Differential Topology (MAST90029)

Graduate courseworkPoints: 12.5On Campus (Parkville)

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Overview

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Semester 1
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This subject extends the methods of calculus and linear algebra to study the topology of higher dimensional spaces.  The ideas introduced are of great importance throughout mathematics, physics and engineering.  This subject will cover basic material on the differential topology of manifolds.  Topics include: smooth manifolds, tangent spaces, inverse and implicit function theorems; differential forms, integration on manifolds and de Rham cohomology; submersions and fibre bundles; immersions and transversality; examples coming from Lie groups and homogeneous spaces.  Additional topics may include: Morse theory; intersection theory; characteristic classes and Chern-Weil theory; the Thom isomorphism; bordism theory.

 

Intended learning outcomes

On completion of this subject, students should be able to:

  • Demonstrate understanding of the basic notions of Differential Topology, including smooth manifolds, vector bundles, differential forms and integration on manifolds;
  • Calculate with smooth manifolds, smooth maps, and differential forms in local coordinates;
  • Compute global invariants of manifolds;
  • Explain and apply major foundational results in differential topology;
  • Demonstrate knowledge of important examples of Lie groups and homogeneous spaces; and
  • Pursue further studies in differential topology and related areas.

Generic skills

In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

  • Problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
  • Analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
  • Collaborative skills: the ability to work in a team; and
  • Time-management skills: the ability to meet regular deadlines while balancing competing commitments.

Last updated: 4 March 2025