Analysis Seminar: Garrett Tresch (Texas A&M)
- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- 243 Altgeld Hall
- Date
- Dec 4, 2025 2:00 pm
- Speaker
- Garrett Tresch (Texas A&M)
- Contact
- Alexander Tumanov
- tumanov@illinois.edu
- Originating Calendar
- General Events - Department of Mathematics
Garrett Tresch (Texas A&M)
Transportation Cost Spaces and Stochastic Trees
Abstract:
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.