Du 16 au 17 septembre 2010
IRMA
L'objet de ces journées sera de proposer un aperçu de différents développements faisant intervenir le transport optimal, comme outil ou comme objet d'étude.
The aim of this meeting is to report on various developments when optimal Transport is used as a tool or is studied for its own sake.
Les conférences auront lieu en anglais.
Lieu : Salle de conférences IRMA
Organisation/Organization : Nicolas Juillet
Conférenciers invités / Invited speakers :
- Nicola GIGLI (Nice)
- Nathael GOZLAN (Marne-la-Vallée)
- Benoît KLOECKNER (Grenoble)
- Max von RENESSE (Berlin)
- Ludovic RIFFORD (Nice)
- Filippo SANTAMBROGIO (Orsay)
Informations pratiques / Practical information :
- Hôtels
- Comment venir à l'IRMA / How to reach IRMA
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Jeudi 16 septembre 2010
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11:00 - 12:00
Benoît Kloeckner, Grenoble
EMBEDDING QUESTIONS INTO WASSERSTEIN SPACES
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. -
14:00 - 15:00
Nicola Gigli, Nice
HEAT FLOW ON ALEXANDROV SPACES
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). -
15:30 - 16:30
Max-Konstantin Von Renesse, Berlin
AN OPTIMAL TRANSPORT VIEW ON THE SCHROEDINGER EQUATION.
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Vendredi 17 septembre 2010
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09:30 - 10:30
Ludovic Rifford, Nice
OPTIMAL TRANSPORT ON SURFACES.
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. -
11:00 - 12:00
Nathaël Gozlan, Marne-la-Vallée
FROM CONCENTRATION TO LOGARITHMIC SOBOLEV AND POINCARE INEQUALITIES
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). -
14:00 - 15:00
Filippo Santambrogio, Orsay
CROWD MOVEMENT WITH DENSITY CONSTRAINT: A GRADIENT FLOW APPROACH.
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.