MATH41920: Geometry

• Complex Analysis and Analysis in Many Variables and Algebra.

• None

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• 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.

• 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).
• Mbius transformations, inversion, cross-ratios.
• Inversion in space and stereographic projection.
• Conformal models of hyperbolic geometry (Poincar 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.

Subject-specific Knowledge:

• 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.

Subject-specific Skills:

• In addition students will have highly specialised and advanced mathematical skills in the following areas: Spatial awareness.

Key Skills:

• Students will be able to study independently to further their knowledge of an advanced topic.

• 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.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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