Autumn days in Grenoble: "Curvature in metric spaces"

Presentation

These autumn days are part of the the A.N.R. project GeMeCoD. They consisted in two courses:

• “Analysis and geometry on spaces with Ricci curvature bounded from below”]] by Nicola Gigli. Slides:1st course, 2nd course and 3rd course.
• “Ricci curvature and functional inequalities in discrete spaces” by Cyril Roberto. Notes: part 1, part 2.

and talks by:

• Michel Bonnefont: “Measure doubling property and Poincaré inequality in sub-elliptic geometry under a curvature-dimension criterion”
• Abstract: In this talk, I will present a curvature-dimension criterion in sub-elliptic geometry that generalizes the one of Bakry-Emery. With this criterion one can obtain Li-Yau type gradient estimates and a reverse log-sobolev inequality for the semigroup. Finally, in the non-negative curvature case, one can then deduce the measure doubling property, gaussian estimates for the heat kernel and the Poincaré inequality on balls. This is joint work with F. Baudoin and N. Garofalo.
• Abstract: Let $(f_t)_{t \in [0,1]}$ be a smooth family of probability measures on Z. We set $g_t(k)=-\sum_{l \leq k} f_t'(l)$ and $h_t(k)= -\sum_{l \leq k} g_t'(l)$. The family $(f_t)$ is said to be a Benamou-Brenier curve (or BB-curve), if $f_t(k)h_t(k-1)=g_t(k)g_t(k-1)$. During this talk, we will explain why BB-curves can be seen as a discrete version of W_2 geodesics, then we will construct, under a stochastic domination assumption, a BB-curve joining two fixed measures $f_0$ and $f_1$. We will finish by proving the convexity of some functionals along BB-curves.
• Abstract: I will discuss the role of branching geodesics in the theory of Ricci curvature lower bounds in metric measure spaces. I will focus on the $CD(K,N)$ condition by Lott-Sturm-Villani and on the $RCD^*(K,N)$ condition by Ambrosio-Gigli-Savaré-(Erbar-Kuwada-Mondino-Rajala-Sturm). Some of the results have been obtained in collaborations with Ambrosio, Gigli, Mondino, and Sturm.
The $CD(K,N)$ condition is satisfied, for instance, by $\mathbb{R}^n$ equipped with any norm and the Lebesgue measure. Thus a $CD(K,N)$ space can be have a lot of branching geodesics. Regardless of branching, one can still select quite good geodesics in the space of probability measures by means of minimizing the entropy. These geodesics can be used as test plans in the Sobolev-space theory and in proving local Poincaré inequalities. Recently it was noticed that even some non-convex (in the Euclidean sense) closed subsets of $\mathbb{R}^n$ (equipped with the supremum norm and the Lebesgue measure) satisfy $CD(K,N)$. This observation shows that $CD(K,N)$ does not have the local-to-global property.
The $RCD^*(K,N)$ condition is more strict and it excludes non-Euclidean normed spaces. Via the EVI-formulation one can see that the $RCD^*(K,N)$ spaces are essentially non-branching. Consequently, the Brenier-McCann Theorem (stating the existence of optimal transportation maps from absolutely continuous measures) holds in $RCD^*(K,N)$ spaces and a notion of exponentiation in $RCD^*(K,N)$ spaces can be given.
• Abstract: In this talk, I will explain how solving transport problems between measures on graphs can be used to get new results on their $l^p$ cohomology. The (reduced) $l^p$ cohomoglogy of an infinite graph of bounded valency is the quotient of the space of functions with $l^p$ gradients by the (closure of) the gradients of functions in $l^p$. It turns out this quotient carries interesting geometric information about the graph, e.g. when $p=1$ this identifies to a function space on ends, e.g. when the graph is hyperbolic it identifies to a certain space of functions on the hyperbolic boundary. Cheeger & Gromov showed the reduced $l^2$-cohomology of (Cayley graphs of) amenable groups is trivial. That the same can be said for other $p$ remains the topic of a conjecture. Finding bounds for $W_1$ and some less traditionnal metrics allows to extend the range where this conjecture holds ( e.g. in certain wreath products and for all $p$, e.g. in all amenable groups for all $p<2$, e.g. in groups of intermediate growth for all $p$ , …).
• Laurent Veysseire: “Coarse Ricci curvature for Markov processes and Poincaré inequality”
• Abstract: In this talk, I will introduce Ollivier's notion of coarse Ricci curvature for Markov chains on metric spaces, and its generalization to continuous time Markov processes. If the process is reversible and if this curvature is bounded from below by a positive constant $k$, $k$ is a lower bound for the spectral gap of the generator, or equivalently, we have a Poincaré inequality with constant $\frac{1}{k}$. In the case of diffusions on Riemannian manifolds, we can get a better bound on the spectral gap: the harmonic mean of the curvature. To define the coarse Ricci curvature, one can choose the metric independently of the generator. I will give some examples of diffusions where the choice of a metric different from the canonical one associated with the generator provides a better bound for the spectral gap.
• Christian Léonard: “Entropic interpolations on graphs”
• Abstract: Entropic interpolations are entropy minimizers under marginal constraints. They are regular approximations of displacement interpolations. On a Riemannian manifold, they allow to implement a rigorous version of the informal Otto calculus and to recover the main results of the Bakry-Emery and synthetic Ricci lower bound theories. On a discrete metric graph, they lead to new natural candidates for curvature whose relevance still need to be tested. We introduce the dynamics of the entropic interpolations, discuss the related notions of curvature on graphs and address some open problems. This talk is unfortunately cancelled. If you want to read a related preprint, here it is: http://www.cmap.polytechnique.fr/~leonard/papers/lazyRW.pdf

