Mathematical Physics and Harmonic Analysis Seminar

Date: March 31, 2023

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Terry Harris, Cornell University

Title: Projections and intersections in the first Heisenberg group.

Abstract: In this talk, I will discuss some recent work on the Hausdorff dimension of projections and intersections in the first Heisenberg group. In Euclidean space, it is known that projections of sets onto k-dimensional subspaces almost surely do not decrease Hausdorff dimension, and that projections of sets of dimension greater than k have projections almost surely of positive k-dimensional area. It has been conjectured that these theorems extend to "vertical projections" in the Heisenberg group. This conjecture is still open, but was recently solved in a significant part of the range by Fassler and Orponen, using a "point-plate incidence" method. I will outline some of my recent work, which also uses the point-plate incidence method, and which proves the "positive area" part of the conjecture. One connection of this talk to harmonic analysis is that it uses the (endpoint) trilinear Kakeya inequality, which grew out of multilinear Fourier analysis inspired by the Fourier restriction and Kakeya conjectures.