CAREER: Analysis of Operators on Rough Sets

从统计力学到材料科学，分形学和自相似结构无处不在。这个项目试图分析一些基本的物理物体是如何与分形结构相互作用的。例如，假设我们知道一个与空间中的物体相关的场(例如，它的引力场)是良好的(比如说，具有有界的量级)，那么我们可以从物体的几何形状中推断出什么呢？场上的条件是否排除了身体表面具有分形型结构？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。更准确地说，人们感兴趣的是信号可以在多大程度上从一小组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计划，为他们未来在STEM领域的职业生涯进行培训。PI将组织一年一度的本科生研究研讨会，让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成果，与其他有兴趣从事数学科学研究的人建立一个网络，并提供关于研究生水平研究机会的信息。这个项目涉及当问题背后的几何图形具有粗糙或分形性质时，对操作员的分析。考虑的第一个问题是：从与其相关的Calderon-Zygmund算子是有界的知识中可以推导出关于度量的什么？在这种情况下，度量是否具有分形结构，或者它的支撑点是否必须包含在(可数个)适当维度的Lipschitz子流形中？分析中的这个基本问题在变分法、自由边界的研究和调和测量的几何中都有应用。尽管在过去的三十年里进行了密集的研究，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，这一建议旨在纠正这一点。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它在一个“粗”子集上的值唯一地重建？虽然这个问题是不确定性原则的经典形式，但它还没有被很好地理解，特别是在几个维度上。这个奖项反映了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。

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.