Members’ Seminar

(joint work with Assaf Naor) A key problem in metric geometry asks: given metric spaces $X$ and $Y$, how well does $X$ embed in $Y$? In this talk, we will consider this problem for the case of the Heisenberg group and explain its connections to geometric measure theory and computer science. The Heisenberg group $H$ is a metric space that is hard to embed in $\mathbb{R}^n$. Cheeger and Kleiner used a metric derivative based on surfaces in $H$ to show that it also fails to embed into $L_1$. We will use uniform rectifiability to analyze surfaces in $\mathbb{H}$, provide sharp bounds on the distortion of embeddings of $H$ and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem.