NOTE: The lectures of the course have been cancelled, at least for the time being. However, the course can be passed by self-study. The lecture notes are online, and will be (mildly) updated during the next few weeks. There are also weekly exercises; the first set is online, see the material section below, and the first exercise sheet also contains more precise information on passing the course. In brief: you'll need to complete a certain number of exercises, plus write an essay. If the situation improves sufficiently by the end of the course, there may also be an oral presentation (based on the written one).

What is a fractal? The picture on the left is an example. It is a highly non-smooth compact set, which appears to have interesting structures at many, even very small, scales. The picture on the left also hints at a a classical question in fractal geometry: what is the relation between the "size" of a fractal, and the "sizes" of its projections? Fractals also occur in nature: think of the cauliflower, or the coastline of the Turku archipelago, or the brain, or the set of all galaxies.

Some of the topics covered on the course will include (depending a little on the background of the participants):

- Hausdorff measures and dimension, box dimension (these are tools to quantify the "size" of fractals)

- Basic examples of fractals where we can calculate the dimension exactly: self-similar sets

- Frostman's lemma, and other techniques for finding lower bounds for the "size" of fractals

- Projection theorems of Besicovitch and Marstrand (then you will know what the picture stands for)

- Rectifiable and unrectifiable sets

- Fourier transforms of measures (yes, the Fourier transform has very geometric applications)

- Possible additional topics: distance sets, Kakeya sets, visible parts of fractals

Pre-requisites for this course: Measure and integration, Real Analysis I (or equivalent background)