MTH3130 · 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 cell 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

Learning outcomes

1. Apply the basic definitions, concepts, examples, theorems and proofs of topology.
2. Construct and recognize topological spaces in various guises.
3. Apply some of the most famous theorems of topology such as the classification of surfaces and the Seifert-van Kampen theorem.
4. Demonstrate advanced problem solving and theorem proving skills.
5. Be aware of the scope of applications of topology in other areas of mathematics and the natural sciences.
6. Demonstrate advanced skills in the written and oral presentation of mathematical arguments that enable mathematical concepts, processes and results to be communicated effectively.
7. Work both individually and collectively with staff and fellow students on the synthesis of mathematical knowledge and the application of mathematical skills to problem solving.

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.