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Faculty Handbook Archive

# Archive Module Description

This page is for the academic year 2021-22. The current handbook year is 2022-23

## Department: Mathematical Sciences

### MATH4141: GEOMETRY IV

Type Level Credits Availability Module Cap Open 4 20 Not available in 2021/22 Durham

#### Prerequisites

• Complex Analysis II (MATH2011) AND Analysis in Many Variables II (MATH2031) AND Algebra II (MATH2581) AND a minimum of 40 credits of Mathematics modules at Level 3.

• None.

#### Excluded Combination of Modules

• Geometry III (MATH3201).

#### 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).
• MÃ¶bius transformations, inversion, cross-ratios.
• Inversion in space and stereographic projection.
• Conformal models of hyperbolic geometry (PoincarÃ© 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.
• (NB the syllabus is identrical to GEOMETRY III (A) which is taught in parallel).

#### Learning Outcomes

Subject-specific Knowledge:
• 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.
Subject-specific Skills:
• In addition students will have highly specialised and advanced mathematical skills in the following areas: Spatial awareness.
Key Skills:

#### 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

Activity Number Frequency Duration Total/Hours Lectures 42 2 per week for 20 weeks and 2 in term 3 1 Hour 42 Problems Classes 8 Four in each of terms 1 and 2 1 Hour 8 Preparation and Reading 150 Total 200

#### Summative Assessment

Component: Examination Component Weighting: 90%
Element Length / duration Element Weighting Resit Opportunity
Written examination 3 Hours 100%
Component: Continuous Assessment Component Weighting: 10%
Element Length / duration Element Weighting Resit Opportunity
Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students. 100%

#### 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