MTH4130 · Topology: The mathematics of shape

Overview

From point-set topology to manifolds: sets, topological spaces, basis of topology, and properties of spaces such as compact, connected, and Hausdorff. Maps between spaces and their properties, including continuity, homeomorphism, and homotopy. Constructing spaces via subspace, product, identification, and delta complexes. Manifolds. Additional topics from algebraic and low-dimensional topology may include fundamental group and Seifert-van Kampen theorem, classification of surfaces, and topics in knot theory. Throughout, examples of spaces will include Euclidean spaces, surfaces (real projective plane, Klein bottle, Mobius strip), complexes, function spaces, and others.

Areas of study: Applied mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Critically evaluate and synthesise definitions, concepts, examples, theorems, and proofs of topology.
2. Design and construct topological spaces in various guises, recognising their underlying structures and properties
3. Apply some of the most famous theorems of topology, such as the classification of surfaces and the Seifert-van Kampen theorem, to solve complex problems.
4. Exhibit mastery in advanced problem-solving and theorem-proving techniques, demonstrating creativity and rigour in the approach to complex topological problems
5. Explore and assess the extensive applications of topology across advanced areas of mathematics and the natural sciences, identifying potential interdisciplinary connections and contributions.
6. Communicate sophisticated mathematical arguments and concepts related to topology with clarity and precision, both in written and oral forms, suitable for academic and professional contexts.
7. Collaborate effectively with staff and fellow students in the synthesis of advanced mathematical knowledge and the application of topological methods to complex problem-solving scenarios, demonstrating leadership and initiative in group settings

Workload

Three 1-hour seminars; One 2-hour applied class (in weeks 2-12) and 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.