Lecturer: Helena Verrill

Term(s): Term 1

Status for Mathematics students: List A (MA243 is for 2nd years, MA343 for 3rd years provided MA243 has not been taken in a previous year).

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:

MA148 Vectors and Matrices

• 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:

MA145 Mathematical Methods and Modelling 2:

• Euclidean distance and norm
• Parametrised curves

Synergies: The following modules go well together with Geometry:

• MA266 Multilinear Algebra
• MA3D9 Geometry of Curves and Surfaces

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

• MA3J2 Combinatorics II
• MA3F1 Introduction to Topology
• MA448 Hyperbolic Geometry
• MA4H4 Geometric Group Theory
• MA4A5 Algebraic Geometry

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.

Books:

M Reid and B Szendröi, Geometry and Topology, CUP, 2005

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

Additional Resources