Multiscale Methods in Quantitative Geometry

该NSF奖为一个项目提供资金，该项目旨在开发新的方法来处理具有许多不同比例的复杂几何体的对象。海岸线、云层和树叶都是自然界中出现的此类物体的例子。著名的海岸线悖论指出，由于海岸线在许多不同的尺度上是如此崎岖，它没有定义的长度。PI计划开发和研究新的方法来建造这样的多尺度物体，并将它们分解成易于处理的碎片。分解物体的新方法通常会带来数学上的巨大进步，在数学中，简单的问题可以有复杂的、多尺度的解决方案。该项目的三管齐下的方法包括研究如何分解复杂的物体，用简单的碎片建造复杂的物体，以及测量多尺度的复杂性。除了这项研究，PI还将通过咨询、研讨会、迷你课程和阅读小组来培训研究生和博士后，并在他的网页上发布材料供公众访问。该项目旨在研究与地图和具有多尺度结构的表面(即Lipschitz和Holder地图或分形图)有关的度量几何、几何测度论和调和分析问题，这些问题很难用仿射地图或平面来逼近。一个焦点是研究有非光滑解但没有光滑解的几何问题，如纳什嵌入定理。这些问题的解决方案通常使用多尺度或自相似结构来打破平滑地图必须满足的规则，而这个项目的目的之一就是了解这种现象发生的时间和原因。另一个关注点是量化和统一的可纠正性。一致可纠正性是欧氏空间中研究奇异积分和几何测度论的有力工具。最近的工作已经使得在海森堡群中找到表面的定量可纠正性的精确界限成为可能，该项目将探索将一致可纠正性的概念扩展到海森堡群并使用它们来解决欧几里德空间和海森堡群中的几何问题的可能性。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。

This NSF award provides funding for a project to develop new methods for working with objects with complicated geometry at many different scales. Coastlines, clouds, and leaves are examples of such objects that occur in nature. The famous coastline paradox states that since a coastline is so rough at so many different scales, it has no defined length. The PI plans to develop and study new methods to build such multiscale objects and to break them down into easily handled pieces. New ways to decompose objects often lead to great advances in mathematics, where simple problems can have complicated, multiscale solutions. The project’s three-prong approach includes studying ways to break down complex objects, build complex objects out of simple pieces, and to measure multiscale complexity. In addition to the research, the PI will train graduate students and postdocs in the techniques developed by this project through advising, seminars, minicourses, and reading groups and disseminate material on his webpage for public access.The project aims to study problems in metric geometry, geometric measure theory, and harmonic analysis related to maps and surfaces with multiscale structure, that is, objects like Lipschitz and Holder maps or fractals, that are hard to approximate by affine maps or by planes. One focus is the study of geometric problems that have non-smooth solutions but no smooth solutions, such as the Nash Embedding Theorem. Solutions to these problems often use multiscale or self-similar structure to break rules that smooth maps have to satisfy, and one aim of this project is to understand when and why this phenomenon occurs. Another focus is quantitative and uniform rectifiability. Uniform rectifiability has been a powerful tool for studying singular integrals and geometric measure theory in Euclidean space. Recent work has made it possible to find sharp bounds on the quantitative rectifiability of surfaces in the Heisenberg group, and this project will explore the possibility of extending notions of uniform rectifiability to the Heisenberg group and using them to solve problems in the geometry of Euclidean space and the Heisenberg group.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.