Ergodic and combinatorial methods in fractal geometry

分形几何包括研究自相似和自仿射集合、源于动力学的集合以及一般的任何集合(通常在欧氏空间中)，其中Hausdorff/Packing维度是一个有趣的性质。最近关于自相似集和自仿射集的许多进展都借鉴了Mergodic理论的思想和方法；许多在组合学(离散几何)中解决的悬而未决的问题在分形几何中有有趣的对应物。研究的主要部分将集中在自相似集和管零性的一个版本上，目的是寻找具有少量窄管的自相似集的有效覆盖，并研究某些双Lipschitz不变量的结果。这项研究完全符合数学科学主题的数学分析研究领域。

Fractal geometry includes the study of self-similar and self-affine sets, sets arising from dynamics and in general any sets (usually in Euclidean spaces) forwhich Hausdorff/packing dimension is an interesting property. Many recent advances about self-similar and self-affine sets use ideas and methods fromergodic theory; many solved an unsolved problems in combinatorics (discrete geometry) have interesting counterparts in fractal geometry. A main part of theproposed research will focus on self-similar sets and a version of the tube-null property, aiming at finding efficient coverings of self-similar sets with a smallnumber of narrow tubes and investigating the consequences for certain biLipschitz invariants. This research fits in the Mathematical Analysis research area, wholly within the Mathematical Sciences theme.