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Title:
Optimal transport: old and new
Speaker:
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
Link:
Workshop: