Multiscale Methods in Quantitative Geometry

该NSF奖项为一个项目提供资金，该项目旨在开发在许多不同尺度下处理复杂几何形状物体的新方法。云层、树叶和云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都是如此粗糙，因此它没有定义的长度。PI计划开发和研究新的方法来构建这种多尺度对象，并将它们分解为易于处理的部分。分解对象的新方法通常会导致数学的巨大进步，简单的问题可以有复杂的多尺度解决方案。该项目的三管齐下的方法包括研究分解复杂对象的方法，用简单的部件构建复杂对象的方法，以及测量多尺度复杂性的方法。除研究外，研究所所长亦会透过提供意见、举办研讨会、小型课程、阅读小组及在其网页上发放资料，以培训研究生及博士后，使他们掌握本计划所发展的技术。本计划旨在研究与多尺度结构的地图及表面有关的度量几何、几何测度理论及调和分析等问题，像Lipschitz和保持器映射或分形这样的对象，很难用仿射映射或平面来近似。一个重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光滑映射必须满足的规则，本项目的一个目的是了解这种现象何时以及为什么会发生。另一个焦点是定量和统一的可校正性。一致可求长性是研究欧氏空间奇异积分和几何测度理论的有力工具。最近的工作使得在海森堡群中找到曲面的可求长性的精确界限成为可能，该项目将探索将统一可求正性的概念扩展到海森堡群的可能性，并利用它们来解决欧几里德空间和海森堡群的几何问题。该奖项反映了NSF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。

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.