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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758578
• 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：较小的部分类似于整体。自然界中存在着大量的分形特征：山、肺、水系、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。由于其不规则性，分形图不能用长度、面积或体积等经典概念来测量。取而代之的是，人们开发了一些“分维”来估计它们的大小和不规则程度。分维的计算是一个具有重要理论意义和应用价值的问题。例如，分维可以让我们区分健康生长和癌变生长。我的研究重点是更深入地了解分形物的数学性质和自相似现象。分数维对象出现在整个数学中，程序的核心部分是发现和探索更经典的数学部分的新链接，包括调和分析、遍历理论、数论和组合学。通常的情况是，自相似对象很容易定义，但具有极其丰富和复杂的性质。伯努利卷积(BCS)就说明了这一点，BCS是一类自相似质量分布，自20世纪30年代以来一直被研究，自那以来一直与动力学、数论和信息论中的问题联系在一起。BCS展示了最简单的自相似形式：它们由两个缩小的自己的精确副本组成。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(它们的分维很小)，但通常它们是相当光滑的。理解和量化这一现象是一个重要而活跃的问题。我将以我在这个问题上的最新成果为基础，对自相似的更一般和更灵活的形式有更深入的理解。另一个引起许多著名数学家注意的问题是集合的分维与集合中点所跨距离的集合的分维之间的关系。更广泛地说，人们想要了解分维如何与物体中图案的出现相关。我计划将我以前开发的解决这个一般问题的方法与调和分析和组合学中一些令人兴奋的新发展相结合，以期全面解决这个高度活跃的领域中的一些突出猜想。