Institut de Recherche Mathématique Avancée, UMR 7501

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16 septembre 2010

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11h00

Benoît Kloeckner - Grenoble

Optimal transportation enables the construction, from any metric space $X$, of a new ``Wasserstein'' metric space made of sufficiently concentrated probability measure on $X$. We are interrested in the geometry of this space of measures, more specificaly in the way one can embed some particular spaces, e.g. Euclidean, in the Wasserstein space.
We shall show a bi-Lipschitz embedding of $X^n$ and prove that as soon as $X$ is sufficiently negatively curved, the plane does not embed isometrically.

14h00

Nicola Gigli - Nice

I will prove that on a compact finite dimensional Alexandrov space with curvature bounded below the gradient flow of the Dirichlet energy w.r.t L^2 coincides with the gradient flow of the relative entropy w.r.t W_2. The proof does not use PDE techniques but relies only on metric arguments. I will also show how from this identification it easily follows that the heat kernel is Lipschitz (from a joint work with Kazumasa Kuwada and Shin-ichi Ohta).

15h30

Max-Konstantin Von Renesse - Berlin

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17 septembre 2010

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09h30

Ludovic Rifford - Nice

We address the problem of regularity of optimal transport maps on surfaces. We give necessary and sufficient conditions for the so-called Transport Continuity Property and discuss a list of examples.

11h00

Nathaël Gozlan - Marne-la-Vallée

We give a new proof of the fact that Gaussian concentration implies
the logarithmic Sobolev inequality when the curvature is bounded from below
and also that exponential concentration implies Poincaré inequality under null curvature condition.
Our proof holds on non smooth structures such as length
spaces and provides a universal control of the constants. We also give a new
proof of the equivalence between dimension free Gaussian concentration and
Talagrand’s transport inequality(joint work with C. Roberto and P-M. Samson).

14h00

Filippo Santambrogio - Orsay

One of the simplest model to handle the flow of people willing to exit a room is the following: at every point $x$, the velocity $U(x)$ that individuals at $x$ would like to realize is given (say, it is their maximal speed in the direction of the door). Yet, the incompressibility constraint (say, $\rho\leq 1$ prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities (i.e., there is a constraint on the divergence on the set $\{\rho=1\}$). The unknwon in this macroscopical model is the evolution of a density $\rho(t,x)$. If a gradient structure is given, $U -\nabla D$ where D is the distance to the door, the problem turns out to be a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density) nor geodesically convex (at least in the interesting cases), which requires for an ad-hoc study of the convergence of a discrete scheme.