Study information

Fractal Geometry - 2024 entry

MODULE TITLE CREDIT VALUE Fractal Geometry 15 MTHM004 Dr Jimmy Tseng (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 20
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 Module: MTH2001 or MTH2008

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 (1 lecture);

- Basic topology and metric spaces (3 lectures);

- Basic Measure Theory (2 lectures);

- The Cantor Set (1 lecture);

- Box dimension (5 lectures);

- Hausdorff dimension (6 lectures);

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

- Winning sets (3 lectures);

- Examples from Diophantine approximation (3 lectures);

- Examples from Dynamical systems and ergodic theory (1 lecture);

- Problem-solving sessions (5 lectures) .

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 127 0
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
Not Applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 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

Reassessment will be by written exam only. For deferred candidates, the module mark will be uncapped.  For referrals, the module mark will be capped at 50%.

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

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Falconer, K. Fractal Geometry 2nd edition Wiley 2003 978-0470848623
Set Edgar, G.A. Measure, Topology and Fractal Geometry Springer 1990 000-0-387-97272-2
Set Mattila, P. Geometry of Sets and Measure in Euclidean Space 1st Cambridge 1995 0-521-46576-1
Set Pesin, Y. and Climenhaga, V. Lectures on Fractal Geometry and Dynamical Systems 1st AMS 2009 978-0-8218-4889-0
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2001, MTH2008
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 12th March 2024 Tuesday 12th March 2024
KEY WORDS SEARCH None Defined

Please note that all modules are subject to change, please get in touch if you have any questions about this module.