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：较小的部分类似于整体。分形特征在自然界中比比皆是：山脉，肺部，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建模。 由于它们的不规则性，分形不能用长度、面积或体积等经典概念来测量。相反，一些“分形维数”已经被开发出来，以估计其大小和不规则程度。分形维数的计算是一个重要的理论和应用问题。例如，分形维数可以让我们区分健康和癌症生长。我的研究计划侧重于获得分形对象的数学特性和自相似现象的更深入的理解。分形对象出现在整个数学，该计划的核心部分是发现和探索新的联系，更经典的数学部分，包括调和分析，遍历理论，数论和组合学。通常情况下，自相似对象很容易定义，但具有极其丰富和复杂的属性。伯努利卷积（英语：Bernoulli convolutions）是自20世纪30年代以来一直被研究的一类自相似质量分布，并与动力学、数论和信息论中的问题联系起来。BC显示了自相似性的最简单形式：它们由两个缩小的自身精确副本组成。然而，它们的性质是出了名的难以理清。众所周知，BC有时是粗糙的（它们具有小的分形维数），但通常它们是相当光滑的。对这一现象的认识和量化是一个重要而活跃的问题。我将基于我最近在这个问题上的成就，对更一般和更灵活的自相似形式有更深入的理解。另一个问题，已引起注意的许多领先的数学家关注之间的关系，分形维数的一套和收集的距离跨越点在该集。更一般地说，人们希望了解分形维数如何与物体中图案的出现有关。我计划将我以前开发的联合收割机方法与谐波分析和组合学中一些令人兴奋的新发展结合起来，以解决这个普遍问题，目的是全面解决这个高度活跃的领域中一些杰出的问题。

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.