups in high-dimensional hyperbolic spaces, where you will be able to look for new polyhedra tessellating the space. PrerequisitesAlgebra II; Galois Theory, Groups and Geometry III (co-requisite).ResourcesThe main source for the introduction will be the book "Reflection Groups and Coxeter Groups" by J.E. Humphreys.You can also look at the notes "Coxeter groups" by G. Heckman. A more advanced book "Geometry and Topology of Coxeter groups" by M.W. Davis will be used later (here is the library link to a paper version). A bit more combinatorial/geometric approach can be found in the book "Geometric and topological aspects of Coxeter groups and buildings" by A. Thomas. Wikipedia article on Coxeter groups. Basics on geometric reflection groups can be found in the book "Geometry II" by E. Vinberg (ed.). Further sources can be found here |
email: Pavel Tumarkin