This intermediate course is designed as a comprehensive introduction to one
of the most active and attractive fields of contemporary mathematics with
a remarkable range of applications to Lie groups, geometry, physics and
coding theory.
Prerequisites:
basic algebra, linear algebra, elementary hyperbolic geometry
and an interest in learning new things by solving a variety of problems.
Curriculum:
- Discrete transformation groups.Fundamental domain. Quotients by discrete
groups.
- Cocompact discrete groups. Some instructive classical examples.
- Discrete reflection groups in spherical,euclidean and hyperbolic
geometries.
- Polytopes with nonobtuse angles. Coxeter polytopes as fundamental domains.
- Classification of the finite reflection groups.
- Abstract Coxeter groups and their braid -friends.
- Basics on the hyperbolic geometry in the Klein model.
- Finite Coxeter polytopes in hyperbolic space and associated cocompact
discrete reflection groups.
- Circle packings and Andreev`s theorem (after Thurston)
- Any cocompact reflection group is a Coxeter group.
- Tits representation .
- The root system attached to any Coxeter group.
- Classification of finite and affine root systems.
- Combinatorial theory of Coxeter groups.
Textbooks
- S.Lang, Undergraduate algebra, Springer 1989 (first three chapters).
- S.Lang, Algebra,Addis.-Wesley ,1970 (chapters 13,14).
- H.Coxeter, Introduction to geometry, Wiley&Sons, 1970.
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