Fractal Geometry - 2024 entry

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 

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 (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)
.

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%. 

Basic reading:

ELE - http://vle.exeter.ac.uk/

Reading list for this module: