While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting because it explicitly investigated the geodesics of Wasserstein space to produce solutions to a type of parabolic PDE.
I have some questions and I know this is probably something I should reach out to a researcher about, but I'm in another very different field (computational neuroscience) and few people around me that have the ability to address these questions with sufficient detail -- so I figure I'd try my luck and ask here.
Are there any open questions that would require a detailed study of the geodesics in Wasserstein space to understand the existence of PDE solutions (as in Agueh's thesis) or are these results mostly covered now by the general theory in Ambrosio's book on Gradient Flows?
Are there any open questions about analysis on Wasserstein space itself? I think one game people try to play is defining Laplacians on this space. Anything else?
Are there any PDE problems that have not yet been looked at through the Wasserstein perspective?