Metrics and Singular Integrals

数学中最基本的概念之一是距离的概念。在不同的语境中，不同的距离概念是相关的。例如，当平行停车时，目标是将汽车水平移动到停车位，但这比将汽车向前或向后行驶相同距离要花费更多的时间。因此，很自然地，停车时这样的水平距离比直线行驶时的类似距离要远。这种类型的距离度量被称为“亚黎曼”距离，在数学的几个领域起着决定性的作用。这个项目涉及这些距离的概括及其在开放性问题上的应用。首席研究员将考虑谐波分析中的两个重要问题，其中这些距离是关键组成部分，目标是对这些距离有更深入的了解，以在开放问题上取得进展。研究这些广义次黎曼距离的工具是从几个不同的数学领域借来的：微分方程、几何和谐波分析。然后将它们应用于其他数学领域的问题：几个复变量和奇异积分。因此，该项目汇集了来自几个数学领域的想法，以解决重要的开放式问题。通过定义亚黎曼度量的广义概念，本项目介绍了两个主要的探究领域，其中几何方法可以应用。第一个领域是研究一类有限型域上的Bergman和Szego投影。为了研究这一点，首席研究员计划在全纯范畴中发展一个广义的尺度概念。这是纽兰德-尼伦堡定理的一个定量版本。这是亚黎曼几何定量理论的全纯类比，已被证明在分析和几何的许多领域是有用的。第二个研究领域是提供奇异Radon变换的测量理论方法。奇异氡变换是一个活跃的研究领域，已经有50年了，尽管工作主要集中在光滑的下垫面。这个项目介绍了奇异Radon变换固有的几何，通过它可以从纯粹的测量理论的角度研究这些算子。该项目旨在使用该框架来解决有关曲线的问题，其中某些类型的曲率消失到无限阶，这是一个已被证明无法用当前方法解决的主题。该项目还探讨了在反问题中出现的一种新的微分方程。在某些情况下，这个微分方程表现出存在性而不是唯一性，而在另一些情况下，它表现出唯一性而不是存在性。这个微分方程的研究将是一个本科研究项目的主题，并将使用开发的方法来研究一个特殊情况。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。

One of the most basic concepts in mathematics is the notion of distance. In different contexts, different notions of distance are relevant. For example, when parallel parking, the goal is to move a car horizontally into a parking space, but this takes a lot more time than driving the car the same distance forwards or backwards. It is natural, therefore, to treat such a horizontal distance in parking as being further than the comparable distance in driving straight forward. Distance measures of this type are known as "sub-Riemannian" distances and have played a decisive role in several areas of mathematics. This project concerns generalizations of such distances and their applications to open questions. The principal investigator will consider two important questions in harmonic analysis where such distances are a key component, with the goal of using a deeper understanding of these distances to make progress on open questions. The tools to study these generalized sub-Riemannian distances are borrowed from several different areas of mathematics: differential equations, geometry, and harmonic analysis. They are then applied to questions from other areas of mathematics: several complex variables and singular integrals. The project thus brings together ideas from several areas of mathematics to attack important open questions.By defining a generalized notion of a sub-Riemannian metric, this project introduces two primary areas of inquiry where geometric methods can be applied. The first area is the study of the Bergman and Szego projections on a class of domains of finite type. To study this, the principal investigator plans to develop a generalized notion of scaling in the holomorphic category. This is presented as a quantitative version of the Newlander-Nirenberg theorem. This is the holomorphic analog of the quantitative theory of sub-Riemannian geometry, which has proven to be useful in many areas of analysis and geometry. The second area of study is providing a measure-theoretic approach to singular Radon transforms. Singular Radon transforms have been an active area of research for fifty years, though work has mostly focused on results where the underlying surfaces are smooth. This project introduces a geometry that is intrinsic to a singular Radon transform, through which one can study these operators from a purely measure-theoretic standpoint. The project aims to use this framework to address questions regarding curves where certain types of curvature vanish to infinite order, a topic which has proven unamenable to current methods. The project also explores a new kind of differential equation that arises in inverse problems. In some settings, this differential equation exhibits existence but not uniqueness, while in others it exhibits uniqueness but not existence. Study of this differential equation will be the topic of an undergraduate research project and will use methods developed to study a special case.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.