Spring School

Discrete Ricci curvature

Paris, Institut Henri Poincaré (I.H.P.), May 18-22, 2015

#### Presentation

This Spring School will consist in two courses given by professors Jürgen Jost and Christian Leonard on discrete Ricci curvature. The aim is to bring researchers from different communities (Geometry, Probability, Analysis) on the common topic of the Ricci curvature and its consequences. Follow the abstracts of the courses, practical informations and registration.

 Jürgen Jost (Max Plank Institute, Germany) Curvature concepts for metric spaces and their implications and applications.

The concept of curvature was originally introduced to express the deviation of a curve from being straight, or of a surface in space from being planar, that is, flat. Gauss then discovered that curvature had both an extrinsic aspect, expressing how a surface curves in space, and an intrinsic one, how quickly geodesics emanating from the same point converge or diverge. Positive Gauss curvature leads to convergence, as on a sphere, negative curvature to divergence as in the hyperbolic plane. Riemann then developed the general theory of manifolds equipped with a metric tensor. The curvature tensor of such a Riemannian manifold encodes its geometric invariants and expresses the deviation from Euclidean, that is, flat space. In the 20th century, it was found that such a deviation from being Euclidean could also be quantified for more general metric spaces, and various notions of gneralized sectional curvature inequalities were proposed. In Riemannian geometry, sectional curvature is evaluated on tangent planes, that is, two-dimensional spaces, and generalized sectional curvature therefore refers to the distance between pairs of geodesics. It is quantified by convexity properties of that distance function between geodesics. Another important notion of curvature in Riemannian geometry is that of Ricci curvature, the average of the sectional curvatures of all tangent planes containing some given direction. In order to extend the concept of Ricci curvature to more general metric spaces, one therefore needs a measure in order to perform some suitable analogue of such an averaging. An important aspect of Ricci curvature is that it can control the eigenvalues of the Laplace operator.

In my lectures, building upon the notions of curvature in Riemannian geometry as just sketched, I shall develop such abstract notions of curvature for metric spaces and other mathematical objects. The guiding principle will always be to express and quantify some deviation from a flat reference situation. This could be the divergence or convergence of geodesics, as already sketched above; the transportation cost of the measure of some distance ball to another ball, compared with the distance between their centers; the probability that two independent stochastic motions get closer to each other as they evolve; or eigenvalue bounds for Laplace operators.

Perhaps somewhat surprisingly, such notions of curvature are even meaningful for discrete structures like graphs. In fact, graphs are ideally suited for explaining the meaning of curvature and the consequences of curvature inequalities in the simplest possible setting, without analytical complications.

 Christian Leonard (Paris Ouest Nanterre La Défense University, France) Some aspects of lower bounded curvature of metric spaces with applications to random walks on graphs.

The aim of the Lott-Sturm-Villani (LSV) theory is to extend from Riemannian manifolds to geodesic spaces the notion of lower bounded Ricci curvature as well as several consequences in terms of concentration of measure and convergence to equilibrium of heat flows. Optimal transport plays a crucial role. In particular, it leads to the notions of displacement interpolation and gradient flow on the space of probability measures which are fundamental tools for this approach.

The lectures will introduce the bases of the LSV theory on a geodesic space. Then, several recent attempts to extend the LSV theory to the framework of discrete metric graphs will be developed. The main approaches that will be considered are based on displacement interpolations and gradient flows on sets of probability measures. Representations in terms of stochastic processes (Brownian motion on a manifold and random walk on a graph) will be emphasized. This will lead us to some entropic inequalities and their consequences in terms of convergence to equilibrium of random walks.

Further details: Several connections between probability theory, analysis and the notion of curvature are known for a long time. Geometers consider the Laplace-Beltrami operator as a fundamental tool for deriving global properties of Riemannian manifolds, analysts look at it as the infinitesimal generator of the heat semigroup leading to the heat equation: a parabolic partial differential equation, while the Laplace operator is associated with the Brownian motion, a fundamental probabilistic notion.

Let us sketch a typical consequence of these interesting connections. Since the diameter of a Riemannian manifold with positive curvature depends decreasingly on the curvature (Bonnet-Myers theorem), it is not surprising that the speed of exploration of the manifold by a Brownian motion increases with the curvature. Conversely, estimating the speed of convergence to the volume measure of a Brownian motion allows to derive some upper bounds for the diameter.

Besides the Bonnet-Myers theorem, one can also derive functional inequalities such as Poincaré or logarithmic Sobolev inequalities which encode the speed of convergence of the heat flow to the volume measure as well as the concentration profile of this measure. The aim of the Lott-Sturm-Villani (LSV) theory is to extend this type of results from Riemannian manifolds to geodesic spaces. The basic ingredient of this approach is the quadratic optimal transport. It allows for defining the Wasserstein distance (of order 2) on the space P(X) of probability measures on the geodesic state space X which in turn allows for defining relevant notions of geodesic paths (McCann displacement interpolations) and gradient flows on P(X). When the state space X is a Riemannian manifold, some dynamical properties of these geodesic motions on P(X) are intimately connected to lower bounds of the Ricci curvature of X. This suggests the definition of a natural extension of lower bounded Ricci curvature on geodesic spaces which leads to results on the concentration of measure or the convergence to equilibrium of heat flows.

