Mathematics

MTHM004 - Fractal Geometry (2018)

Back | Download as PDF
MODULE TITLEFractal Geometry CREDIT VALUE15
MODULE CODEMTHM004 MODULE CONVENERDr Jimmy Tseng (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
Number of Students Taking Module (anticipated) 22
DESCRIPTION - summary of the module content

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

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.
 

SYLLABUS PLAN - summary of the structure and academic content of the module

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)

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33.00 Guided Independent Study 127.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures including example classes
Guided Independent Study 127 Lecture and assessment preparation
     

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
       
       
       
       
       
SUMMATIVE ASSESSMENT (% of credit)
Coursework 0 Written Exams 100 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
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
         
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
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
       
       

 

RE-ASSESSMENT NOTES

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.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
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