Metrics and Singular Integrals

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

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.