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Unit of study_

MATH5340: Topics in Topology

Topology is the mathematical theory of the "shape of spaces". It gives a flexible framework in which the fabric of space is like rubber and thus enables the study of the general shape of a space. The spaces often arise indirectly: as the solution space of a set of equations; as the parameter space for a family of objects; as a point cloud from a data set; and so on. This leads to strong interactions between topology and a plethora of mathematical and scientific areas. The love of the study and use of topology is far reaching, including the use of topological techniques in the phases of matter and transition which received the 2016 Nobel Prize in Physics. This unit introduces you to a selection of topics in pure or applied topology. Topology receives strength from its areas of applications and imparts insights in return. A wide spectrum of methods is used, dividing topology into the areas of algebraic, computational, differential, geometric and set-theoretic topology. You will learn the methods, key results, and role in current mathematics of at least one of these areas, and gain an understanding of current research problems and open conjectures in the field.

Code MATH5340
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Assumed knowledge:
Familiarity with metric spaces (e.g., MATH4061 or equivalent) and algebraic topology (e.g., MATH4311 or equivalent). Please consult with the coordinator for further information

At the completion of this unit, you should be able to:

  • LO1. Demonstrate a coherent and advanced understanding of the key concepts of fundamental group, covering spaces, homology and cohomology.
  • LO2. Apply the fundamental principles and results of algebraic topology to solve given problems.
  • LO3. Distinguish and compare the properties of different types of topological spaces and maps between them.
  • LO4. Formulate topological problems in terms of algebraic invariants and determine the appropriate framework to solve them.
  • LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
  • LO6. Devise computational solutions to complex problems in algebraic topology.
  • LO7. Compose correct proofs of unfamiliar general results in algebraic topology.

Unit outlines

Unit outlines will be available 1 week before the first day of teaching for the relevant session.