Aug 26, 2013
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Some non local versions of the Monge Ampere equation
Luis Caffarelli (University of Texas, Austin)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- I will discuss some non-local versions of the Monge Ampere equation based on its optimal control character.
Existence, regularity and possible extensions
- Supplements
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v1131
5.3 MB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Optimal transport: old and new
Robert McCann (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The Monge-Kantorovich optimal transportation problem is to pair producers
with consumers so as to minimize a given transportation cost.
When the producers and consumers are modeled by probability densities
on two given manifolds or subdomains, it is interesting to try to understand
the analytical, geometric and topological features of the optimal pairing as a
subset of the product manifold. This subset may or may not be the graph of a
map.
This minicourse contrasts some recent developments concerning Monge's original
version of this problem, with a capacity constrained variant
in which a bound is imposed on the quantity transported between each given
producer and consumer. In particular, we give a new perspective on
Kantorovich's linear programming duality and expose how more subtle questions
relating the structure of the solution are intimately connected to the
differential topology and geometry of the chosen transportation cost.
In the later lectures, we shall illustrate how different aspects of curvature
(sectional, Ricci and mean) enter into the
- Supplements
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v1132
478 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Determining the regularity of a measure via the Wasserstein distance
Tatiana Toro (MSRI / Simons Laufer Mathematical Sciences Institute (SLMath))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- In this talk we will discuss several results with a common theme:" the regularity of a measure is determined
by the extent to which the measure can be is approximated by flat measures". This notion of approximation involves the Wasserstein distance, and to some extent is reminiscent of other approximation coefficients which appear in the Geometric Measure Theory literature. These work is joint with J. Azzam and G. David
- Supplements
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v1133
218 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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04:00 PM - 05:00 PM
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MSRI/Evans Talk: Swarming by Nature and by Design
Andrea Bertozzi (University of California, Los Angeles)
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- Location
- Evans Hall
- Video
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- Abstract
- The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community.
This talk will cover a survey of the speakers research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments.
We conclude with a discussion of some interesting problems for the mathematics community.
- Supplements
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05:00 PM - 06:30 PM
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Pizza Dinner
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- Location
- Laval's Pizza
- Video
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- Abstract
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- Supplements
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Aug 27, 2013
Tuesday
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09:30 AM - 10:30 AM
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Optimal transport and lower Ricci curvature bounds
Nicola Gigli (Université Nice Sophia-Antipolis)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- To provide an introduction to the field of synthetic treatment of lower Ricci curvature bounds via optimal transport. The course will cover the basic definitions and the crucial aspects of the theory, showing applications to both analysis and geometry.
- Supplements
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v1134
252 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Prescribed-divergence problems in optimal transportation
Filippo Santambrogio (Université de Paris XI)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The classical Monge problem (but only with cost $|x-y|$, and not $|x-y|^2$) has an equivalent counterpart which is the minimization of the $L^1$ norm of a vector field $v$, subject to the constraint $\nabla\cdot v=\mu-\nu$. This is a minimal flow problem introduced by M. Beckmann in the '50s without knowing the relation with the works by Kantorovich. It has recently come back into fashion because of its possible variants, where the cost rather than being linear as in the $L^1$ norm can be made convex (thus taking into account for congestion effects) or concave (favoring joint transportation). Also, well-posedness of this problem and regularity issues about the optimal $v$ have brought many questions about the so-called transport density, a measure of the local amount of traffic during the transportation which is naturally associated to these 1-homogeneous transport problems.
I will present the problem using in particular some recent approach based on the flow by Dacorogna-Moser, and give the main results on the transport density
- Supplements
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v1135
536 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Optimal transport: old and new
Robert McCann (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- The Monge-Kantorovich optimal transportation problem is to pair producers
with consumers so as to minimize a given transportation cost.
When the producers and consumers are modeled by probability densities
on two given manifolds or subdomains, it is interesting to try to understand
the analytical, geometric and topological features of the optimal pairing as a
subset of the product manifold. This subset may or may not be the graph of a
map.
This minicourse contrasts some recent developments concerning Monge's original
version of this problem, with a capacity constrained variant
in which a bound is imposed on the quantity transported between each given
producer and consumer. In particular, we give a new perspective on
Kantorovich's linear programming duality and expose how more subtle questions
relating the structure of the solution are intimately connected to the
differential topology and geometry of the chosen transportation cost.
