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## ECM3732 - Applications of Geometry and Topology (2018)

MODULE TITLE | Applications of Geometry and Topology | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM3732 | MODULE CONVENER | Prof Mitchell Berger (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
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DURATION: WEEKS | 0 | 11 | 0 |

Number of Students Taking Module (anticipated) | 51 |
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On this module, you will have the opportunity to study mathematical topics involving geometry, topology, and their applications in science and technology. Firstly, you will become familiar with the mathematical description of curves and surfaces, and the idea of topological equivalence. Secondly, you will learn about various topics from geometry and topology, such as knot theory, classification of surfaces, and the shape of bubbles and soap films. You will then learn about the applications of geometry and toplogy, including the geometry of DNA molecules, the shape of the universe, and the topology of magnetic fields.

Prerequisite module: ECM1706, ECM2706 or equivalent

This module intends to develop your sense of shape, geometry, and topology. By taking it, you will gain a better understanding of possible geometrical structures and their mathematical description. The module covers some applications of knot theory and braid theory in detail. You will also have the opportunity to learn about current cosmological speculations concerning the shape of the universe.

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

**Module Specific Skills and Knowledge:**

1 demonstrate a working knowledge of the mathematical representation of geometrical objects.

**Discipline Specific Skills and Knowledge**:

2 reveal an understanding of the key concepts of geometry and topology, and appreciate their relevance to many areas of mathematics.

**Personal and Key Transferable/ Employment Skills and Knowledge:**

3 display enhanced problem-solving skills.

4 show competence in modelling geometric objects in computer graphics.

5 demonstrate self management and time management skills.

- curves and surfaces: parameterised curves and surfaces, manifolds, connectivity, genus, Euler’s Formula;

- basic geometry and topology of curves: tangent, normal, and binormal vectors, curvature and torsion, geometrical phase, linking number and crossing number;

- ribbons: twist, writhe. DNA geometry;

- knots: knots and links, knot invariants, DNA topology;

- braids: the braid group, relation to knots and links;

- applications: mixing theory, dynamics, solar flares;

- bubbles: surface curvature, minimal surfaces.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures |

Guided independent study | 117 | Coursework preparation, private study |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable | |||

Coursework | 20 | Written Exams | 80 | Practical Exams |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours - Summer Exam Period | All | None |

Coursework assignment 1: Basic geometrical ideas | 10 | 10 hours | All | Written and verbal |

Coursework assignment 2: Knots, links and Braids | 10 | 10 hours | All | Written and verbal |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Banchoff, Thomas F, Lovett, Stephen | Differential geometry of curves and surfaces | A K Peters | 2010 | 978-1568814568 | [Library] | |

Extended | Oprea, John | The mathematics of soap films: explorations with Maple | 1 | AMS Bookstore | 2000 | 0821821180 | [Library] |

Extended | Carlson, Stephan C | Topology of surfaces, knots and manifolds: a first undergraduate course | 1 | Wiley, New York | 2001 | 0471355445 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | ECM1706, ECM2706 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Wednesday 27 February 2019 |

KEY WORDS SEARCH | Geometry; topology; knot theory. |
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