In my most recent paper, I prove that when one of the marginal probabilities is suitably regular, the optimal transport problem is locally quantitatively stable with respect to perturbation of both marginals. My results also have direct implications for the statistical estimation of transport maps.
In my first paper, I gave the first quantitative bounds for almost-uniqueness of the dual optimal transport problem, as well as vastly generalising the sharp hypotheses for uniqueness to allow for lower-dimensional marginal measures. This was a result of a novel geometric insight regarding the structure of the set of optimal potentials.
During my M2 research internship, I studied the stability properties of the fully discrete optimal transport problem. This has intimate connections with linear programming, and taking this novel approach to stability led to the uniqueness results in my first paper.