In the later lectures, we shall illustrate how different aspects of curvature
(sectional, Ricci and mean) enter into the problem, and discuss applications
to economics if time permits
- Supplements
-
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v1136
354 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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Regularity in optimal transportation
Xu-Jia Wang (Australian National University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The potential functions in the optimal transportation satisfy a Monge-Ampere type equation. When the cost function c(x, y)=|x-y|^2, it is the standard Monge-Ampere equation, and has been studied by many people. For more general cost functions, Ma, Trudinger and myself obtained the regularity under a condition denoted as A3. Loeper showed that a weaker form of the condition, denoted as A3w, is necessary. The regularity under A3w was studied by Figalli, Kim, McCann. Most recently, Li, Santambrogio and myself also studied the regularity in Monge's mass transfer problem. In this talk I will discuss the latest development in this direction.
- Supplements
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v1137
189 KB application/pdf
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04:30 PM - 06:30 PM
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Reception
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- Location
- --
- Video
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- Abstract
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- Supplements
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Aug 28, 2013
Wednesday
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09:00 AM - 10:00 AM
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Prescribed-divergence problems in optimal transportation
Filippo Santambrogio (Université de Paris XI)
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- Location
- SLMath:
- Video
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- Abstract
- The classical Monge problem (but only with cost $|x-y|$, and not $|x-y|^2$) has an equivalent counterpart which is the minimization of the $L^1$ norm of a vector field $v$, subject to the constraint $\nabla\cdot v=\mu-\nu$. This is a minimal flow problem introduced by M. Beckmann in the '50s without knowing the relation with the works by Kantorovich. It has recently come back into fashion because of its possible variants, where the cost rather than being linear as in the $L^1$ norm can be made convex (thus taking into account for congestion effects) or concave (favoring joint transportation). Also, well-posedness of this problem and regularity issues about the optimal $v$ have brought many questions about the so-called transport density, a measure of the local amount of traffic during the transportation which is naturally associated to these 1-homogeneous transport problems.
I will present the problem using in particular some recent approach based on the flow by Dacorogna-Moser, and give the main results on the transport density
- Supplements
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v1138
520 KB application/pdf
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10:00 AM - 10:30 AM
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Tea
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- Location
- --
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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Optimal transport in non-commutative probability
Jan Maas (Rheinische Friedrich-Wilhelms-Universität Bonn)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 present transportation inequalities in this setting and obtain non-commutative versions of results by Bakry--Emery and Otto--Villani
- Supplements
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v1139
251 KB application/pdf
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11:35 AM - 12:00 PM
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Gradient Flow in the 2-Wasserstein Metric: a Crandall and Liggett type proof of the exponential formula
Katy Craig (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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v1140
372 KB application/pdf
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12:05 PM - 12:30 PM
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Numerical Computation of Soft Harmonic Maps
Adrian Butscher (Max-Planck-Institut für Informatik)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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v1141
6.42 MB application/pdf
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Aug 29, 2013
Thursday
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09:30 AM - 10:30 AM
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Optimal transport and lower Ricci curvature bounds
Nicola Gigli (Université Nice Sophia-Antipolis)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- To provide an introduction to the field of synthetic treatment of lower Ricci curvature bounds via optimal transport. The course will cover the basic definitions and the crucial aspects of the theory, showing applications to both analysis and geometry.
- Supplements
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v1142
247 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Partial regularity of optimal transport maps
Guido De Philippis (Hausdorff Research Institute for Mathematics, University of Bonn)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- We prove that for general smooth cost functions on the Euclidean space, or for the cost given by the squared distance on a Riemannian manifold, optimal transport maps between smooth densities are smooth outside a closed set of measure zero.(joint work with Alessio Figalli)
- Supplements
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v1143
343 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Optimal transport: old and new
Robert McCann (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- The Monge-Kantorovich optimal transportation problem is to pair producers
with consumers so as to minimize a given transportation cost.
When the producers and consumers are modeled by probability densities
on two given manifolds or subdomains, it is interesting to try to understand
the analytical, geometric and topological features of the optimal pairing as a
subset of the product manifold. This subset may or may not be the graph of a
map.
