Introductory description

Fractals are geometric forms that possess structure on all scales of magnification. Examples are the middle third Cantor set, the von Koch snowflake curve and the graph of a nowhere differentiable continuous function.
The main focus of the module will be the mathematical theory behind fractals, such as the definition and properties of the Hausdorff dimension, which is a number quantifying how ``rough'' the fractal is and which reduces to the usual dimension when applied to Euclidean space. However, more recent developments will be included, such as iterated function systems (used for image compression) where we study how a fractal is approximated by other compact subsets.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The course will cover: Iterated function schemes, Julia sets, Sierpinski triangles, Circle packings, Box Dimension, Hausdorff dimension, Iterated Function Schemes, Moran's Theorem, Potential theoretic methods, Falconer's theorem, Projection and Slice theorems, Multifractal analysis, Number theory applications, including Besicovich's Theorem, Numerical estimates.

Learning outcomes

By the end of the module, students should be able to:

• By the end of this module students will be familiar with the different constructions of fractal sets, including several explicit constructions, and the different notions of dimension available to describe them. By the end of this module students will be familiar with the different constructions of fractal sets, including several explicit constructions, and the different notions of dimension available to describe them.
• They will also have the knowledge to estimate these values and to quantify the notion of being fractal.
• Students will learn the connection between these sets and values, and other areas of mathematics.

Indicative reading list

K. Falconer, Fractal geometry: mathematical foundations and applications, Wiley, 1990 or 2003. (We shall cover much of the first half of this book.)

Subject specific skills

The students will develop skills in real analysis and probability which with be motivated by problems in Fractal geometry, but be invaluable in other areas of pure and applied mathematics.

Transferable skills

The students will acquire confidence in dealing with apparently complicated problems which nonetheless have a simple underlying solution.