CAREER: Analysis of Operators on Rough Sets

职业：粗糙集算子分析

基本信息

批准号：
2049477
负责人：
Benjamin Jaye
金额：
$ 40万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2049477&HistoricalAwards=false
关键词：
CAREER Analysis Operators Rough Sets

项目摘要

Fractals and self-similar structures permeate the physical word, from statistical mechanics to material sciences. This project seeks to analyze how some fundamental physical objects interact with fractal structures. For instance, suppose we know that a field associated to a body in space (for instance its gravitational field) is well-behaved (say, has bounded magnitude), then what can we infer about the geometry of the body? Does the condition upon the field preclude the surface of the body from having a fractal type structure? A second topic considered concerns understanding signals whose frequencies have a self-similar or fractal behavior. More precisely, one is interested in the extent to which the signal can be reconstructed uniquely from a "small" set of values. The principal investigator (PI) will incorporate both graduate and undergraduate students into the research program, training them for future careers in STEM fields. The PI will organize annual undergraduate research symposia that will bring together undergraduate students interested in mathematical research across southeastern states to present their undergraduate research, build a network with others interested in pursuing research in the mathematical sciences, and provide information about research opportunities at the graduate level.This project concerns the analysis of operators when the geometry underlying the problem is rough or fractal in nature. The first question considered is: What can be deduced about a measure from the knowledge that a Calderon-Zygmund operator associated to it is bounded? Under these circumstances, can the measure have a fractal structure, or must its support be contained in (a countable number of) Lipschitz submanifolds of appropriate dimension? This basic question in analysis has found applications in the calculus of variations, the study of free boundaries, and the geometry of harmonic measure. Despite intense study over the last thirty years, the tools that serve as a bridge from the analytic condition on the field to the geometric structure of the measure are currently underdeveloped, something that this proposal aims to rectify. The second question is: How sparse must the support of the Fourier transform of a function be to ensure that the function can be uniquely reconstructed by its values on a "thick" subset? Although this question is a classical form of the uncertainty principle, it is not yet well understood, especially in several dimensions.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.

分形和自相似结构渗透到物理世界中，从统计力学到材料科学。该项目旨在分析一些基本物理对象如何与分形结构相互作用。例如，假设我们知道与空间中的物体相关的场（例如它的引力场）表现良好（例如，具有有限的大小），那么我们可以推断出物体的几何形状吗？场上的条件是否阻止物体表面具有分形型结构？考虑的第二个主题涉及理解频率具有自相似或分形行为的信号。更准确地说，人们感兴趣的是可以在多大程度上从“小”值集唯一地重建信号。首席研究员 (PI) 将把研究生和本科生纳入研究计划，为他们提供 STEM 领域未来职业培训。 PI 将组织年度本科生研究研讨会，将东南部各州对数学研究感兴趣的本科生聚集在一起，展示他们的本科生研究，与其他有兴趣从事数学科学研究的人建立网络，并提供有关研究生研究机会的信息该项目涉及当问题背后的几何形状本质上是粗糙或分形时的算子分析。考虑的第一个问题是：根据与测度关联的 Calderon-Zygmund 算子是有界的知识，可以推断出什么关于测度的信息？在这种情况下，测度是否可以具有分形结构，或者它的支持必须包含在（可数个）适当维数的利普希茨子流形中？这一基本分析问题已在变分计算、自由边界研究和调和测度几何中得到应用。尽管过去三十年进行了深入的研究，但作为从现场分析条件到测量几何结构的桥梁的工具目前尚未开发出来，本提案旨在纠正这一点。第二个问题是：函数的傅里叶变换的支持必须有多稀疏才能确保该函数可以通过其在“厚”子集上的值唯一地重建？尽管这个问题是不确定性原理的经典形式，但尚未得到很好的理解，特别是在几个方面。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。