Book of abstracts

Giovanni Alberti (Università di Pisa)

Title: Optimal transport and the (missing?) connection to normal currents

Abstract : The Lebesgue measure on the unit cube Q in \R^d can be transported to any probability measure μ on Q by a suitable transport maps f from Q to Q. Therefore one can select one transport map f_μ for each probability measure μ and ask how regular this selection can be.

For instance, in dimension d=1, choosing the optimal transport map provides a selection which is Lipschitz from the space of probability measures endowed with the 1-Wasserstein distance to the space of maps from Q to Q, endowed with the L^1$distance.

This result does not hold for d>1. Actually, for d>1 there exists no Lipschitz selection, because the existence of a such a selection would have consequences on the structure of normal currents of codimension d that are known to be false for d>1. (This connection was actually my original motivation for looking at this question.)

Paul Pegon (Université Paris Dauphine)

Title: Branched Transport and Optimal Quantization

Abstract: Branched transport is a variant of optimal transport in which the cost is strictly subadditive with respect to the mass m to be transported, thus favoring grouped transportation. The trajectories of the particles then form a transport network featuring branching points. This variational model was introduced in an attempt to model various artificial or natural networks (communication and transportation networks, plant root systems, river networks, etc.). In this talk, I will present branched transport, then introduce the problem of optimal quantization of measures, which consists in finding the best approximation of a measure by an atomic measure with a fixed number of atoms. We studied this problem by replacing the Wasserstein distance usually employed with the distance induced by branched transport. I will present several results concerning the asymptotic behavior as the number of atoms tends to infinity, as well as uniformity properties of optimal quantizers. This is joint work with Mircea Petrache.

Andrea Marchese (Università di Trento)

Title: Old questions and new perspectives in Branched Transport: stability, anisotropy, and robustness

Abstract: Branched transport is a variant of the classical Monge-Kantorovich problem in which the trajectories of moving particles are part of the minimization process. The cost functional favors the sharing of paths, leading to the formation of network structures. Geometric Measure Theory provides plenty of powerful and flexible tools to study this problem.

In this seminar, I will present two new models. The first introduces anisotropy, assigning different costs to trajectories depending on their orientation. The second favors redundancy of paths, with the goal of increasing the robustness of the network against possible damage while preserving efficiency. Finally, I will discuss a recent extension to all regimes of the stability property for the mailing problem. Based on joint works with Martina Bellettini, Luigi De Masi, Jakub Krukowski, and Annalisa Massaccesi.

Averil Aussedat (Università di Pisa)

Title: Weird measures near the edges of Otto's manifold in dimension one

Abstract : It is natural to conjecture that elements of the Wasserstein tangent cones are characterized by the initial speed of the curve that they induce. This is indeed true in many cases, but counterexamples exist. In this talk, we identify the 1D measures on which the characterization fails.  

Séverine Rigot (Université Côte d'Azur)

Title: Monotone sets

Abstract: Monotone sets have been introduced about fifteen years ago by J. Cheeger and B. Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into L^1 to the classification of its monotone subsets. In this talk, I will explain further motivations for studying monotone sets in the wider setting of Carnot groups. In the Euclidean setting (i.e. Abelian Carnot groups), monotone sets are nothing more than half-spaces. Our interest in the non Abelian setting stem from the relationship between monotone sets and codimension 1 quantitative rectifiability, as well as minimal hypersurfaces and subharmonic functions. This talk is intended to be accessible to anyone without specific knowledge about Carnot groups. Parts of the talk are based on joint works with E. Le Donne and D. Morbidelli. 

Adolfo Arroyo Rabasa (Università di Pisa)

Title: Structure of 1-dimensional currents in metric spaces: Smirnov SBV-representations and the flat chain conjecture

Abstract : The theory of currents provides a powerful framework for studying geometric and variational problems where classical oriented surfaces are insufficient. Metric currents generalize this theory to spaces lacking a smooth manifold structure. A central open question in this setting is the Flat Chain Conjecture (FCC), which asserts that the space of metric currents coincides with the closure (in the mass norm) of all normal currents. While the FCC has been resolved for 1-dimensional and top-dimensional cases in Euclidean space, it has remained elusive in the general metric setting. 

In this talk, I will present a new approach based on the Kantorovich—Rubinstein norm that provides necessary and sufficient conditions for the 1-dimensional FCC in general metric spaces. This framework allows us to establish a Smirnov-type decomposition for 1-dimensional metric currents into SBV curves, providing a sharp structural description of these objects and bridging the gap between metric current theory and optimal transport.

This is joint work with Guy Bouchitté (IMath Toulon).

Kexin Lin (Université Claude Bernard Lyon 1)

Title: Existence of a solution of the TV Wasserstein gradient flow

Abstract : The total variation(TV) Wasserstein gradient flow has been first proposed in a paper by Duering and Schoenlieb in 2012 for the purpose of denoising image densities. This gradient flow has been studied via its Jordan-Kinderlehrer-Otto scheme in 2019 by Carlier and Poon, but unfortunately a full proof of convergence is only presented in the one-dimensional case. Only in this case they are able to prove the preservation of the lower bounds in order to pass the limit in the JKO scheme. In a collaboration with Filippo Santambrogio, we prove that on the flat torus in any dimension, there exists a solution of the TV Wasserstein gradient flow, only assuming that the initial density ρ_0 is bounded from below and above by strictly positive constants. This solution preserves upper and lower bounds of the densities, and shows a certain decay of the BV norm (of the order of t^{-1/3} for t\to 0 and of the order of t^{-1} as t\to\infty). The main tool consists in considering an approximated TV-JKO scheme which artificially imposes a lower bound on the density and allows to find a continuous-in-time solution regular enough to prove that the lower bounds of the initial datum propagates in time, and study on this approximated equation the decay of the BV norm. 

Guido De Philippis (Università degli Studi di Padova)

Title: Caffarelli Contraction Theorem: old and new

Abstract : Caffarelli contraction theorem asserts that if a measure is "more" log-concave then the Gaussian, then there always exists a 1-Lipschitz map transporting it to the Gaussian. This allows for transferring various functional inequalities valid for the Gaussian measure to other measures. In this  talk I will  revise the theorem and some of its proof and show some recent improvements we have been obtained recently in collaboration with Yair Shenfeld and their applications. I will also mention some future developments.