This minicourse contrasts some recent developments concerning Monge's original
version of this problem, with a capacity constrained variant
in which a bound is imposed on the quantity transported between each given
producer and consumer. In particular, we give a new perspective on
Kantorovich's linear programming duality and expose how more subtle questions
relating the structure of the solution are intimately connected to the
differential topology and geometry of the chosen transportation cost.
In the later lectures, we shall illustrate how different aspects of curvature
(sectional, Ricci and mean) enter into the problem, and discuss applications
to economics if time permits
- Supplements
-
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v1144
338 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Optimal transport and dynamics of expanding circle maps
Benoît Kloeckner (Université Grenoble Alpes (Université de Grenoble I - Joseph Fourier))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- - Expanding circle maps are arguably the simplest examples of discrete-time dynamical systems on manifolds exhibiting a chaotic behavior. The goal of the talk will be to explain in this simple context how tools from optimal transport can shed new light on dynamical systems. More precisely, we will compute the derivative of the action on measures of an expanding circle map, at its absolutely continuous invariant measure, and study its spectral properties. Optimal transport is used to define the derivative, following the intuition of Otto and the framework of Gigli
- Supplements
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v1145
190 KB application/pdf
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Aug 30, 2013
Friday
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09:30 AM - 10:30 AM
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Prescribed-divergence problems in optimal transportation
Filippo Santambrogio (Université de Paris XI)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The classical Monge problem (but only with cost $|x-y|$, and not $|x-y|^2$) has an equivalent counterpart which is the minimization of the $L^1$ norm of a vector field $v$, subject to the constraint $\nabla\cdot v=\mu-\nu$. This is a minimal flow problem introduced by M. Beckmann in the '50s without knowing the relation with the works by Kantorovich. It has recently come back into fashion because of its possible variants, where the cost rather than being linear as in the $L^1$ norm can be made convex (thus taking into account for congestion effects) or concave (favoring joint transportation). Also, well-posedness of this problem and regularity issues about the optimal $v$ have brought many questions about the so-called transport density, a measure of the local amount of traffic during the transportation which is naturally associated to these 1-homogeneous transport problems.
I will present the problem using in particular some recent approach based on the flow by Dacorogna-Moser, and give the main results on the transport density
- Supplements
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v1146
522 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- --
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Optimal transport and lower Ricci curvature bounds
Nicola Gigli (Université Nice Sophia-Antipolis)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- To provide an introduction to the field of synthetic treatment of lower Ricci curvature bounds via optimal transport. The course will cover the basic definitions and the crucial aspects of the theory, showing applications to both analysis and geometry.
- Supplements
-
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v1147
233 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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02:00 PM - 03:00 PM
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Monotonicity Formulas for Bakry-Emery Ricci Curvature
Guofang Wei (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- In the recent papers, Colding and Colding-Minicozzi introduced several new monotonity formulas for manifolds with nonnegative Ricci curvature. We extend these to Bakry-Emery Ricci curvature and obtain some new families of monotonocity formulas. This is a joint work with Bingyu Song and Guoqiang Wu
- Supplements
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v1148
743 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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The Schrödinger problem: a probabilistic analogue of optimal transport. Application to discrete metric graphs
Christian Leonard (Université de Paris X (Paris-Nanterre))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- To fit in with the spirit of an introductory workshop, the first part of the talk intends to be expository. The final part will be devoted to some advanced results about discrete metric graphs.
In 1931, Schrödinger addressed a problem of statistical physics nature which amounts to minimize the relative entropy of Markov random processes on a state space X subject to prescribed initial and final marginal distributions. The time marginal flow of the minimizing Markov process interpolates between the prescribed marginals on the space Proba(X) of probability measures on X.
Large deviations arguments show that slowing the processes down towards a no-motion process leads to a dynamical Monge-Kantorovich problem whose transport cost function is related to the random dynamics. The corresponding entropic interpolations are smooth paths on Proba(X) which converge to some displacement interpolation. For instance, if one takes the Brownian motion as reference process, the limiting interpolation is McCann's one.
Considering random walks on a discrete set X with a graph structure and passing to the slowing down limit allows to define natural displacement interpolations on Proba(X) that are geodesics with respect to an intrinsic graph distance. This opens the way for investigating curvature of graphs by tracking Lott-Sturm-Villani theory
- Supplements
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v1149
2.21 MB application/pdf
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