On Problems in and Connections between Analysis, Geometry and Combinatorics

这个项目涉及调和分析、几何测量和组合学中各种思想之间的相互作用，以及在数据科学中的应用，以有限点构型的概念为中心。基本的问题是，给出一个足够大的点集(这里的“大”是取决于情况)，它是否包含一个等边三角形、链或其他给定类型的构型。来自不同数学领域的思想的相互作用，有时是在研究大数据的特定问题的推动下，将贯穿始终。正如过去的情况一样，由此产生的共生将继续创造一个活的概念网络，导致相应领域的技术和想法之间产生富有成效的相互作用。在这一系列研究中产生的想法将用于为本科生开展暑期研究项目。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Falconer距离猜想，该猜想指出，如果欧几里德空间的紧致子集的Hausdorff维度至少是环境维度的一半，则该距离集的勒贝格度量为正。这项工作将继续进行，目标是使用解耦和算术考虑产生的方法组合来实现最终的猜想。另一个目标是建立主要研究者先前与合作者研究的构形问题的完整图景，以表明在欧几里德空间和黎曼流形中的一组足够大的Hausdorff维中可以找到足够高但不太高的构形的类似副本。在研究这些问题时开发的技术已被证明在数据科学中的降维研究中非常有用。这个奖项反映了NSF的法定使命，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。

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.