MT5870 Hyperbolic Geometry

Academic year

2024 to 2025 Semester 2

Further information on which modules are specific to your programme.

Key module information

SCOTCAT credits

15

The Scottish Credit Accumulation and Transfer (SCOTCAT) system allows credits gained in Scotland to be transferred between institutions. The number of credits associated with a module gives an indication of the amount of learning effort required by the learner. European Credit Transfer System (ECTS) credits are half the value of SCOTCAT credits.

SCQF level

SCQF level 11

The Scottish Credit and Qualifications Framework (SCQF) provides an indication of the complexity of award qualifications and associated learning and operates on an ascending numeric scale from Levels 1-12 with SCQF Level 10 equating to a Scottish undergraduate Honours degree.

Availability restrictions

Module runs in alternating even years, 2020/21, 2022/23, 2024/25, etc.

Planned timetable

10am, Monday (odd weeks), Wednesday, Friday

This information is given as indicative. Timetable may change at short notice depending on room availability.

Module coordinator

Prof J M Fraser

Prof J M Fraser
This information is given as indicative. Staff involved in a module may change at short notice depending on availability and circumstances.

Module description

We study two-dimensional hyperbolic space, which is a fundamental example of a non-Euclidean metric space. Hyperbolic space has a rich structure and many counter-intuitive properties. This module will focus on the geometry of hyperbolic space, including a detailed study of the geodesic structure, the group of isometries, and the actions of Fuchsian groups. Fuchsian group actions lead to beautiful tilings (or tessellations) as well as fractal 'limit sets'. We will combine ideas from analysis, geometry, and group theory, with a strong emphasis on visual intuition.

Relationship to other modules

Pre-requisites

BEFORE TAKING THIS MODULE YOU MUST PASS MT2505 AND PASS MT3502 AND PASS MT3503

Anti-requisites

YOU CANNOT TAKE THIS MODULE IF YOU TAKE MT5830

Assessment pattern

2-hour Written Examination = 100%

Re-assessment

Oral examination = 100%

Learning and teaching methods and delivery

Weekly contact

2.5 lectures (x 10 weeks), 1 tutorial (x 10 weeks)

Intended learning outcomes

  • Understand the Poincaré disc and upper half plane as models of 2-dimensional hyperbolic space
  • Understand the basic geometry of hyperbolic space and be able to perform calculations involving, for example, geodesics, hyperbolic circles, and hyperbolic triangles
  • Understand the group of isometries and be able to classify isometries by fixed points, trace, and standard form
  • Understand and work with Fuchsian groups
  • Appreciate the interplay between Fuchsian groups and tilings of hyperbolic space
  • Understand and be able to derive several properties of limit sets of Fuchsian groups