As a discrete space is not geodesic, the LSV theory doest not apply to the discrete setting. The aim of these lectures is to present several variants of the LSV strategy designed to obtain similar results for discrete metric graphs.

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The school is part of the the A.N.R. project GeMeCoD.

Participation of postdocs and PhD students is strongly encouraged.

#### Schedule

Lectures will be held in the main amphitheater (Hermite) of the I.H.P.

Coffee/tea will be available before each slot.

Slot\Day Mon 18 Tue 19 Wed 20 Thu 21 Fri 22
10:00-10:50 Jost Jost Jost Jost Jost
11:00-12:00 Jost Jost Jost Jost Jost
12:00-14:00 BreakBreakBreakBreakBreak
14:00-14:50 Leonard Leonard Leonard Leonard Leonard
15:00-16:00 Leonard Leonard Leonard Leonard Leonard

#### Organizers

Nathaël Gozlan (Paris-Est), Cyril Roberto (Paris-Ouest), Pascal Romon (Paris-Est), Paul-Marie Samson (Paris-Est)

With the crucial support of Audrey Patout and Christiane Lafargue.

Contact: springschool2015ihp@gmail.com

#### Participants / Registration

Participants Affiliation
Gozlan, Nathaël Université Paris-Est Marne la Vallée
Roberto, Cyril Université Paris-Ouest Nanterre la Défense
Romon, Pascal Université Paris-Est Marne la Vallée
Samson, Paul-Marie Université Paris-Est Marne la Vallée
Fathi, Max UPMC
Han, Bangxian Université Paris-Dauphine
Liu, Shiping Durham University, UK
Melbourne, James University of Minnesota
10 - Won, Yong Sul Imperial College London
Calderon, Eran Technion (institute of technology, Israel)
Tetali, Prasad School of Mathematics, Georgia Institute of Technology, Atlanta, GA, U.S.A.
Paseka, Olga Lomonosov Moscow State University, Russia
Ralli, Peter Georgia Institute of Technology
Kaur, Gursharn Indian Statistical Institute, New Delhi, India
Baird, Paul Université de Bretagne Occidentale
Papilin, Sergey Lomonosov Moscow State University, Russian Federation
Salamon, Joe University of California, San Diego
Tran, Tat Dat Max Planck Institute for Mathematics in the Sciences, Leipzig
20- Wang, Jian Max Planck Institute for Mathematics in the Sciences
Jolany, Hassan University of Lille1
Skorokhod, Natalia Technion, Israel
Chang, Yinshan Max Planck Institute for Mathematics in the Sciences, Leipzig
Huou, Benoit IMT, Toulouse
SHU, Yan Université Paris-Ouest Nanterre la Défense
Xiao, Yunlong Max Planck Institute for Mathematics in the Sciences, Leipzig
Zamora, Sergio Universidad Nacional Autónoma de México
Li, Anshui Mathematical Institute, Utrecht University, The Netherlands
30 - Fillastre, François Université Cergy-Pontoise
Seppi, Andrea Università di Pavia, Italy
Münch, Florentin Friedrich Schiller Universität Jena
Conforti, Giovanni BMS, Potsdam
Joulin, Aldéric Institut de Mathématiques de Toulouse
Kerkyacharian, Gerard LPMA
Rossi, Maurizia University of Rome Tor Vergata
Monasse, Pascal Université Paris-Est Marne la Vallée
Rousset, Mathias INRIA and CERMICS ENPC
Kahn, Jonas CNRS
40 - Collins , Emil Paris 6
Urquijo, Ramón Max Planck Institute for Mathematics in the Sciences
Sosa, Gerardo Max Planck Institute for Mathematics in the Sciences, Leipzig
de Buyer, Paul MODAL'X - Université Paris Ouest
Debin, Clément Institut Fourier (Grenoble)
Giona Veronelli Paris 13
ETBER, Ali BMCE Capital
Al Reda, Fatima Laboratoire Mathématiques d'Orsay
De Marco, Stefano Ecole Polytechnique
Tamanini, Luca Université Paris Ouest
Kell, Martin IHES
Hubard, Alfredo INRIA/Marne la Vallee

Financial support: The school may have some funding for (a limited number of) students (essentially PhD student and postdocs). If you wish to ask for a financial support please enter “yes” in the registration form below (notice that this information will not be public). For administrative reason the financial support will be guaranteed only if the participant give official bills (hotel. In particular no handmade letter from your uncle or cousin attesting that you spent a week in Paris at his/her home! Nothing is required for the meals.).

Depending on the sponsor that will make the reimbursement, the rule might be different. Please wait before buying your ticket and/or booking your hotel/lodging. We will put you in contact with the administration very soon.

The amount of money will approximately be of 500 euros.

Registration form

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