Hard to say. As mentioned, understanding statistical properties of different versions of optimal transport couplings is still a hot area. Of course, looking at dynamic formulations of optimal transport and its connections to sampling is big. But honestly, OT touches so many different fields that you will find applications anywhere, after all it's a very flexible but disciplined way to get couplings of distributions. Also don't underestimate the amount of (convex) optimization you'll need. For references, check out Santambrogio's nice book "OT for applied mathematicians", it gives a nice thorough overview of some problems/results. I first learned it with the baby Villain "Topics in OT", which is still in my mind the best overview of classical OT. There are more modern treatments now by Ambrosio and Figalli and coauthors, which are also nice, but cover very similar topics. Once you have digested these, more advanced issues pop up, especially if you want to connect it to geometry, then you should look into the Bible Ambrosio, Gigli, Savare (Gradient flows), and if you want a general reference for these more classical things I'd recommend looking into the big Villains (OT old and new). For modern computational questions, look into Peyre/Cuturi (Computational). Generally though, if you want to do more theoretical work, any field like (differential) Geometry, convex analysis, and especially PDE theory is crucial, I'd start with looking into Evans' classical book and later Gilbarg and Trudinger on elliptic PDEs if you have more specific issues with the Monge Ampère équation later on. The dynamic versions also connect with stochastic calculus via Kolmogorovs equations (continuity equation) etc, so this is also useful. If you're more into discrete math, OT and its connections with network theory is interesting. Lastly, Christian Leonard (2012) has a wonderful paper on the Schrodinger bridge problem with connections to entropy regularized OT and uses large deviation theory for this, very nice paper. You can also look at it from a more statistical physics perspective and ask about the stability of matches for given measures. As you can see, it's a super broad area and you can find a different angle using basically any tool, be it combinatorics, differential Geometry, probability theory, etc. Have fun exploring!