|Unit level:||Level 3|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH20122 - Metric Spaces (Compulsory)
To obtain an understanding and appreciation of the theory of fractals as metric objects. To be comfortable performing basic calculations involving the geometry of fractals including computing their dimension. To understand the rudiments of measure theory and construct Hausdorff measure in Euclidean space.
Roughly speaking a fractal is a set in a metric space which exhibits non-trivial geometry at arbitrarily small scales. Such objects abound in various areas of mathematics and this course will provide an introduction to the mathematics of fractals in a rigorous context. We will consider several standard examples of fractals, such as self-similar sets like the middle third Cantor set and Sierpinski triangle, and we will study various properties of these fractals with an emphasis on quantifying their ‘size’. This will be done by considering two related notions: dimension and measure. We will begin with the relatively simple, but very useful, box dimension and build up to the more sophisticated Hausdorff dimension. Hausdorff dimension is defined via a beautiful family of measures, known as Hausdorff measures, which we will construct carefully. The pre-requisite for the course is metric spaces although we will mostly work in familiar examples of metric spaces such as the unit interval, rather than abstract spaces. However, the flavour of the course will be similar to that of metric spaces with real analysis underpinning everything.
On successful completion of this course unit students will be able:
- define the Hausdorff measure, Hausdorff dimension and box dimension of a set, explaining why any limits involved in the definition are well defined. State and prove lemmas from the course involving these concepts,
- give upper bounds for the Hausdorff and box dimensions of sets by constructing suitable sequences of covers,
- give lower bounds for the Hausdorff and box dimension of sets using the mass distribution principle, Lipschitz maps, and separation arguments,
- define an attractor of an iterated function system and prove that such attractors exist and are unique,
- state and prove the formula for the Hausdorff dimension of self-similar sets.
- Other - 10%
- Written exam - 90%
Assessment Further Information
End of semester examination (2 hours) 90% and mid-semester in-class test (10%)
- Introduction and motivating examples (1 lecture)
- Iterated function systems and self-similarity (4 lectures)
- Box dimension (5 lectures)
- Construction of measures (3 lectures)
- Hausdorff measure and dimension (5 lectures)
- Further examples, such as Julia sets and fractals in number theory (4 lectures)
Good books on fractal geometry and measure theory include:
1) K J. Falconer, Fractal geometry: mathematical foundations and applications, 3rd Ed. Wiley, 2015.
2) P. Mattila, Geometry of sets and measure in Euclidean space: Fractals and rectifiability, Cambridge, 1995.
3) Pesin and Climenhaga, Lectures on Fractal Geometry and Dynamical Systems, 2008
Falconer is the closest book to the course. Mattila is more advanced, but could interest enthusiastic students.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours