Topics in Fractals Geometry
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
Fractal geometry is a branch of mathematics concerned with the study of fractals. A fractal is a geometric object that displays interesting behaviour at arbitrarily small scales. Examples abound in the natural world - fractal phenomena can be found from the leaves of a fern to vast mountain ranges.
The project will study the dimension theory of various constructions in fractal and multifractal geometry using a range of tools from mathematical analysis. In particular we shall use methods from geometric measure theory and dynamical systems, especially focusing on ideas from ergodic theory.
We shall also consider applications to other areas of mathematics where ideas from fractal geometry may prove useful, such as analytic number theory. Ideas from harmonic analysis (especially Fourier analysis), probability theory, functional analysis and hyperbolic geometry may also be employed in completing the project.
The project will study the dimension theory of various constructions in fractal and multifractal geometry using a range of tools from mathematical analysis. In particular we shall use methods from geometric measure theory and dynamical systems, especially focusing on ideas from ergodic theory.
We shall also consider applications to other areas of mathematics where ideas from fractal geometry may prove useful, such as analytic number theory. Ideas from harmonic analysis (especially Fourier analysis), probability theory, functional analysis and hyperbolic geometry may also be employed in completing the project.
Organisations
People |
ORCID iD |
| Kenneth Falconer (Primary Supervisor) | |
| Lawrence Lee (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509759/1 | 01/10/2016 | 30/09/2021 | |||
| 1948081 | Studentship | EP/N509759/1 | 27/09/2017 | 26/09/2021 | Lawrence Lee |
| Description | We have found some unexpected phenomena in the theory of self-affine multifractals. A fractal can be thought of as a geometric object which displays interesting geometric features at arbitrarily small scales and multifractals generalise this notion to more complex systems. We have been studying something called the Lq-spectrum, which helps quantify how complicated a multifractal is. We have found that for a certain class of multifractal in two dimensions, the Lq-spectrum can behave in rather unexpected ways. We have also calculated the Lq-spectrum for a new class of multifractal. |
| Exploitation Route | This research will help to drive forward the theory of multifractals, which are abundant in nature (they can be found in areas such as geophysics, fluid dynamics and meteorology). |
| Sectors | Other |