Topics in Fractals Geometry

分形几何是研究分形的数学分支。分形是一种几何物体，它在任意小尺度上表现出有趣的行为。自然界中有很多这样的例子——从蕨类植物的叶子到广阔的山脉，都可以找到分形现象。该项目将使用一系列数学分析工具研究分形和多重分形几何中各种结构的维数理论。特别地，我们将使用几何测量理论和动力系统的方法，特别是侧重于遍历理论的思想。我们还将考虑分形几何思想在其他数学领域的应用，例如解析数论。谐波分析（尤其是傅立叶分析）、概率论、泛函分析和双曲几何的思想也可以用于完成项目。

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.