CAREER: The Geometry of Fractals Meets Fourier Analysis

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

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.