Module MATH41920: Geometry

Department: Mathematical Sciences

MATH41920: Geometry

Prerequisites

• Complex Analysis and Analysis in Many Variables and Algebra.

Corequisites

• None

Excluded Combination of Modules

• None

Aims

• To give students a basic grounding in various aspects of plane geometry.
• In particular, to elucidate different types of plane geometries and to show how these may be handled from a group theoretic viewpoint.

Content

• Euclidean geometry: isometry group, its generators, conjugacy classes.
• Discrete group actions: fundamental domains, orbit space.
• Spherical geometry.
• Affine geometry.
• Projective line and projective plane. Projective duality.
• Hyperbolic geometry: Klein disc model (distance, isometries, perpendicular lines).
• Möbius transformations, inversion, cross-ratios.
• Inversion in space and stereographic projection.
• Conformal models of hyperbolic geometry (Poincaré disc and upper half-plane models).
• Elementary hyperbolic geometry: sine and cosine rules, area of a triangle.
• Projective models of hyperbolic geometry: Klein model and hyperboloid model.
• Types of isometries of the hyperbolic plane. Horocycles and equidistant curves.
• Additional topics: hyperbolic surfaces, 3D hyperbolic geometry.

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Geometry.
• have an understanding of specialised and complex theoretical mathematics in the field of Geometry.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Isometries and affine transformations of the plane.
• Spherical geometry.
• Mobius transformations.
• Projective geometry.
• Hyperbolic geometry.

• In addition students will have highly specialised and advanced mathematical skills in the following areas: Spatial awareness.

• Students will be able to study independently to further their knowledge of an advanced topic.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University