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Harmonic Analysis on Fractals (Summer Semester 2023)

Course materials are usually updated on Fridays to include the contents of the following week. You can download the files on ILIAS.

Lecture: Tuesday 11:30-13:00 20.30 SR 2.66

Course contents

This course aims to be an accessible introduction to fractals and selected aspects of their modern harmonic-analytic theory.

We first discuss examples of fractals and their dimension theory:

  • fractals in nature,
  • Cantor sets and Bernoulli convolutions,
  • number-theoretic fractals,
  • Brownian motion,
  • Kakeya sets,
  • Hausdorff dimension, box dimension and intermediate dimensions.

We then introduce the Fourier transform and use it to study further properties of fractals, in particular their Fourier dimension.

In the last part of the course we study some topics of recent research interest in harmonic analysis:

  • Fourier restriction theorems on fractals,
  • fractal uncertainty principles.