Lecturer: Adam Epstein

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 Lectures

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• MA259 Multivariable Calculus
• General notions of metric, topology, and continuity as presented in MA222 Metric Spaces

Useful background: 

• MA243 Geometry
• MA3B8 Complex Analysis

Synergies:

• MA3D9 Geometry of Curves and Surfaces
• MA4H4 Geometric Group Theory
• MA4C0 Differential Geometry

Leads to:

Content: An introduction to hyperbolic geometry, mainly in dimension two, with emphasis on concrete geometrical examples and how to calculate them. Topics include: basic models of hyperbolic space; linear fractional transformations and isometries; discrete groups of isometries; connections to complex analysis.

Aims: To introduce the beautiful interplay between geometry, algebra and analysis which is involved in a detailed study of the Poincaré model of two-dimensional hyperbolic geometry.

Objectives: To understand:

• The non-Euclidean geometry of hyperbolic space
• Tesselations and groups of symmetries of hyperbolic space
• Hyperbolic geometry on surfaces

Books:

J.W. Anderson, Hyperbolic Geometry, Springer Undergraduate Math. Series.

S. Katok, Fuchsian Groups, Chicago University Press.

S. Stahl, The Poincaré Half-Plane, Jones and Bartlett.

A. Beardon, Geometry of Discrete Groups, Springer.

J. Lehner, Discontinuous Groups and Automorphic Functions. AMS.

L. Ford, Automorphic Functions, Chelsea (out of print but in library).

J. Stillwell, Mathematics and its History, Springer.

Additional Resources