Differential Topology and Geometry (MAST90029)

Graduate courseworkPoints: 12.5Not available in 2020

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Overview

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This subject extends the methods of calculus and linear algebra to study the geometry and 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 including integration on manifolds, and give an introduction to Riemannian geometry. Topics include: Differential Topology: smooth manifolds, tangent spaces, inverse and implicit function theorems, differential forms, bundles, transversality, integration on manifolds, de Rham cohomology; Riemanian Geometry: connections, geodesics, and curvature of Riemannian metrics; examples coming from Lie groups, hyperbolic geometry, and other homogeneous spaces.

 

Intended learning outcomes

After completing this subject, students will gain:



  • an understanding of the basic notions of Differential Topology, including smooth manifolds, vector bundles, differential forms and integration on manifolds;
  • an understanding of the basic notions of Riemannian Geometry, including connections, curvature and geodesics;
  • the ability to work with smooth manifolds, smooth maps, differential forms and Riemannian metrics;
  • the ability to do geometric calculations in local coordinates;
  • a knowledge of important examples of Lie groups and symmetric spaces;
  • the ability to pursue further studies in this 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;
  • time-management skills: the ability to meet regular deadlines while balancing competing commitments.

Last updated: 3 November 2022

Eligibility and requirements

Prerequisites

Both of the following, or equivalent.

Code Name Teaching period Credit Points
MAST20009 Vector Calculus
Semester 2 (On Campus - Parkville)
Semester 1 (On Campus - Parkville)
 12.5
MAST30026 Metric and Hilbert Spaces
Semester 2 (On Campus - Parkville)
 12.5

Corequisites

None

Non-allowed subjects

MAST90054

Inherent requirements (core participation requirements)

The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.

Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home

Last updated: 3 November 2022

Assessment

Due to the impact of COVID-19, assessment may differ from that published in the Handbook. Students are reminded to check the subject assessment requirements published in the subject outline on the LMS

DescriptionTimingPercentage

Up to 60 pages of written assignments (three assignments worth 20% each, due early, mid and late in semester)

  • 60 pages
Throughout the teaching period60%

A written examination

  • 2 hours
During the examination period40%

Last updated: 3 November 2022

Dates & times

Not available in 2020

Time commitment details

170 hours

Last updated: 3 November 2022

Further information

  • Texts

    Prescribed texts

    None

    Recommended texts and other resources

    N. Hitchin. Differentiable Manifolds, available online at: people.maths.ox.ac.uk/~hitchin/hitchinnotes/hitchinnotes.html
    M. P. do Carmo, Riemannian Geometry, Birkhäuser (1992).

  • Related Handbook entries

    This subject contributes to the following:

    TypeName
    CourseMaster of Science (Mathematics and Statistics)
    CoursePh.D.- Engineering
    CourseDoctor of Philosophy - Engineering
    CourseMaster of Philosophy - Engineering
    Informal specialisationMathematics and Statistics
  • Available through the Community Access Program

    About the Community Access Program (CAP)

    This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.

    Please note Single Subject Studies via Community Access Program is not available to student visa holders or applicants

    Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.

  • Available to Study Abroad and/or Study Exchange Students

    This subject is available to students studying at the University from eligible overseas institutions on exchange and study abroad. Students are required to satisfy any listed requirements, such as pre- and co-requisites, for enrolment in the subject.

Last updated: 3 November 2022