On Problems in and Connections between Analysis, Geometry and Combinatorics

该项目关注调和分析，几何测量和组合学中各种思想之间的相互作用，并以有限点配置的概念为中心应用于数据科学。基本的问题是，给定一个足够大的点集（这里的“大”取决于情况），它是否包含等边三角形，链或给定类型的另一种配置。来自不同数学领域的思想的相互作用，有时是由大数据研究中的特定问题所激发的，将在整个过程中得到强调。与过去的情况一样，由此产生的共生关系将继续创造一个活跃的概念网络，导致相应领域的技术和思想之间的富有成效的相互作用。在这条研究线产生的想法将用于运行本科生暑期研究计划。研究生和博士后研究人员将参与这项工作的各个方面。关键的基本主题是法尔科纳距离猜想，它说，如果一个紧凑的子集欧几里德空间的豪斯多夫维数至少一半的环境尺寸，然后勒贝格措施的距离集是积极的。这项工作将继续进行，目的是最终的猜想使用的方法产生的去耦和算术的考虑相结合。另一个目标是建立一个完整的图片的配置问题的主要研究者以前研究的合作者，在显示一个类似的副本配置足够高，但不是太高，一个复杂的水平可以发现在一组足够大的豪斯多夫维数在欧几里得空间和黎曼流形。在研究这些问题时开发的技术在数据科学中的降维研究中非常有用。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进行评估，被认为值得支持。

This project concerns the interactions among a variety of ideas in harmonic analysis, geometric measure, and combinatorics, with applications to data science, centered around the concept of finite point configurations. The basic question is, given a sufficiently large set of points (here ‘large’ is situation dependent), does it contain an equilateral triangle, a chain, or another configuration of a given type. The interplay of ideas from different areas of mathematics, sometimes motivated by specific questions in the study of large data, will be emphasized throughout. As was the case in the past, the resulting symbiosis will continue creating a live network of concepts that leads to the productive interplay among the techniques and ideas in the corresponding fields. The ideas generated in this line of research will be used to run summer research programs for undergraduate students. Graduate students and postdoctoral researchers will be involved in all aspects of this work. The key underlying theme is the Falconer distance conjecture which says that if a compact subset of Euclidean space has the Hausdorff dimension at least half the ambient dimension, then the Lebesgue measure of the distance set is positive. This work will be continued, aiming towards the ultimate conjecture using a combination of methods arising from decoupling and arithmetic considerations. Another goal is to establish a complete picture of the configuration question the principal investigator has previously studied with collaborators, in showing that a similar copy of configuration of sufficiently high, but not too high, a level of complexity can be found in a set of sufficiently large Hausdorff dimension in Euclidean space and Riemannian manifolds. The techniques that have been developed while studying these questions have proven extremely useful in the investigation of dimension reduction in data science. This avenue will be explored further.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.