分形几何研究不规则几何形体，是具有很强应用背景的交叉学科。本项目研究分形几何的一些非常基本而重要的前沿问题，它们涉及到分形几何的基本理论以及与其它学科的交叉，相对独立但密切相关，它们对分形几何自身的发展以及应用前景均有重要意义。这些问题包括分形几何理论的一些基本问题:一些重要集类的维数与测度，Moran集类；带重叠结构的自相似集的结构；非正则测度的重分形机理；分形集的双倍李普希兹等价；自相似序列与代换动力系统；因子结构及其应用；非周期链的薛定愕方程的算子谱的分形结构；与实数的各种展式（包括beta展式和连分式图展式）相关的度量数论中的问题；分形在其它学科的应用。上述不同的课题彼此独立但密切相关，它们不仅在分形几何的研究中有重要意义，而且在应用中有很大潜力。

Fractal geometry studies objects that are irregular or non-smooth in mathematics, science and the nature. The applications and interplays of fractal geometry appears in many disciplines of science...The goal of this project is to investigate both fundamental theoretical problems in fractal geometry, and their applications in mathematics as well as other scientific fields. These problems, which represent the recent developments of fractal geometry, include: Dimension theory of typical fractal sets; Moran classes; self-similar structures with overlaps; multi-fractal analysis of non-regular measures; bi-Lipschitz equivalence of fractal sets; self-similar sequences and substitution dynamical systems; structure of factors and application, spectra of discrete Schrodinger operators; metric number theory related to beta-expansions; applications in other scientific fields...These topics are relatively independent and also closely related to each other. They are not only significant in fractal geometry, but also have high potential in applications.