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MA243 Geometry

Lecturer: Christian Boehning and Gavin Brown

Term(s): Term 1

Status for Mathematics students: List A for Mathematics

Commitment: 30 lectures plus weekly worksheets

Assessment: The weekly worksheets carry 15% assessed credit; the remaining 85% credit by 2-hour examination.

Prerequisites: None, but an understanding of MA125 Introduction to Geometry will be helpful.

Leads To: Third and fourth year courses in Algebra and Geometry, including:  MA3D9 Geometry of Curves and Surfaces, MA3E1 Groups and Representations, MA4A5 Algebraic Geometry, MA4E0 Lie Groups, MA473 Reflection Groups, MA4H4 Geometric Group Theory, MA448 Hyperbolic Geometry and others

Content: Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of mathematics and physics, and offers a whole range of views on the nature and meaning of the universe.

Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full  n\times n matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3-space. The study of the group of motions throws light on the chosen model of the world.

Aims: To introduce students to various interesting geometries via explicit examples; to emphasize the importance of the algebraic concept of group in the geometric framework; to illustrate the historical development of a mathematical subject by the discussion of parallelism.

Objectives: Students at the end of the module should be able to give a full analysis of Euclidean geometry; discuss the geometry of the sphere and the hyperbolic plane; compare the different geometries in terms of their metric properties, trigonometry and parallels; concentrate on the abstract properties of lines and their incidence relation, leading to the idea of affine and projective geometry.

Books:

M Reid and B Szendröi, Geometry and Topology, CUP, 2005 (some Chapters will be available from the General office).

E G Rees, Notes on Geometry, Springer

HSM Coxeter, Introduction to Geometry, John Wiley & Sons

Additional Resources

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Year 1 regs and modules
G100 G103 GL11 G1NC

yr2.jpg
Year 2 regs and modules
G100 G103 GL11 G1NC

yr3.jpg
Year 3 regs and modules
G100 G103

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Year 4 regs and modules
G103

Archived Material
Past Exams
Core module averages