A fractal and its projection

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)

### Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 9.3.2020
14:15 - 16:00
Tue 10.3.2020
14:15 - 16:00
Thu 12.3.2020
12:15 - 14:00
Mon 16.3.2020
14:15 - 16:00
Tue 17.3.2020
14:15 - 16:00
Thu 19.3.2020
12:15 - 14:00
Mon 23.3.2020
14:15 - 16:00
Tue 24.3.2020
14:15 - 16:00
Thu 26.3.2020
12:15 - 14:00
Mon 30.3.2020
14:15 - 16:00
Tue 31.3.2020
14:15 - 16:00
Thu 2.4.2020
12:15 - 14:00
Mon 6.4.2020
14:15 - 16:00
Tue 7.4.2020
14:15 - 16:00
Thu 16.4.2020
12:15 - 14:00
Mon 20.4.2020
14:15 - 16:00
Tue 21.4.2020
14:15 - 16:00
Thu 23.4.2020
12:15 - 14:00
Mon 27.4.2020
14:15 - 16:00
Tue 28.4.2020
14:15 - 16:00

### Material

Pertti Mattila: Geometry of Sets and Measures in Euclidean Spaces
Kenneth Falconer: Geometry of Fractal Sets
Kenneth Falconer: Fractal Geometry (a very solid, but slightly "popularised" fractal book).

## Other

• Tuomas Orponen
• Eino Rossi
Lecture on self-similar sets

### Conduct of the course

Solving exercises will be mandatory for credit. In addition, depending e.g. on the number of participants, there will either be an exam, or a presentation on a topic extending the course material (containing both oral and written parts).