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Francesco Nobili
A classical characterization of Sobolev functions relies on absolute continuity along almost every line parallel to the coordinate axes. Over the past decades, this viewpoint has inspired extensive research in the analysis of metric spaces, revealing deep connections between optimal transport, Sobolev calculus, and Ricci curvature.
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After a brief review of these ideas, I will present an... -
12. Coupling methods for lattice gases and nonlinear Fokker–Planck equations with local interactionsGiovanni Conforti
It is well known that the hydrodynamic limit of several classes of stochastic interacting particle systems is described by nonlinear Fokker–Planck equations with local interactions. This is the case, for example, for basic stochastic lattice gas models such as the zero-range process and the simple exclusion process. The aim of this talk is to introduce a simple probabilistic approach that...
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Averil Aussedat
The geometric tangent cone to the Wasserstein space is known to be isometric to a set of plans since the work of Gigli (2008). It is a nonlocal object, in that restricting the measure on a subset might change its tangent cone substantially. However we show that the subset of centred elements, those with barycenter 0 in each fiber, is stable by restriction. Following this we compute a...
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Gabriele Steidl
In score-based diffusion models, the underlying probability path is governed by an Ornstein-Uhlenbeck (OU) process,
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connecting a target data variable with a Gaussian.
This variance-preserving OU path is driven by the Wiener process,
and it is known that due to its unbounded variation,
the OU path may admit certain singularities concerning its score at times close to the data.
To... -
Clarice Poon
In this talk, I will discuss a particular inverse problem arising from optimal transport (OT). In many modern pipelines, however, the transport cost is not known a priori and must be inferred from data. This leads to inverse optimal transport (iOT): recover the ground cost (or metric parameters) from an observed optimal coupling. On the other hand, modern computational pipelines typically...
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Giacomo Cozzi (Università di Padova)
The sliced Wasserstein distance was introduced as an analogue of the Wasserstein distance, based solely on one-dimensional transports. This makes it computationally efficient and well suited to applications.
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We prove that the squared sliced Wasserstein distance from a fixed target measure is (\lambda)-geodesically convex, provided that the target satisfies a geometric condition controlling... -
Krzysztof Ciosmak
Classical localisation in optimal transport decomposes transport into monotone couplings along transport rays. In this talk, I will present a second-order analogue: transport rays are replaced by the leaves of the 1-Lipschitz gradient Du of an optimal dual potential u in C^(1,1)(R^n), while monotone couplings are replaced by bimartingale couplings. This framework applies to the three-marginal...
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Emanuela Radici
Nonlocal aggregation equations with nonlinear mobility arise naturally
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in the study of interacting particle systems subject to congestion and density
constraints. Their evolution can be formally interpreted as a gradient
flow of the interaction energy with respect to a generalized transport geometry
induced by the mobility.
In this talk, I will investigate the long-time behaviour of... -
Guillaume Carlier
In this talk, I will consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. Relying on a notion of discrepancy adapted to a phase space of positions and velocities, built upon...
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Charles Bertucci
I will distinguish two geometries on Wasserstein spaces, one coming from the linear structure of signed measures, and one coming from the geometry of the base space. I will explain why a Radon-Nikodym property holds in both geometries, and draw their usual consequences. I will then spend some time explaining what I believe are natural expression of two "geometric" objects in this second...
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Federico Renzi (Scuola Normale Superiore)
The RCD condition provides a notion of Riemannian-like metric measure space satisfying synthetic Ricci curvature lower bounds. In this talk I will present a notion of RCD condition localized on an open subset of the ambient space, with variable curvature lower bound, by requiring infinitesimal Hilbertianity and displacement convexity of suitable entropy functionals on neighbourhoods of every...
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Andrea Mondino
The talk introduces a framework for studying the geometry of lightlike (null) hypersurfaces in spacetimes that may be singular or lack a smooth structure. Building on tools from optimal transport and modern Lorentzian geometry, we develop a flexible, synthetic approach to describing such hypersurfaces without relying on the usual smoothness assumptions.
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Within this framework, we propose a... -
Tapio Rajala
We will revisit the regularity properties of (quasi)minimizers and consider two types of minimizers involving the perimeter. One minimization is for the perimeter among sets with fixed volume, leading to isoperimetric sets and the other is a minimization of the perimeter with an additive term penalizing the deviation in measure from a given set.
In essentially non-branching MCP(K,N)-spaces,...
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Shouhei Honda
In this talk, we introduce a structure theory of metric measure spaces with Ricci curvature bounded below and positive injectivity radii. In particular, we show that such spaces are weighted Riemannian manifolds of W^{1,p}-regularities for any finite p. Applications include a resolution to a conjecture of V. Kapovitch via a metric smoothing. This is a joint work with Ruobing Zhang (US San...
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Karl-Theodor Sturm
We derive an explicit formula for the sectional curvature of the space M_0(X) of finite measures on a Riemannian manifold X. The space M_0(X) is equipped with the Hellinger-Kantorovich metric HK. Even in the case X = Rn, the curvature is comprised of two parts: the ‘lifted part’ is negative, and the ‘twisted part’ is positive. It will be analyzed in detail for the multidimensional torus.
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Our... -
Nataliia Monina
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Mathias Braun
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Yann Brenier
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Sinho Chewi
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Fanch Coudreuse
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Martin Huesmann
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Anna Korba
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Robert McCann
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Alpar Meszaros
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Gudmund Pammer
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Maxime Sylvestre
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Dario Trevisan
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Prof. Gero Friesecke (Technische Universität München)
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Vanessa Ryborz (Universität Wien, Austria)
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Guy Bouchitté
Inspired by Monge-Kantorovich optimal transport theory, we present a notion of tangency to a measure based on the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures with respect to the $W_1$-distance. For any Radon measure $\mu$ in $Rd$, a tangent bundle $T_\mu$ can be defined...
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