Optimal Measures and Point Configurations: A Harmonic Analysis Approach

Optimal point configurations and distributions arise in numerous areas in science applications including sampling of data, signal processing, equilibria of particles and charges, crystal formation, population distribution, discretization of surfaces, and coding theory. They are also extremely useful in numerical analysis and in fields that call for numerical computation of high dimensional integrals such as finance, often yielding better results than Monte Carlo methods. Studied within the framework of this project will be optimal point configurations and measures through the prism of harmonic analysis. The project will also study fundamental questions regarding uniform distribution of energy minimizers; determination of optimal distributions; discrete clustering phenomena; the interplay between geometry and uniform distribution; connections between discrepancy and energy minimization. In addition, the project will study related questions in applied harmonic analysis, compressed sensing, and frame theory. The results and topics of this project will be disseminated through publications and presentations, and young mathematicians will actively be mentored.

The quality of a point distribution may be expressed or measured in a variety of different ways depending on the problem at hand. Examples include lattices, cubature formulas, codes, packings, and random point processes. This project is focused on two central inter-connected concepts, both of which quantify the uniformity of distributions: discrepancy and energy minimization. The former compares a given distribution with the uniform measure by looking at the errors on a certain class of test sets or test functions. The latter interprets points as 'particles', which interact according to a certain potential. While progress in these subjects often relies on methods and ideas of harmonic analysis, many of these connections have only been discovered recently. This leads to natural cross-fertilization: analysis provides the necessary tools, and at the same time it is enriched with new questions. The project will study several grand topics related to discrepancy and energy optimization including clustering of minimizers, attractive-repulsive energies and their relations to signal processing and applied harmonic analysis (tight frames, SIC-POVMs, mutually unbiased bases), discrete and metric geometry (equiangular lines, distance integrals), tessellations of spheres and their relation to discrepancy, one-bit compressed sensing, and embedding of metric spaces, direct interplay between energy and discrepancy and its applications to various questions and conjectures, the long standing open question of the exact asymptotics of discrepancy with respect to axis-parallel rectangles, as well as sibling questions in harmonic analysis, approximation and probability theory. Ubiquitous connections between the subjects bring together a variety of questions into an integral project.

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.