CAREER: The Geometry of Fractals Meets Fourier Analysis

职业：分形几何与傅立叶分析的结合

• 批准号: 2142221
• 负责人: Yumeng Ou
• 金额: $ 50万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142221&HistoricalAwards=false
• 关键词: CAREER; Geometry; Fractals; Meets; Fourier

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.

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。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学的问题之间的深层联系。分形图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。分形学在自然科学和工程中有着广泛的应用。这个项目旨在通过探索傅立叶分析中的现代思想来推进我们对分形学的理解。傅立叶分析是数学中的一个领域，通过将函数分解成小块来研究函数的性质。在这两个学科的交界处最著名的悬而未决的问题之一是Kakeya猜想，该猜想断言在每个方向上包含单位线段的分形集不能太小(就维度而言)。众所周知，这个猜想与分析、偏微分方程式和数论中的中心问题密切相关。近年来，组合数学的工具和思想也应运而生，导致了该领域的许多突破。研究人员计划通过使用和发明一系列跨学科工具来推进对这类问题的研究。这项研究将加深对分形学和傅立叶分析的理解，阐明其他领域的重要问题，并可能导致物理、生物、工程和计算机科学的发现。此外，这项研究预计将被纳入研究人员的教育计划，以培养新一代研究人员和教育工作者，并将研究带给更广泛的受众。具体地说，研究人员计划举办多个学习指南编写讲习班，组织一个专门针对小型工作组的在线研究平台，并创建一个用于谐波分析领域的YouTube频道。这些活动不仅旨在推进研究项目，也为学生和早期职业数学家提供了独一无二的机会。该项目的广泛目标是在公开问题上取得进展，如Kakeya猜想和Falconer距离猜想(关于确保一个分形集产生许多不同距离的维阈值)，开发新的工具来统一不同领域的思想，并探索它们在其他相关领域的进一步应用，如研究偏微分方程组和波动方程的解的正则性以及数论中丢番图方程的解的数目。有三个相互关联的长期目标：(A)了解包含在分形集中的几何构形(推广距离)的存在、数量和分布；(B)研究几何测度论中出现的算子的稀疏控制及其在有限点构形问题中的应用；(C)研究傅里叶分析中与Kakeya猜想相关的关键算子，如(加权)傅立叶限制/扩张算子、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，打开系统研究分形学精细几何结构的大门，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联系，并发现这些领域和外部的新联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Sparse bounds for the bilinear spherical maximal function

• DOI: 10.1112/jlms.12715
• 发表时间: 2022-03
• 期刊: Journal of the London Mathematical Society
• 作者: Tainara Borges;B. Foster;Yumeng Ou;J. Pipher;Zirui Zhou