# MA243 Geometry

Lecturer: Helena Verrill

Term(s): Term 1

Status for Mathematics students: List A for Mathematics

Commitment: 30 lectures plus homework and quizzes

Assessment: The homework and quizzes carry 15% assessed credit; the remaining 85% credit by 2-hour examination

Formal registration prerequisites: None

Assumed knowledge: Basics of linear algebras:

• Vector spaces
• Bases and dimension
• Linear maps
• Rank and nullity
• Represent linear maps by matrices
• Euclidean inner product
• Eigenvalues and eigenvectors

Useful background: Familiarity with the basic language of geometry:

• Euclidean distance and norm
• Parametrised curves

Synergies: The following modules go well together with Geometry:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

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 SzendrĂ¶i, Geometry and Topology, CUP, 2005

J G Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, 2006