Schedule

Slot\Day Wed 16 Thu 17 Fri 18
9:30-10:00 Course 1: Nicola Gigli (2/3) Course 1: Nicola Gigli (3/3)
10:00-10:30
10:30-11:00 Coffee
11:00-11:30 Course 1: Nicola Gigli (1/3) Coffee Course 2: Cyril Roberto (3/3)
11:30-12:00 Talk 3: Antoine Gournay
12:00-12:30 Lunch
12:30-13:00 Lunch Lunch
13:00-13:30
13:30-14:00
14:00-14:30 Course 2: Cyril Roberto (1/3) Course 2: Cyril Roberto (2/3)
14:30-15:00
15:00-15:30
15:30-16:00 Coffee Coffee
16:00-16:30 Talk 1: Michel Bonnefont Talk 4: Laurent Veysseire
16:30-17:00
17:00-17:30 Talk 2: Erwan Hillion Talk 5: Tapio Rajala
17:30-18:00

Practical informations

Directions

The conference will take place on the university campus, at the Institut Fourier, room 4.

How to get there: http://www-fourier.ujf-grenoble.fr/-Venir-a-l-Institut-Fourier-.html

Accomodation

Hotels in Grenoble Residences hotels downtown Grenoble
Hôtel des Alpes Apparthotel Marie-Curie
Hôtel Suisse et Bordeaux Residhome Appart Hotel Caserne de Bonne
Hôtel de l'Institut Park&Suites Elégance Grenoble
Hôtel d'Angleterre Resid'hotel Grenette
Hôtel de l'Europe Resid'hotel Central Gare
Touring Hôtel
Grand Hôtel Hotels on the university campus
Hôtel Lux Hôtel Ibis
Hôtel Acacia Hôtel Formule 1
Hôtel Bastille Hôtel B&B
Hôtel Les Patinoires
Hôtel Gallia
Hôtel Alizé

Participants / Registration

Registrations are closed

Participants Affiliation
Rossignol, Raphaël Institut Fourier (Grenoble 1)
Pajot, Hervé Institut Fourier (Grenoble 1)
Colin de Verdière, Yves IF
Truc, Françoise Institut Fourier
Leonard, Christian Universite Paris Ouest
Dahmani, Francois Institut Fourier
Prandi, Dario École Polytechnique
Hillion, Erwan University of Bristol
Gournay, Antoine Université de Neuchâtel
Rajala, Tapio University of Jyväskylä
MUNNIER, Vincent Ecole Centrale de LILLE
Roberto, cyril Paris Ouest Nanterre
FENEUIL, Joseph Université de Grenoble
jabrane , saïd icj lyon1
Gentil, Ivan Université Lyon 1
Bonnefont, Michel IMB
Correia dos Santos, Matheus Unicamp
HAN, Bangxian Université Paris Dauphine
COQUIO, AGNES Institut Fourier (Grenoble 1)
Kloeckner, Benoît Institut Fourier (Grenoble 1)
Russ, Emmanuel Institut Fourier (Grenoble I)
Guillin, Arnaud Universite Blaise Pascal
Bouyrie, Raphaël Institut de Mathématiques de Toulouse
charlot, grégoire Institut Fourier
Besson, Gérard CNRS/Université de Grenoble
Petit, Camille University of Jyväskylä
Ogundare, Babatunde Obafemi Awolowo University, Ile-Ife, Nigeria
Funar, Louis CNRS/Universite de Grenoble
Duchon, Jean Institut Fourier (CNRS)
Gozlan, Nathael U. Paris-Est Marne-la-Vallée
ZEGLAOUI, AHMED USTHB - ALGER
Ahidar, Adil Institut de Mathématiques de Toulouse
Huou, Benoit Institut de Mathématiques de Toulouse
Valtorta, Daniele EPFL
Veysseire, Laurent Technion (Haïfa, Israël)
SHU, Yan Paris Ouest Nanterre
Rimoldi, Michele Università dell'Insubria
Bowditch, Brian University of Warwick
Gigli, Nicola Paris 6
Delplancke, Claire IMT
Gallot, Sylvain Institut Fourier
Débin, Clément Institut Fourier
Otera, Daniele Vilnius