New connections between Fractal Geometry, Harmonic Analysis and Ergodic Theory

分形几何、调和分析和遍历理论之间的新联系

• 批准号: RGPIN-2020-04245
• 负责人: Shmerkin, Pablo
• 金额: $ 2.7万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716855
• 关键词: New; connections; between; Fractal; Geometry

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.

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：较小的部分类似于自然界中的整体：山脉、肺部。 、河流系统、森林树木覆盖、肿瘤甚至城市都可以使用分形几何工具进行建模。 由于分形的不规则性，因此无法用长度、面积或体积等经典概念来测量，而是开发了许多“分形维数”来估计其大小和不规则性程度。分形维数的计算是一个重要问题。例如，分形维数可以让我们区分健康和癌性生长。 我的研究计划侧重于更深入地了解分形对象的数学特性以及数学中出现的自相似现象，该计划的核心部分是发现和探索与更经典部分的新联系。数学，包括调和分析、遍历理论、数论和组合学。 通常情况下，自相似物体定义起来很简单，但具有极其丰富和复杂的属性，伯努利卷积 (BC) 就说明了这一点，伯努利卷积是一类自相似质量分布，自 20 世纪 30 年代以来一直在研究。 BC 与动力学、数论和信息论中的问题相关联，它们表现出最简单的自相似形式：它们由两个按比例缩小的精确副本组成，但众所周知，它们的属性很难分开。众所周知，BC 有时是粗糙的（它们具有较小的分形维数），但通常它们相当平滑，理解和量化这种现象是一个重要且活跃的问题，我将在我最近在这个问题上取得的成就的基础上获得更深入的了解。理解更普遍和灵活的自相似形式。 引起许多领先数学家注意的另一个问题涉及集合的分形维数与集合中的点跨越的距离的集合之间的关系，更一般地，人们想了解分形维数与出现的关系。我计划将我之前为解决这个普遍问题而开发的方法与调和分析和组合学中一些令人兴奋的新发展结合起来，以期全面解决这个高度活跃领域中的一些突出猜想。