# Optimal transportation and applications (10w5025)

Arriving in Banff, Alberta Sunday, April 18 and departing Friday April 23, 2010

## Organizers

Alessio Figalli (ETH Zurich)

Yuxin Ge (Universite Paris Est Creteil)

Young-Heon Kim (University of British Columbia)

Robert McCann (University of Toronto)

Neil Trudinger (Australian National University)

## Objectives

The purpose of the workshop is to bring together several subgroups working in the different features of mass transportation theory,to report on recent progress,identify the questions and issues which will drive the next wave of research, and offer an opportunity for the exchange of ideas and cross-fertilization. The proposed gathering will include scientists working in geometrical aspects of the theory of elliptic Monge-Ampere equations, geometers, analysts, applied mathematicians and selected experts in economics, meteorology, and design. Special attention will be giving to the inclusion of young researchers, minorities and women.

The smoothness of optimal transport maps is an important issue in transportation theory since it gives information about qualitative behavior of the map, as well as simplifying computations and algorithms in numerical and theoretical implementations. Thanks to the results of Brenier and McCann, it is well known that the potential function of the map satisfies a Monge-Ampere type equation, an important fully nonlinear second order elliptic PDE arising in differential geometry. In the case of the quadratic cost function in Euclidean space, pioneering papers in this field are due to Delanoe, Caffarelli and Urbas. Very recently, Ma, Trudinger and Wang [2005] discovered a mysterious analytical condition (by now called (A3S) or the Ma-Trudinger-Wang condition) to prove regularity estimates for general cost functions. Costs functions which satisfy such a condition are called regular. At this point, Loeper [2007] gave a geometric description of this regularity condition, and he proved that the distance squared on the sphere is a uniformly regular cost, giving the first non-trivial example on curved manifolds. The Ma-Trudinger-Wang tensor is reinterpreted by Kim-McCann [2007] in an intrinsic way, and they show that it can be identified as the sectional curvature tensor on the product manifold equipped with a pseudo-Riemannian metric with signature $(n,n)$. Finally, recent results of Loeper-Villani [2008] and Figalli-Rifford [2008] show that the regularity condition on the square distance of a Riemannian manifold implies geometric results, like the convexity of the cut-loci. This new progress opens several directions in optimal transportation theory, especially its geometric aspects.

Links from optimal transport to geometric analysis, including to the theory of Ricci curvature and Ricci flow, have begun to emerge because of recent works of Lott, Villani, Sturm, Topping and McCann. The possibility to define useful analogs of such concepts in a metric measure space setting has been a tantalizing goal, only partly realized so far. Still this progress, together with the original contribution due to Otto on the formal Riemannian structure of the Wasserstein space and its application on PDE, is having a strong impact on the research community. Optimal transportation has also provided a new and simpler way to establish sharp geometric inequalities like the isoperimetric theorem, optimal Sobolev inequalities and optimal Gagliardo-Nirenberg inequalities. The proposed workshop will facilitate the interplay between the theory of optimal transport, geometric analysis and nonlinear partial differential equations.

On the other hand, among the numerous applications of optimal transport, we concentrate on economics, meteorology, and design problems:

1. Mass transportation duality is useful in formulating the problem of existence, uniqueness and purity for equilibrium in hedonic models. Recent works of Ekeland, Chiappori, McCann and Nesheim have shown that optimal transportation techniques are powerful tools for the analysis of matching problems and hedonic equilibria. Work of Rochet and Carlier also exposed applications to the principal agent problem - a central paradigm in microeconomic theory, which models the optimal decision problem facing a monopolist whose must act based on statistical information about her clients.

Although existence has generally been established in such models, characterization of the solutions, including uniqueness, smoothness, and comparative statics remain pressing open questions. Transportation theory has a wide range of further potential applications in econometrics, urban economics, adverse selection problems and nonlinear pricing. These deserve to be highlighted for the upcoming generation of mathematical researchers by experts in these fields.

2. Geophysical dynamics seeks to understand the evolution of the atmosphere and oceans, which is fundamental to weather and climate prediction. It has been shown that mass transportation theory can be applied to fluid dynamical problems, for instance those governing the large-scale behaviour of the atmosphere and oceans (Cullen [2006]). Here discontinuous solutions find important applications as models for atmospheric fronts, where the point is to analyze the geometry and dynamics of the discontinuity. The theory can also be given a geometrical interpretation, which has led to important extensions in its applicability, and can be used to investigate the qualitative impact of geographical formations, such as mountain ranges. A related open problem to which mass transportation is relevant is the incorporation moisture and thermodynamics into the dry dynamics, to model, e.g., rainstorms.

3. Finally, transportation has a number of promising applications in

engineering design - ranging from the construction of reflector antennas or shapes which minimize wind resistance, to problems in computer vision. Oliker and X-J Wang have pioneered the use of transportation theory in reflector design, while Plakhov has been exploring novel applications in aerodynamics.

