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Modul

Harmonic Analysis on Fractals [M-MATH-106287]

Credits
3
Recurrence
Einmalig
Duration
1 Semester
Language
English
Level
4
Version
1

Responsible

Organisation

  • KIT-Fakultät für Mathematik

Part of

Bricks

Identifier Name LP
T-MATH-112742 Harmonic Analysis on Fractals 3

Competence Goal

After the course, students will be able to discuss

  • examples of fractals and their properties,
  • different notions of fractal dimension and their relationships,
  • the interaction between metric and harmonic-analytic properties of fractals,
  • selected recent developments in the harmonic analysis of fractals.

Prerequisites

none

Content

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

We first introduce 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,
  • Fourier transforms of measures and Fourier dimension.

Then we study topics of recent research interest in harmonic analysis:

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

Recommendation

Some basic knowledge of functional analysis is recommended.