Topics in Fractals Geometry

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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.