Ryan Bushling, University of Washington

Thursday, October 26, 2023 - 12:30pm to 1:20pm

PDL C-401

Let \$E\subseteq \mathbb{R}^d\$. The *Falconer distance problem* is to relate the dimension of to the size of its *distance set* \$\Delta (E):=\{\|x-y\|:x,y\in E\}\$. After surveying the problem and some of the progress that has been made toward its solution, we consider the role of curvature and arrive at a bound on the dimension of pinned distance sets with respect to arbitrary norms—one that is sharp for all polyhedral norms. Its proof is grounded in *effective methods*, leading us to study *Kolmogorov complexity* and its surprising role in geometric measure theory. With the basic tools in hand, we prove the distance set estimate and sketch the ideas underlying its sharpness.