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.

分形几何形状是一个相当新的数学领域，研究对象的对象与经典几何形状不同，在各个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。大自然中的分形特征：山，肺部，河流系统，森林，肿瘤甚至城市的树覆盖物可以用分形几何形状进行建模。 由于它们的不规则性，无法用经典音符（例如长度，面积或体积）来测量分形。取而代之的是，已经开发了许多“分形维度”来估计其规模和不规则程度。分形维度的计算在理论和应用中都是重要的问题。例如，分形维度可能使我们能够区分健康和取消增长。 我的研究计划着重于对分形对象的数学特性和自相似性现象进行更深入的了解。整个数学过程中都出现了分形对象，该程序的核心部分是发现和探索了与数学更古典部分的新链接，包括谐波分析，千古理论，数字理论和组合学。 通常，自相似的对象易于定义，但潜在的属性非常丰富和复杂。伯努利卷积（BCS）说明了这是一类自1930年代以来已经研究的自相似质量分布，此后与动态，数理论和信息理论的问题有关。 BCS展示了自相似性的最简单形式：它们由两个缩小的精确副本组成。但是，它们的特性很难分离。众所周知，BCS有时很粗糙（它们具有较小的分形维度），但通常它们很光滑。理解和量化这一现象是一个重要而活跃的问题。我将基于我最近在这个问题上取得的成就，以更深入地理解更一般和灵活的自我相似性。 引起许多主要数学家注意的另一个问题是，集合的分形维度与集合中点所跨越的距离的分形维度之间的关系。更普遍地，人们想了解分形维度与对象中模式的出现如何相关。我计划将我以前开发的方法结合在一起，以解决这个总体问题和谐波分析和组合学方面的一些令人兴奋的新发展，以旨在为这个高度活跃的领域中的某些出色的猜想提供完整的解决方案。