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。这个项目试图分析一些基本的物理对象如何与分形结构相互作用。例如，假设我们知道一个与空间中一个物体相关的场（例如它的引力场）表现良好（例如，具有有限的大小），那么我们可以从物体的几何形状中推断出什么？场上的条件是否排除了身体表面具有分形型结构？第二个考虑的主题是理解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在多大程度上可以从一组“小”值中唯一地重建出来。首席研究员（PI）将把研究生和本科生纳入研究项目，为他们在STEM领域的未来职业生涯提供培训。PI将组织年度本科生研究专题讨论会，汇集东南各州对数学研究感兴趣的本科生，展示他们的本科研究，与其他对数学科学研究感兴趣的人建立网络，并提供有关研究生水平研究机会的信息。这个项目关注的是当问题背后的几何是粗糙的或分形的性质时算子的分析。考虑的第一个问题是：从与之相关的Calderon-Zygmund算子有界的知识中可以推导出关于测度的什么？在这种情况下，度量是否具有分形结构，或者它的支持必须包含在适当维数的（可数的）Lipschitz子流形中？这个分析中的基本问题在变分法、自由边界的研究和调和测度的几何中都有应用。尽管在过去的三十年里进行了大量的研究，但作为从现场分析条件到测量的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。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通过其在“厚”子集上的值唯一地重构？虽然这个问题是测不准原理的经典形式，但它还没有被很好地理解，特别是在几个维度上。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.