MATH30620: Topology

• Complex Analysis, Analysis in Many Variables and Algebra

• None

• Algebraic Topology

• To provide a balanced introduction to Point Set, Geometric and Algebraic Topology, with particular emphasis on surfaces and knots.

• Topological Spaces and Continuous Functions: Topology on a set, open sets, closed sets, limit points and closure, examples of topologies.
• Compactness and Connectedness.
• Topological groups and group actions.
• The Orthogonal groups. The Fundamental Group: calculation for circle, homotopy type, homotopy equivalence.
• Generators and relations of groups, Tietze theorem, Van Kampen's theorem.
• Compact surfaces, classical knots, basic knot invariants.

Subject-specific Knowledge:

• By the end of the module students will: be able to solve novel and/or complex problems in Topology.
• have a systematic and coherent understanding of theoretical mathematics in the field of Topology.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Topological spaces.
• Topological Groups and group actions.
• Fundamental group, homotopy type.
• Group presentations and Van Kampen's Theorem.
• Surfaces and Knots.

Subject-specific Skills:

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial awareness.

Key Skills:

• 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 predictable and unpredictable 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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