MATH3201: GEOMETRY III

• Complex Analysis II (MATH2011) AND Analysis in ManyVariables II (MATH2031) AND Algebra II (MATH2581).

• None.

• None.

• To give students a basic grounding in various aspects of planegeometry.
• In particular, to elucidate different types of plane geometries andto 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).
• Moebius transformations, inversion, cross-ratios.
• Inversion in space and stereographic projection.
• Conformal models of hyperbolic geometry (Poincare 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 solvenovel and/or complex problems in Geometry.
• have a systematic and coherent understanding of theoreticalmathematics in the field of Geometry.
• have acquired a coherent body of knowledge of these subjectsdemonstrated 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 specialised mathematicalskills in the following areas which can be used with minimal guidance:Spatial awareness.

Key Skills:

• Lectures demonstrate what is required to be learned and theapplication of the theory to practical examples.
• Assignments for self-study develop problem-solving skills andenable students to test and develop their knowledge andunderstanding.
• Formatively assessed assignments provide practice in theapplication 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 acquiredand the ability to solve unpredictable problems of somecomplexity.

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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