Brenier, building on earlier work by Arnold, introduced a celebrated variational formulation of the incompressible Euler equations of fluid dynamics. This formulation, after time discretization, is a prototype example of multi-marginal optimal transport, yet the question whether it exhibits mass-splitting (or equivalently, whether there exist solutions which are not of Monge form) has remained open.

In this talk -- after introducing Arnold's and Brenier's formulations of Euler and explaining the connection to optimal transport -- we resolve this question by giving a mass-splitting example in one spatial dimension. Moreover we present a related and very simple fully discrete example of mass-splitting which reveals a transparent underlying mechanism. Reference: arXiv:2601.02616 (2026)

Optimal transport provides an intuitive and robust way to compare probability measures with applications in many areas of mathematics.
This holds in particular for the Wasserstein-2 distance with its formal Riemannian structure.
While entropic regularization of optimal transport has several favourable effects, such as improved statistical sample complexity, it destroys this metric structure. The de-biased Sinkhorn divergence is a partial remedy, as it is positive, definite, and its sublevel sets induce the weak* topology. However, it does not satisfy the triangle inequality. We resolve this issue by considering the Hessian of the Sinkhorn divergence as a Riemannian tensor and study the induced distance. In this talk we outline the key steps of this construction, the corresponding induced notion of tangent space, some early results on the distance, and open directions for future work.

In this talk, we introduce acceleration matching, a data driven approach to multimarginal generative modelling. The main purpose of this method is to guarantee smoothness of the generative flow in time, a feature that is lost when sequentially applying standard flow matching. The main idea is learned an acceleration fields, instead of a velocity field, thus replacing Brownian motion with kinetic Brownian motion in the choice of interpolants.