Conference on Harmonic Analysis and Fractal Sets

Ohio State University, The, Columbus OH

Investigators

Abstract

The award supports a March 2023 conference at the Ohio State University (Columbus campus) on the topics of harmonic analysis and fractal geometry. Fractals are geometric objects which are highly irregular, exhibiting detail at arbitrarily small scales. Such sets arise naturally in mathematics, and have applications to a diverse array of problems in science and engineering. Harmonic analysis emerged in the late 20th century as a tool for the study of the geometry of fractal sets, and in return fractal geometry has motivated many interesting questions in harmonic analysis. This event brings together researchers to find new synergies between existing methods and to formulate new approaches to build on recent progress at the intersection of these fields. The focus of the conference is on geometric properties of fractal sets using tools from harmonic analysis. A classical problem in this area is the Falconer distance problem, which asks how large the Hausdorff dimension of a compact set must be to ensure that it determines a positive Lebesgue measure set of pairwise distances. More generally, one may inquire when high dimensional sets determine more complicated patterns, leading to a host of fascinating questions about finite point configurations. These problems can be studied via bounds for convolution operators, bringing to bear tools from harmonic analysis. Another classical example is the Kakeya problem, which asks when sets containing a line segment in every direction must have full Hausdorff dimension. The Kakeya problem, in turn, is closely tied to Fourier restriction theory. A recurring theme throughout the conference is the relationship between the Fourier transform and notions of structure such as Hausdorff dimension. https://u.osu.edu/hafs2023/ 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.