CBMS Conference: Analysis, Geometry, and Partial Differential Equations in a Lower-Dimensional World

Florida State University, Tallahassee FL

Investigators

Abstract

The CBMS conference "Analysis, geometry, and PDEs in a lower-dimensional world" will take place in Florida State University on 17-21 of August 2020. During the conference, there will be discussed recent groundbreaking advances pertaining to connections between analytic, partial differential equations, and geometric properties of multi-dimensional sets. These problems have surprising and intricate applications across several areas of physics, materials science, and engineering, and the audience will have a unique chance to have a direct exposure to the ways in which seemingly abstract concepts and results at the cutting edge of pure mathematics can immediately influence state-of-the-art engineering of photonic devices and the physics behind them. When advertising the conference, we will make special efforts to encourage underrepresented groups to attend. The main speaker Dr. Mayboroda is a well-respected expert, and interacting with such an accomplished female researcher is important for junior participants and will boost their confidence in pursuing careers in mathematics. The last few years have marked groundbreaking advances pertaining to connections between analytic, PDE, and geometric properties of (n-1)-dimensional sets in R^n. This includes the first converse to the 1916 F. and M. Riesz theorem regarding absolute continuity of harmonic measure with respect to Hausdorff measure and an array of beautiful characterizations of rectifiability, from boundedness of the Riesz transform on L2 (a much thought-after solution of the David-Semmes conjecture) to Carleson measure estimates and similar properties of harmonic functions. We will quickly survey those and concentrate on a mysterious world of lower-dimensional sets when virtually none of these characterizations is available and search for meaningful and powerful analogues revealed some completely new phenomena: a prominent role of degenerate PDEs, new non-linear integration operators in place of classical singular integrals, a new smooth distance function, an existence, for every lower-dimensional set, of a special "elliptic" operator whose harmonic measure is absolutely continuous with respect to the Hausdorff measure on this set. More information about the conference can be found on the website: https://cbms2020.math.fsu.edu 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.