CAREER: The Geometry of Fractals Meets Fourier Analysis

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该项目旨在了解傅里叶分析领域与几何测量理论中有关分形的问题之间的深层联系。分形是指一种源于自然形状的集合，如海岸线、雪花、晶体和DNA，它们在不同的尺度上显示出自相似的结构。分形的研究在自然科学和工程中有许多应用。该项目旨在通过探索傅里叶分析的现代思想来推进我们对分形的理解，傅里叶分析是通过将函数分解成小块来研究函数性质的数学领域。这两个学科之间最著名的开放问题之一是Kakeya猜想，它断言一个分形集合在每个方向上都包含一个单位线段，它一定不会太小（就维数而言）。这个猜想被认为与分析、偏微分方程和数论中的中心问题密切相关。近年来，组合学的工具和思想也进入了人们的视野，这导致了该领域的许多突破。研究者计划通过使用和发明一系列跨学科工具来推进这类问题的研究。这项研究将加深对分形和傅立叶分析的理解，揭示其他领域的重要问题，并可能导致物理学、生物学、工程学和计算机科学的发现。此外，这项研究预计将被纳入研究者的教育计划，以培养新一代的研究人员和教育工作者，并将这项研究带给更广泛的受众。具体来说，研究者计划举办多个研究指南写作研讨会，组织一个致力于小型工作组的在线研究平台，并为谐波分析领域创建一个YouTube频道。这些活动不仅旨在推进研究项目，而且为学生和早期职业数学家提供了独一无二的机会。该项目的总体目标是在开放性问题上取得进展，如Kakeya猜想和Falconer距离猜想（在维度阈值上确保分形集产生许多不同的距离），开发新的工具来统一来自不同领域的想法，并进一步探索其在其他相关领域的应用，如研究Schrödinger解的规律性和偏微分方程中的波动方程，以及数论中丢芬图方程的解的数量。有三个相互关联的长期目标：(A)理解分形集合中包含的几何构型（广义距离）的存在、数量和分布；(B)研究几何测度理论中算子的稀疏支配及其在有限点组态问题中的应用；(C)研究与Kakeya猜想相关的傅里叶分析中的关键算子，如（加权）傅里叶限制/扩展算子、Bochner-Riesz乘子和Kakeya极大函数。该项目有望促进对这些核心问题的理解，为分形精细几何结构的系统研究打开大门，开发新的跨学科技术，揭示连续和离散问题之间的更深层次的联系，并发现这些领域之间的新联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The project aims to understand the deep connections between the field of Fourier analysis and questions in geometric measure theory concerning fractals. Fractals refer to a type of sets originating from natural shapes such as coastlines, snowflakes, crystals, and DNA, which display a self-similar structure across different scales. The study of fractals has numerous applications in natural sciences and engineering. This project aims to advance our understanding of fractals by exploring modern ideas in Fourier analysis, a field in mathematics that studies properties of a function by decomposing it into small pieces. One of the most famous open questions at the interface of these two subjects is the Kakeya conjecture, asserting that a fractal set that contains a unit line segment in every direction must not be too small (in terms of dimension). This conjecture is known to be closely related to central questions in analysis, partial differential equations, and number theory. In recent years, tools and ideas from combinatorics also came into the picture, which led to many breakthroughs in the field. The investigator plans to advance the study of this type of problems by using and inventing an array of interdisciplinary tools. This research will deepen the understanding of fractals and Fourier analysis, shed light on important questions in other fields, and potentially lead to discoveries in physics, biology, engineering, and computer sciences. Moreover, the research is expected to be integrated into the investigator's educational plan to nurture a new generation of researchers and educators, and to bring the research to a broader audience. Specifically, the investigator plans to run multiple study guide writing workshops, to organize an online research platform dedicated to small working groups, and to create a YouTube channel for the field of harmonic analysis. These activities are designed to not only advance the research projects, but also provide one-of-a-kind opportunities for students and early career mathematicians.The broad aim of the project is to make progress towards open questions such as the Kakeya conjecture and the Falconer distance conjecture (on the dimensional threshold ensuring a fractal set to generate many distinct distances), to develop novel tools unifying ideas from different fields, and to explore their further applications in other related fields such as for the study of regularity of solutions to Schrödinger and wave equations in partial differential equations and number of solutions to Diophantine equations in number theory. There are three interconnected long-term goals: (A) understand the existence, amount, and distribution of geometric configurations (generalizing distances) contained in fractal sets; (B) investigate sparse domination for operators arising in geometric measure theory and their applications in finite point configuration problems; (C) study key operators in Fourier analysis related to the Kakeya conjecture such as (weighted) Fourier restriction/extension operators, Bochner-Riesz multipliers, and the Kakeya maximal function. The project is expected to advance the understanding of these central problems, to open the door to the systematic study of refined geometric structures of fractals, to develop novel interdisciplinary techniques, to reveal deeper connections between continuous and discrete questions, and to discover new connections among these fields and beyond.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.