Lecturer: Emmanuele Dotto
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 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.
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