Wednesday November 19th
14:00 – 15:00 Thomas Borsoni : Quantum optimal transport for observability in crystals (joint work with
Virginie Ehrlacher and François Golse) – Slides
The aim of this talk is to present a quantitative observability inequality for the von Neumann equation which describes the evolution of the state of an infinite number of electrons in a crystalline material,
uniform in the reduced Planck constant. Following the method of [Golse, Paul, 2022] where a similar result was proved for a system composed of a finite number of electrons, the approach relies on a stability argument between the quantum (von Neumann) and classical (Liouville) dynamics and uses an optimal transport-like pseudo-distance between quantum and classical densities. Our contribution aims at extending this approach in order to treat systems with an infinite number of electrons yields in a periodic setting, relying on the Bloch decomposition, notions of periodic Schrödinger coherent state, periodic Töplitz operator and periodic Husimi densities.
15:00 – 16:00 David Bourne : Semi-discrete convex order: From convex geometry to the steel industry and back again – Slides
In this talk I will describe an application of semi-discrete optimal transport theory in computational materials science, and how in turn this very applied problem led to new pure results in convex geometry, including necessary and sufficient conditions for the existence of convex partitions (Laguerre tessellations) with cells of prescribed volumes and barycentres. Along the way we’ll make links with weak optimal transport, convex order, and higher-order numerical methods for Wasserstein gradient flows. This talk is based on joint work with Thomas Gallouët, Quentin Mérigot, Andrea Natale, Mason Pearce and Steve Roper.
16:30 – 17:30 Gudmund Pammer : Towards a Brenier theorem on (P2(…P2(H)…), W2) and adapted transport
In this talk we discuss recent advances towards Brenier-type results on iterated Wasserstein spaces
P2N(H) = P2(…P2(H)…) over a separable Hilbert space H. We construct a full-support probability measure Λ in P2N(H) that is transport-regular. A key ingredient is a novel characterisation of optimal couplings on P2(P2(H)) via convex potentials on the Lions lift, and, more generally, on P2N(H) via a new adapted variant of the Lions lift that respects the nested structure. A primary motivation comes from adapted transport: here, our results yield a first Brenier theorem for the adapted Wasserstein distance.
This talk is based on joint work with Mathias Beiglböck and Stefan Schrott.
17:30 – 18:30 Averil Aussedat : Local characterization of tangent plans that are martingale plans – Slides
Lott showed that if mu is the Hausdorff measure on a C2 submanifold, then the optimal transport plans that are martingale plans can split mass only in the normal directions to the manifold. This talk provides the generalization of this result to any measure with finite second moment : given a measure, we identify d+1 sets behaving as k-dimensional DC manifolds, and show that if an optimal transport plan with first marginal mu is, additionally, a martingale plan, then it must be concentrated on the normal directions to these sets. A converse is given if one allows the plans to be merely tangent, i.e. limits of optimal plans in L2(mu).
Thursday November 20th
9:00 – 10:00 Aurélien Alfonsi : Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case (Joint work with B. Jourdain) – Slides
Abstract: We first show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension d, the quadratic Wasserstein projection of a probability measure μ on the set of probability measures dominated by ν in the convex order is non-expansive in μ and Hölder continuous with exponent 1/2 in ν. When μ and ν are Gaussian, we check that this projection is Gaussian and also consider the quadratic Wasserstein projection on the set of probability measures ν dominating μ in the convex order. In the case when d ≥ 2 and ν is not absolutely continuous with respect to the Lebesgue measure where uniqueness of the latter projection was not known, we check that there is always a unique Gaussian projection and characterize when non Gaussian projections with the same covariance matrix also exist. Still for Gaussian distributions, we characterize the covariance matrices of the two projections.
10:30 – 11:30 Marco Veneroni : Generalized Wasserstein barycenters – Slides
We study the existence of Wasserstein barycenters for a continuous distribution of probability measures on a Hilbert space, allowing for negative weights. Barycenters are characterized, as usual, as minimizers of a functional on P2(H). In the case where the positive-weight measure is a singleton, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the L2-barycenter of the quantiles on the cone of nonincreasing functions in L2(0,1). Finally, we show that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure. This talk is based on a recent joint work with Francesco Tornabene and Giuseppe Savaré.
11:30 – 12:30 Yating Liu : Functional convex order for the McKean-Vlasov processes with and without common noise – Slides
Abstract: This presentation introduces functional convex order results for two McKean–Vlasov processes X and Y, with and without common noise. Under suitable convexity and monotonicity assumptions on the dynamics of X, together with a matrix order condition on the volatility, we show that the convex order of the initial distributions can be propagated to the entire paths of the processes. Specifically, for any convex functional F defined on the path space, the expected value of F(X) is less than or equal to that of F(Y). Similarly, for any convex functional G defined on the product space involving the process path and its marginal distribution, the expected value of G applied to X and its law does not exceed that of G applied to Y and its law, under appropriate conditions. In addition, we present applications of these results to mean-field control and mean-field games.
