4 Mai 2026: Charlotte Dietze et Alessandro Pinzi

lundi 4 mai 2026, 10:00 → 12:00 Europe/Paris

Salle Maryam Mirzakhani (201) (Institut Henri Poincaré)

10h-11h: Charlotte Dietze

 

Title: Spectral theory for singular Riemannian metrics

Abstract: We prove eigenvalue asymptotics and concentration of eigenfunctions of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of sound waves in gas planets. The talk is based on joint works with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat, and with Larry Read.


11h-12h: Alessandro Pinzi

Title:  Optimal transport between laws of random probability measures

 

Abstract: In this talk, I will present recent results concerning the Optimal Transport problem between (laws of) random measure, i.e. probability measures over probability measures. Given a lower semicontinuous cost function cost: X1 × X2 → [0,+∞], we can associate to it the optimal transport cost C : P(X1) × P(X2) → [0,+∞]. We can iterate this construction to build the iterated OT cost C : P(P(X1)) × P(P(X2)) → [0,+∞]. Clearly, the general theory of OT applies to this setting, giving a characterization of the optimal couplings through duality and C-concavity. However, this setting allows for a refinement of the problem:

C(M1,M2) = min {∫ ∫ cost(x1, x2)dπ(x1, x2)dP(π) : P ∈ RΓ(M1,M2) } ,

 where the set RΓ(M1,M2) ⊂ P(P(X1 × X2)) is the set of random couplings, suitably defined such that the marginals of its elements are some given M1 ∈ P(P(X1)) and M2 ∈ P(P(X2)). These are more general objects than usual couplings Π ∈ Γ(M1,M2) ⊂ P(P(X1) × P(X2)), and I will present how to characterize them in terms of Kantorovich C-optimal potentials. I will also discuss the strict Monge problem, that is a more general version of the classic Monge problem associated to the cost C. Then: in a quite general setting, I will prove a result in the spirit of the Theorem by A. Pratelli; in the Hilbert case X1 = X2 = H, when cost(x1, x2) = |x1−x2|^2, the OT cost C is the L2-Wasserstein-on-Wasserstein distance. Exploiting the Lions’ lifting for the Wasserstein space (P2(H),W2), we will give sufficient conditions on M_1 such that the optimal random coupling is unique and is induced by a solution to the strict Monge problem. The talk is partially based on a joint work with Giuseppe Savaré.