Image registration offers medical applications, in which the goal is establish a common geometric reference frame between two or more diagnostic images captured at different times. Based on the mass transportation theory, Tannenbaum and his group developed powerful algorithms for computing elastic registration and warping maps.

The proposal is timely, both because of interest surrounding the enormous applications of optimal transportation methods, and because of new geometric concepts issued from mass transportation theory. The aim for such meeting between the different subgroups in mass transportation theory is to increase their awareness of each other's work, and also to identify the most important unexplored problems and pave the way for collaborations.

The smoothness of optimal transport maps is an important issue in transportation theory since it gives information about qualitative behavior of the map, as well as simplifying computations and algorithms in numerical and theoretical implementations. Thanks to the results of Brenier and McCann, it is well known that the potential function of the map satisfies a Monge-Ampere type equation, an important fully nonlinear second order elliptic PDE arising in differential geometry. In the case of the quadratic cost function in Euclidean space, pioneering papers in this field are due to Delanoe, Caffarelli and Urbas. Very recently, Ma, Trudinger and Wang [2005] discovered a mysterious analytical condition (by now called (A3S) or the Ma-Trudinger-Wang condition) to prove regularity estimates for general cost functions. Costs functions which satisfy such a condition are called regular. At this point, Loeper [2007] gave a geometric description of this regularity condition, and he proved that the distance squared on the sphere is a uniformly regular cost, giving the first non-trivial example on curved manifolds. The Ma-Trudinger-Wang tensor is reinterpreted by Kim-McCann [2007] in an intrinsic way, and they show that it can be identified as the sectional curvature tensor on the product manifold equipped with a pseudo-Riemannian metric with signature $(n,n)$. Finally, recent results of Loeper-Villani [2008] and Figalli-Rifford [2008] show that the regularity condition on the square distance of a Riemannian manifold implies geometric results, like the convexity of the cut-loci. This new progress opens several directions in optimal transportation theory, especially its geometric aspects.

Links from optimal transport to geometric analysis, including to the theory of Ricci curvature and Ricci flow, have begun to emerge because of recent works of Lott, Villani, Sturm, Topping and McCann. The possibility to define useful analogs of such concepts in a metric measure space setting has been a tantalizing goal, only partly realized so far. Still this progress, together with the original contribution due to Otto on the formal Riemannian structure of the Wasserstein space and its application on PDE, is having a strong impact on the research community. Optimal transportation has also provided a new and simpler way to establish sharp geometric inequalities like the isoperimetric theorem, optimal Sobolev inequalities and optimal Gagliardo-Nirenberg inequalities. The proposed workshop will facilitate the interplay between the theory of optimal transport, geometric analysis and nonlinear partial differential equations.

On the other hand, among the numerous applications of optimal transport, we concentrate on economics, meteorology, and design problems:

1. Mass transportation duality is useful in formulating the problem of existence, uniqueness and purity for equilibrium in hedonic models. Recent works of Ekeland, Chiappori, McCann and Nesheim have shown that optimal transportation techniques are powerful tools for the analysis of matching problems and hedonic equilibria. Work of Rochet and Carlier also exposed applications to the principal agent problem - a central paradigm in microeconomic theory, which models the optimal decision problem facing a monopolist whose must act based on statistical information about her clients.

Although existence has generally been established in such models, characterization of the solutions, including uniqueness, smoothness, and comparative statics remain pressing open questions. Transportation theory has a wide range of further potential applications in econometrics, urban economics, adverse selection problems and nonlinear pricing. These deserve to be highlighted for the upcoming generation of mathematical researchers by experts in these fields.

2. Geophysical dynamics seeks to understand the evolution of the atmosphere and oceans, which is fundamental to weather and climate prediction. It has been shown that mass transportation theory can be applied to fluid dynamical problems, for instance those governing the large-scale behaviour of the atmosphere and oceans (Cullen [2006]). Here discontinuous solutions find important applications as models for atmospheric fronts, where the point is to analyze the geometry and dynamics of the discontinuity. The theory can also be given a geometrical interpretation, which has led to important extensions in its applicability, and can be used to investigate the qualitative impact of geographical formations, such as mountain ranges. A related open problem to which mass transportation is relevant is the incorporation moisture and thermodynamics into the dry dynamics, to model, e.g., rainstorms.

3. Finally, transportation has a number of promising applications in

engineering design - ranging from the construction of reflector antennas or shapes which minimize wind resistance, to problems in computer vision. Oliker and X-J Wang have pioneered the use of transportation theory in reflector design, while Plakhov has been exploring novel applications in aerodynamics.

Image registration offers medical applications, in which the goal is establish a common geometric reference frame between two or more diagnostic images captured at different times. Based on the mass transportation theory, Tannenbaum and his group developed powerful algorithms for computing elastic registration and warping maps.

The proposal is timely, both because of interest surrounding the enormous applications of optimal transportation methods, and because of new geometric concepts issued from mass transportation theory. The aim for such meeting between the different subgroups in mass transportation theory is to increase their awareness of each other's work, and also to identify the most important unexplored problems and pave the way for collaborations.