The is a followup on a previous BMSG talk given by the speaker last semester. We will discuss recent developments and address some of the questions raised during that talk.

Given a finite metric space $M$ one can define the corresponding transportation cost space $\mathcal{F}(M)$ as the normed linear space of transportation problems on $M$. Roughly speaking, a transportation problem can be understood as a supply/demand configuration on $M$ where the norm of the transportation problem is the lowest cost of transporting goods from locations with a surplus to those with shortages. In this setting, an important line of research is studying the relation between transportation cost spaces and $\ell_1$. A core problem posed by S. Dilworth, D. Kutzarova, and M. Ostrovskii is finding a condition on a metric space $M$ equivalent to $\mathcal{F}(M)$ being Banach-Mazur close to $\ell_1^N$ in the corresponding dimension. In this talk, we discuss our recent work where a partial solution to this problem is obtained by examining tree-like structure within the underlying metric space. Tangential to this result, we have also developed a new technique that, potentially, could serve as a step toward a complete solution to the problem of Dilworth, Kutzarova, and Ostrovskii. We conclude by discussing two applications of this technique: finding an asymptotically tight upper bound of the $\ell_1^N$-distortion of the Laakso graphs, and proving that finite hyperbolic approximations of doubling metric spaces have uniformly bounded $\ell_1^N$-distortion. This is joint work with Ruben Medina.