Topics in Fractals Geometry

分形几何是研究分形的一个分支。分形是一种几何对象，在任意小的尺度上显示出有趣的行为。自然界中的例子比比皆是--从蕨类植物的叶子到广阔的山脉，都可以发现分形现象。 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。特别是，我们将使用几何测度论和动力系统的方法，特别是侧重于遍历理论的思想。我们还将考虑应用到其他领域的数学，从分形几何的想法可能证明是有用的，如解析数论。调和分析（特别是傅立叶分析），概率论，泛函分析和双曲几何的思想也可以用来完成这个项目。

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.