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## MTHM004 - Fractal Geometry (2018)

MODULE TITLE | Fractal Geometry | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | MTHM004 | MODULE CONVENER | Dr Jimmy Tseng (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 |

Number of Students Taking Module (anticipated) | 22 |
---|

Fractal geometry is the study of certain irregular sets (called fractals) which arise naturally in many branches of mathematics such as Dynamical Systems and Ergodic Theory, Diophantine Approximation, and Analysis, and which are used to model natural phenomena in the natural sciences. The importance and ubiquity of these irregular sets is a significant realization of modern mathematics. Unlike the more familiar sets from classical geometry, these irregular sets are not, in general, amenable to the techniques of classical calculus. Instead, new ideas, especially from measure theory, are required to understand their properties.

This module aims to give an introduction to fractals, to develop basic tools for their study, especially various notions of dimension, and to give applications to other fields of mathematics, especially Diophantine approximation, and Dynamical Systems and Ergodic Theory. The basic notions of measure, box dimension, Hausdorff dimension, etc., will be introduced and developed. Important examples of fractals will be introduced and studied. In every section covered in this module, we will start by studying the definitions and then we will present examples and some basic properties. Some important theorems will be stated and proved. With this module you will have the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Pre-requisite - ECM2701 Analysis

The objective of this module is to provide an introduction to the geometry of fractals and to the tools used in in their study. Our main objective will be to give important examples of fractals, to define and develop various notions of dimension and other basic concepts, and to provide proofs of useful theorems.

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

**Module Specific Skills and Knowledge**

1. Recall and apply key definitions in fractal geometry;

2. State, prove and apply core theorems in fractal geometry.

**Discipline Specific Skills and Knowledge**

3. Extract abstract problems from a diverse range of problems

4. Use abstract reasoning to solve a range of problems.

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

5. Think analytically and use logical argument and deduction.

6. Communicate results in a clear, correct and coherent manner.

Review of some background material on set theory and functions (3 lectures)

Basic topology and metric spaces (3 lectures)

Construction of fractals: iterated function systems, self-similar sets (3 lectures)

Box dimension (3 lectures)

Basic measure theory (3 lectures)

Hausdorff dimension (6 lectures)

Winning sets (3 lectures)

Examples from Diophantine approximation (3 lectures)

Examples from Dynamical systems and ergodic theory (3 lectures)

Revision (3 lectures)

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 127.00 | Placement / Study Abroad | 0.00 |
---|

Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures including example classes |

Guided Independent Study | 127 | Lecture and assessment preparation |

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

Coursework | 0 | Written Exams | 100 | Practical Exams | 0 |
---|

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

Written Examination | 100% | 2 hours - Summer Exam Period | All | On Request |

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

As above | Written Examination | 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

**Basic reading:**

**ELE: http://vle.exeter.ac.uk/**

**Web based and Electronic Resources:**

**Other Resources:**

Reading list for this module:

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

Set | Falconer, K. | Fractal Geometry | 2nd edition | Wiley | 2003 | 978-0470848623 | [Library] |

Set | Edgar G.A. | Measure, Topology and Fractal Geometry | Springer | 1990 | 000-0-387-97272-2 | [Library] | |

Set | Pesin, Y. and Climenhaga, V. | Lectures on fractal geometry and dynamical systems | 1st | AMS | 2009 | 978-0-8218-4889-0 | [Library] |

Set | Mattila, P. | Geometry of sets and measure in Euclidean space | 1st | Cambridge | 1995 | 0-521-46576-1 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2701 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 7 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Thursday 08 March 2018 | LAST REVISION DATE | Thursday 28 February 2019 |

KEY WORDS SEARCH | None Defined |
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