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 … For more content click the Read More button below.

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.

Offerings

S1-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Dr Andy Hammerlindl

Unit Coordinator(s)

Dr Andy Hammerlindl

Notes

This Level 4 unit and its Level 3 counterpart MTH3130 share the same core content and learning activities such as seminars and applied classes. However, studies at Level 4 are distinguished from those at Level 3 by a deeper understanding of mathematical theories and their applications, higher levels of critical thinking, and greater autonomy in learning

Learning outcomes

On successful completion of this unit, you should be able to:

Critically evaluate and synthesise definitions, concepts, examples, theorems, and proofs of topology.

Design and construct topological spaces in various guises, recognising their underlying structures and properties

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.

Exhibit mastery in advanced problem-solving and theorem-proving techniques, demonstrating creativity and rigour in the approach to complex topological problems

Explore and assess the extensive applications of topology across advanced areas of mathematics and the natural sciences, identifying potential interdisciplinary connections and contributions.

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.

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

Teaching approach

Peer assisted learning

Each week you will need to prepare and present some material to a small group of 3 or 4 students, and actively participate (by asking questions, and helping out) when the other students in your group present. Being able to clearly explain mathematics is an important skill that needs practice, and nothing cements mathematics in your mind like teaching it. These activities count as a kind of active, peer assisted learning. You will solve problems in small groups during contact hours. Solving these problems together is a kind of active, peer assisted learning, and a good source practice and instant feedback. You will also submit written solutions to problems, to practice careful written explanations of mathematics, and to get feedback on your writing.

Assessment

1 - Continuous assessment

2 - Final assessment - Exam (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions

Seminars

Workload requirements

Workload

Other unit costs

Costs are indicative and subject to change.Miscellaneous items required (unit course reader, printing, stationery) - $120

Availability in areas of study

Applied mathematics
Mathematical statistics
Mathematics
Pure mathematics