Optimal Transport and Geometry

International workshop on
Optimal Transport and Geometry

Montpellier, 22-26 June 2015

The schedule of the workshop is now available.

Mini-courses

Francesco Maggi, Mass transportation and applications to geometric/functional inequalities.

The theory of optimal mass transportation has had a tremendous impact on the field of geometric/functional inequalities, both by providing new proofs and new insights on old inequalities in sharp and non-sharp form, and by leading to discover new inequalities. In particular, the mass transportation approach has shown to be extremely effective in attacking stability questions, although its application to these problems is quite delicate and still largely unexplored. The course will describe the various results known at present, and will try to make the point on possible directions of further investigation.
Karl T. Sturm, Super-Ricci flows of metric measure spaces.

A time-dependent family of Riemannian manifolds is a super-Ricci flow if 2 Ric + \partial_t g \ge 0. This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation. We extend this concept of super-Ricci flows to time-dependent metric measure spaces. In particular, we present characterizations in terms of dynamical convexity of the Boltzmann entropy on the Wasserstein space as well in terms of Wasserstein contraction bounds and gradient estimates. And we prove stability and compactness of super-Ricci flows under mGH-limits.

Talks

Davide Barilari, Volume distorsion along geodesics and curvature in sub-Riemannian geometry.

In this talk we introduce a family of symmetric operators canonically associated with a geodesic on a sub-Riemannian manifold. Under some assumptions this family admits an asymptotic expansion and each term of this expansion defines a geometric invariant. The zero order term is associated with the asymptotic volume distorsion along geodesics, while higher order terms are curvature-like invariant. In the case of Riemannian and Finsler spaces one recovers the notion of sectional and flag curvature, respectively.
Franck Barthe, Gaussian kernels also have Gaussian minimizers.

The title hints to E. Lieb's celebrated results on maximizers of multilinear Gaussian kernels acting on products of L_p spaces. In a joint work with P. Wolff, we put forward a similar principle for minimizers when some of the indices p are less than 1 (possibly negative). This extends and unifies many "reverse" inequalities known in the litterature. Monotone transportation maps play a key role in the proof, where the presence of negative coefficients requires new ingredients.
Fabio Cavalletti, One dimensional localization for metric measure spaces.

The method of one-dimensional localization permits to reduce most of the information on the curvature of the space to information on the geometry of a one dimensional space. We present how to perform this reduction in the framework of metric measure spaces. As an application, we prove that essentially non-branching metric measure spaces verifying the reduced curvature dimension condition CD*(K,N) also verify sharp Brunn-Minkowski inequality. Joint work with Andrea Mondino.
Aude Dalet, Lipschitz-free spaces.

Let M be a pointed metric space and Lip₀(M) the space of Lipschitz functions vanishing at 0. Endowed with the Lipschitz norm this space is a Banach space. Denote F(M) the closed subspace of Lip₀(M)* spanned by the evaluation points and call it the Lipschitz-free space over M. We will first describe the link between optimal transport and Lipschitz-free spaces: the space of probability measures on M, endowed with the optimal transport distance, is an affine convex subset of F(M). Then we will study Lipschitz-free spaces over compact ultrametric spaces and prove that they are isomorphic to the space l₁.
Matthias Erbar, Curvature bounds for configuration spaces.

This talk is concerned with the geometry of configuration spaces, the natural state spaces of infinite particle systems. We consider a system of independent Brownian particles with drift on a manifold M. The configuration space Conf(M) of locally finite counting measures on M inherits a differentiable stucture from the base space in such a way that the natural diffusion on this infinite-dim. Riemannian manifold is the particle system. An invariant measure is given by the Poisson point process on M.
We are interested in curvature properties of the manifold Conf(M). More precisely, we will show that various manifestations of lower bounded (Ricci-)curvature extend from the base space to the configuration space. This includes gradient estimates for heat semigroup and convexity of the entropy along L² transport geodesics. In particular, configuration spaces present a natural new example of infinite-dimensional metric measure spaces with curvature bounds in the sense of Lott--Villani and Sturm.
Ivan Gentil, Contraction in Wasserstein distance for Markov diffusion semigroups.

We prove the equivalence between a dimensional contraction inequality and the lower bound on the Ricci curvature.
Nicola Gigli, Nonsmooth differential geometry.

In this talk I will discuss in which sense general metric measure spaces possess a first order differential structure and how on spaces with Ricci curvature bounded from below a second order one emerges. In this latter framework, the notions of Hessian, covariant/exterior derivative and Ricci curvature are all well defined.
Christian Ketterer, Geometry of metric measure spaces with variable Ricci curvature bounds.

In my talk I will introduce a curvature-dimension condition CD(K,N) for metric measure spaces where K is a lower semi-continous function. The condition admits geometric consequences as a generalized, sharp Bonnet-Myers theorem. It is stable under measured Gromov-Hausdorff convergence, and it is stable w.r.t. tensorization provided a non-branching condition is assumed.
Young-Heon Kim, Multimarginal optimal transport.

We explain some recent progress on multi-marginal optimal transport, where a family of mass distributions are matched in an optimal way. This is based on joint work with Brendan Pass.
Robert McCann, The intrinsic dynamics of optimal transport.

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. With Ludovic Rifford (Nice), we introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probability densities is unique.
Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric measure spaces satisfying Ricci curvature lower bounds.

By using an L¹ localization argument used to reduce to a 1-D problem, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely RCD*(K,N) or more generally essentially non branching CD*(K,N)) the classical Levy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements. This is joint work with Fabio Cavalletti.
Magnus Önnheim, To be announced.
Yi Wang, Alexandrov-Fenchel inequalities on k-convex domains.

We will discuss the classical Alexandrov-Fenchel inequalities for quermassintegrals on convex domains, and report our progress on generalizing the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillon for k = 1 case of the inequalities, uses the method of optimal transport. We will also discuss the sharp higher order isoperimetric inequalities, also using the optimal transport. These are joint works with Alice Chang.