14:00 – 15:00 Charles Bertucci : Some “geometric” tools to study Hamilton-Jacobi-Bellman equations on the Wasserstein space – Slides
Hamilton-Jacobi-Bellman equations on the Wasserstein space arise naturally when studying optimal control of probability measures. Due to the particular nature of the Hamiltonian in such equations, standard techniques involving smooth test functions typically fails. I will show how more geometric tools, which are not so smooth in such an infinite dimensional setting can help with several problems surrounding such equations. I will focus on the super-differential of the squared distance and suitable approximation of it, as well as on parallel transport and entropy methods.
15:00 – 16:00 Gabriele Todeschi : Metric extrapolation and signed Wasserstein Barycenters
Wasserstein barycenters are a very natural and popular way of averaging probability distributions, with numerous applications in data science and image processing. The model is conceived as a Fréchet mean, namely a variational problem where one minimizes the positively weighted sum of the Wasserstein distances squared from N given measures. As this is a convex optimization problem, everything is well understood from a computational and theoretical perspective. The situation is completely different for so-called signed Wasserstein barycenters, where the weights are allowed to be negative. In a recent work, we addressed the case with only two measures, which we called metric extrapolation, for which we now have a solid understanding. In this case in fact the problem enjoys some hidden convexity properties, which allow to rewrite it in equivalent convex formulations and to fully characterize (unique) minimizers. In particular, the problem can be recast as an instance of weak optimal transport. Our aim here is to generalize this work to the case of more general transport costs. This extension relies on a geometric property of the ground space, nonnegative cross-curvature, which provides favorable properties both for the metric extrapolation and its dual formulation. A second key ingredient is a characterization of c-concave functions via a Toland-like duality result. As a specific example, we will consider the Wasserstein over Wasserstein space, which may provide a suitable convex relaxation to the signed Wasserstein barycenters problem. This talk is based on joint work with Thomas Gallouët and Andrea Natale.
16:30 – 17:30 Marc Lambert : Projected Wasserstein gradient flow: Bures gradient flow & Gaussian-particles – Slides
Estimating general multimodal distributions is challenging, but a powerful approach is to view this distribution as the asymptotic limit of a Fokker–Planck equation (FPE). While many traditional methods rely on MCMC to sample on this limiting distribution, an alternative is to directly approximate the solution of the FPE. Since Otto’s seminal work, the solution of the Fokker–Planck equation can be interpreted as the minimization of a relative entropy functional in Wasserstein space, leading to the Wasserstein gradient flow. Following the projected gradient flow framework, we approximate this flow by projecting its gradient onto the tangent space of a finite-dimensional parametric manifold. In particular, choosing the Bures manifold leads to the Bures–Wasserstein gradient flow, which exhibits fast convergence properties for log-concave distribution. Finally, this framework can be lifted to a Gaussian-particle representation, enabling efficient approximation of complex multimodal distributions using multiple particles.
The presentation will be based on the following articles https://arxiv.org/abs/2205.15902 and https://arxiv.org/abs/2506.13613.
17:30 – 18:30 Adrien Vacher : Convergence of a variable metric scheme with application to optimal transport – Slides
In this presentation, we study the convergence of a proximal optimization scheme with variable metric. We prove linear convergence under convexity and relative smoothness and exponential convergence under additional relative strong convexity. After showing that the semi-dual is indeed relatively smooth and strongly convex for a well-chosen variable metric under suitable assumptions, we apply this scheme to compute quadratic optimal transport potentials. Notably, this scheme does not require the measures to be supported on a grid and scale well with the dimension.
Friday November 21st
9:00 – 10:00 Julio Backhoff : Bass Martingales: An Overview – Slides
Bass martingales are increasing, pointwise transformations of Brownian motion. Furthermore, they can be interpreted as the adapted Wasserstein projection of Brownian motion onto the set of martingales meeting initial and terminal marginals.
In this overview talk we discuss the structure and existence of Bass martingales, their connection to weak optimal transport (together with its duality theory), and some recent computational recipes that have been proposed in order to find them.
10:30 – 11:30 Eloi Tanguy : Computing Barycentres of Measures for Generic Transport Costs
Wasserstein barycentres represent average distributions between multiple probability measures for the Wasserstein distance. The numerical computation of Wasserstein barycentres is notoriously challenging. A common approach is to use Sinkhorn iterations, where an entropic regularisation term is introduced to make the problem more manageable. Another approach involves using fixed-point methods, akin to those employed for computing Fréchet means on manifolds. The convergence of such methods for 2-Wasserstein barycentres, specifically with a quadratic cost function and absolutely continuous measures, was studied by Alvarez-Esteban et al.. In this paper, we delve into the main ideas behind this fixed-point method and explore how it can be generalised to accommodate more diverse transport costs and generic probability measures, thereby extending its applicability to a broader range of problems. We show convergence results for this approach and illustrate its numerical behaviour on several barycentre problems.
11:30 – 12:30 Guillaume Carlier : Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization – Slides
Weak optimal transport is a nonlinear version of the classical mass transport of Monge and Kantorovich which has received a lot of attention since its introduction by Gozlan, Roberto, Samson and Tetali, ten years ago. In this talk, I will address weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet. This talk is based on a recent joint work with Hugo Malamut and Maxime Sylvestre