New connections between Fractal Geometry, Harmonic Analysis and Ergodic Theory

分形几何是一个相当新的数学领域，它研究的对象与经典几何不同，在所有尺度上都具有复杂的细节，并且通常具有自相似性：较小的部分与整体相似。自然界中有大量的分形特征：山、肺、河流系统、森林中的树木覆盖、肿瘤甚至城市都可以用分形几何的工具来建模。

Fractal Geometry is a fairly new area of mathematics that studies objects that, unlike those of classical geometry, have intricate detail at all scales and often feature self-similarity: smaller parts resemble the whole. Fractal features abound in nature: mountains, lungs, river systems, tree cover in forests, tumors and even cities can be modeled with the tools of fractal geometry. Because of their irregularity, fractals cannot be measured with classical notions such as length, area or volume. Instead, a number of “fractal dimensions” have been developed to estimate their size and degree of irregularity. The computation of fractal dimensions is an important problem in both theory and applications. For example, fractal dimensions may allow us to differentiate between healthy and cancerous growth. My research program focuses on obtaining a deeper understanding of the mathematical properties of fractal objects and the phenomenon of self-similarity. Fractal objects arise throughout mathematics, and a central part of the program is the discovery and exploration of new links to more classical part of mathematics, including harmonic analysis, ergodic theory, number theory, and combinatorics. It is often the case that self-similar objects are simple to define but possess extremely rich and intricate properties. This is illustrated by Bernoulli convolutions (BCs), a class of self-similar mass distributions that have been studied since the 1930s and have since been linked to problems in dynamics, number theory, and information theory. BCs display the simplest form of self-similarity: they are made up of two scaled down exact copies of themselves. However, their properties are famously hard to disentangle. It is known that BCs are sometimes rough (they have small fractal dimension) but typically they are rather smooth. It is an important and active problem to understand and quantify this phenomenon. I will build upon my recent achievements on this problem to obtain a deeper comprehension of more general and flexible forms of self-similarity. Another problem that has captured the attention of many leading mathematicians concerns the relationship between the fractal dimension of a set and that of the collection of distances spanned by points in the set. More generally, one would like to understand how fractal dimensions relate to the emergence of patterns in an object. I plan to combine methods I previously developed to tackle this general problem with some exciting new developments in harmonic analysis and combinatorics, in order to aim for a full solution to some of the outstanding conjectures in this highly active area.