On Problems in and Connections between Analysis, Geometry and Combinatorics

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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.