Undergraduate Programme and Module Handbook 2018-2019 (archived)
Module MATH4141: GEOMETRY IV
Department: Mathematical Sciences
MATH4141:
GEOMETRY IV
Type |
Open |
Level |
4 |
Credits |
20 |
Availability |
Available in 2018/19 |
Module Cap |
|
Location |
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.
Corequisites
Excluded Combination of Modules
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.
- (NB the syllabus is identrical to GEOMETRY III (A) which
is taught in parallel).
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.
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: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
Written examination |
3 Hours |
100% |
|
Eight written assignments to be assessed and
returned. Other assignments are set for self-study and complete solutions
are made available to students.
■ 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