CAREER: Analysis of Operators on Rough Sets

分形和自相似结构渗透到物理世界，从统计力学到材料科学。 该项目旨在分析一些基本的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联的场（例如它的引力场）是良好行为的（例如，具有有界的大小），那么我们可以推断出物体的几何形状吗？ 能量场的条件是否排除了身体表面具有分形结构？ 考虑的第二个主题涉及理解其频率具有自相似或分形行为的信号。 更确切地说，人们感兴趣的是信号可以从一个“小”的值集唯一地重建的程度。 主要研究者（PI）将把研究生和本科生纳入研究计划，为他们未来在STEM领域的职业生涯进行培训。 PI将组织年度本科生研究座谈会，将汇集东南各州对数学研究感兴趣的本科生，介绍他们的本科生研究，与其他有兴趣从事数学科学研究的人建立网络，并提供有关研究生水平的研究机会的信息。这个项目涉及到当问题的几何基础是粗糙的或分形理论 考虑的第一个问题是：什么可以推断出的措施，从知识的卡尔德龙-Zygmund运营商与它是有界的？ 在这种情况下，测度是否具有分形结构，或者它的支集是否包含在（可数个）适当维数的Lipschitz子流形中？这一基本问题在分析中已发现应用在变分法，研究自由边界，几何调和措施。 尽管在过去的三十年里进行了深入的研究，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还不发达，这是本提案旨在纠正的问题。 第二个问题是：一个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由它在“厚”子集上的值唯一地重构？ 虽然这个问题是不确定性原理的经典形式，但它还没有得到很好的理解，特别是在几个方面。这个奖项反映了NSF的法定使命，并被认为值得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来支持。

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.