Speaker: Fabio Cavalletti (The University of Pavia)
Speaker: Nicola Gigli (SISSA)
Title: Schrodinger problem and geodesics in the Wasserstein space
Abstract: I shall introduce the Shrodinger problem and discuss in which sense it provides a general and `good' way of approximating W_2 geodesics both in smooth and non-smooth spaces. From a work with Luca Tamanini.
Speaker: Hyung Ju Hwang(POSTECH)
Title: Regularity vs singularity in kinetic equations
Abstract: We describe the structure of solutions of kinetic equations in domains with boundaries near the grazing set. Representative equations are the Vlasov-Poisson, the kinetic Fokker-Planck equations. We discuss in particular regularity, singularity, and the behavior of the solutions of these equations with various boundary conditions.
Speaker: Yong-Jung Kim (KAIST)
Title: A traveling wave phenomenon in dynamics of metric
Abstract: A diffusion operator in the Euclidean space is introduced when the walk length is constant with respect to a non-Euclidean metric. We will see that, the dynamics of the metric may produce a traveling wave phenomenon.
Speaker: Bo’az Klartag (Tel-Aviv)
Title: Needle decomposition and Ricci curvature
Abstract: Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. This technique was promoted by Payne-Weinberger, Gromov-Milman and Kannan-Lovasz-Simonovits. In this lecture we will explain what needles are, what they are good for, and how one may construct them by using optimal transportation with the L^1-cost, under lower bounds on the Ricci curvature.
Speaker: Kazumasa Kuwada (Tokyo Institute of Technology)
Title: Estimates of transportation costs between heat distributions
Abstract: On a broad class of spaces, the canonical heat flow can be regarded as a gradient flow of the relative entropy functional on the space of probability measures equipped with the L^2-Wasserstein distance. This fact provides a connection between geometry of the state space described by optimal transport and analysis of heat flows. In this talk, I will show several estimates of transportation costs between heat distributions on metric measure spaces satisfying a Riemannian curvature-dimension condition. Some of them are indeed known to be equivalent to the curvature-dimension condition and others are obtained from them, by passing through the Bakry-'Emery theory.
Speaker: Jan Maas (IST)
Title: Optimal transport of density matrices
Abstract: One of the highlights in optimal transport is the interpretation of diffusion equations as gradient flows of the entropy in the Wasserstein space of probability measures. In this talk we show that this interpretation extends to the setting of non-commutative probability. We construct a class of Riemannian metrics on the space of density matrices, which may be regarded as non-commutative analogues of the 2-Wasserstein metric. These metrics allow us to formulate quantum Markov semigroups as gradient flows of the von Neumann entropy. We discuss analogues of displacement convexity, and present applications to transport inequalities for density matrices. This is joint work with Eric Carlen.
Speaker: Robert McCann (Toronto)
Title: Multi- to one-dimensional matching
Abstract: The stable marriage problem with transferable utility is well-known to be equivalent to a Monge-Kantorovich optimal transportation problem. Motivated by the possibility that the husbands (but not the wives) can be described by a single variable, we consider the problem of transporting a probability density on $R^m$ to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely, by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level-set dynamics to develop a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples.
This represents joint work with Pierre-Andre Chiappori and Brendan Pass.
Speaker: Alpar Richard Meszaros (UCLA)
Title: Density constraints and congestion effects in evolution equations
Abstract: In this talk I will present how optimal transport techniques can be used to study some PDE systems describing the evolution of the density of a population subject to a density constraint. In this context a particular emphasis will be on a Fokker-Planck type equation. From the modeling point of view this equation can describe the movement of a crowd in a bounded domain, when individuals try to follow a given spontaneous velocity field, but are subject to a Brownian diffusion and - to avoid congestion - have to adapt to a density constraint which introduces a pressure term affecting the movement.
In the second part of the talk I will describe how these models can be recovered by a limiting procedure involving porous medium type equations with drifts. Here I will focus on a system of porous medium equations with gradient drifts for several fluids coupled by a common pressure. This can be seen as the gradient flow of a suitable functional on a product of Wasserstein spaces. Connections with the Hele-Shaw and the Muskat problems will be also given. The talk is based on joint works with F. Santambrogio (Orsay) and S. Di Marino (Orsay) and on an ongoing work with I. Kim (UCLA).
Speaker: Connor Mooney (U. Texas)
Title: Singular solutions to the Monge-Ampere equation
Abstract: Strictly convex solutions to the Monge-Ampere equation det D^2u = 1 are smooth. However, there are important examples of singular solutions due to Pogorelov that are not strictly convex. We will discuss recent optimal estimates on the Hausdorff dimension of the singular set, and applications including unique continuation and W^{2,1} regularity.
Speaker: Aldo Pratelli (Erlangen)
Title: About the approximation of maps in nonlinear elasticity
Abstract: We will discuss about the issue of approximating maps in the context of the nonlinear elasticity. The question is non-trivial because the maps one is dealing with are homeomorphisms, and this property is easily lost with the usual approximation via convolution. Then, one has to construct an ad-hoc method for this problem. We will review the general history of the problem, and then we will concentrate on the newest achievements and open problems.
Speaker: Ludovic Rifford (Nice)
Title: The intrinsic dynamics of optimal transport
Abstract: 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. 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 probility densities is unique. This is a joint work with Robert McCann.
Speaker: Filippo Santambrogio (Orsay)
Title: Moment measures via optimal transport
Abstract: We say that a finite positive measure $mu$ on $R^d$ is the moment measure of convex function $u:R^dtoRcup{+infty}$ if $mu$ is the image of the log-concave density $rho=e^{-u}$ through the map $xmapsto Du(x)$.
A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures which can be expressed as the moment measures of suitable convex functions and finds the corresponding $u$ by a variational method in the class of convex functions.
Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost and the notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers.
Speaker: Neil Trudinger (ANU)
Title: Classical solutions of Monge-Ampere type equations
Abstract: We present an overview of recent a priori solution estimates and classical solvability of boundary value problems for Monge - Ampere type equations arising in various applications including optimal transportation and geometric optics.
Speaker: Xu-Jia Wang (ANU)
Title: Monge-Ampere equations arising in geometric optics
Abstract: We discuss a Monge-Ampere type equation arising in geometric optics,
in the design of a reflection surface such as the reflector antenna.
An interesting phenomenon is that the regularity of the reflection surface
depends not only on the light distributions but also on the position of reflector, and
the position and curvatures of the object. It was also discovered that the design of
reflection surface is a linear optimisation problem in the far field case;
and a nonlinear optimisation problem in the near field case.
Speaker: Johannes Zimmer (Bath)
Title: From particles to the geometry of thermodynamic evolution
Abstract: Often one wishes to describe the collective behaviour of an infinite number of particles by the differential equation governing the evolution of their joint density. The theory of hydrodynamic limits addresses this problem. In this talk, the focus will be on linking the particles with the geometry of the macroscopic evolution. In particular, gradient flows will be explained; they can be thought of as steepest descent/ascent of an entropy. Over the last two decades the associated theory of metric gradient flows has flourished and the talk will summarise some now-classic key results and present an extension to a class of nonlinear diffusion models. Large deviations will be used to provide a way to describe the many-particle dynamics by partial differential equations (PDEs); the PDE is a minimiser of the so-called rate function. However, the underlying microscopic process contains more information, notably fluctuations around the minimum state described by the deterministic PDE. Can stochastic terms be derived which model this additional information, in a way that is compatible to the limit passage via large deviations (and the geometric structure, such as the Wasserstein setting)? This will be discussed in the second part.
