JCJC SIMI 1 - JCJC - SIMI 1 - Mathématiques et interactions

Geometry and optimal measure transportation – GMT

Submission summary

The Ricci curvature plays an important role in Geometry and has lately been linked with the Optimal transportation theory. First by
G.~Perelman's proof of the Poincaré conjecture using the Ricci-flow and citing Bakry and Emery's work on
``Diffusions hypercontractives''. Secondly by a serie of papers by D.Cordero-Erausquin, R.McCann and M.Schmuckenschläger importing
optimal transportation in the realm of Riemannian geometry and using it to translate properties of curvature in terms of functional
inequalities. Finally by J.Lott-C.Villani and K.T.Sturm who accurately used optimal transportation to give a "synthetic approach to
the Ricci curvature". This synthetic definition is not only compatible with the usual one, but it also extends some geometric results,
such as Bishop's volume comparison theorem, to spaces satisfying lower bounds on their synthetic Ricci curvature. Since then other
bridges have been built from Optimal Transportation theory, such as the one obtained by P. Bernard-B. Buffoni with the Mather-Mañe
theory in dynamical systems and application of this synthetic Ricci curvature to metric measured spaces. Let us also mention
R.McCann-P.Topping who closed the circle by introducing some optimal transportion into the Ricci flow. One of the interesting
geometric aspect of this synthetic definition of a Ricci curvature bounded from below adapted to metric measured spaces is that it
is compatible with the Gromov-Hausdorff measured topology. A noticeable consequence of this property is that when one takes the limit
with respect to the Gromov-Hausdorff measured topology of a sequence of smooth manifolds with Ricci curvature bounded from below in
the usual sens one obtains a metric measured space with synthetic Ricci curvature bounded from below (compare to Gromov pre-compactness
theorem and Cheeger-Coldings work). However the very definition of this synthetic Ricci curvature makes it difficult to deal with, and
up to now the only spaces known to satisfy it (in a satisfactory sens) are those whose geometry is close enough to the Riemannian ones
to mimic their properties or which are limits of some who does, e.g., Alexandrov spaces, finite dimensional normed vector spaces.
Another important feature of optimal transportation is that it allows one to reprove most of geometric inequalities already known and
linked to functional inequalities, such as the isoperimetric one in the Euclidean space.Another important feature of optimal
transportation is that it allows one to reprove most of geometric inequalities already known and linked to functional inequalities,
such as the isoperimetric one in the Euclidean space. It also allowed specialist in optimal transportation to obtain some new
inequalities, such as inverse Brascamp-Lieb ones or get optimal constant in Sobolev inequalities. However other geometric problems
implying inequalities in an non-Euclidean setting, such as the isoperimetric inequality for Hadamard spaces, which is still an open
problem for dimension bigger than four, have yet to be addressed. The common motivation of the participants to this project is to
input more geometry into optimal transportation in order to get new informations, such as new functional inequalities, geometric ones
or comparisons results concerning volumes or distance between set, regularity of the transport.We intend to investigate optimal
transportation in setting where it is not fully understood, such as in Finsler geometry, where the results are not fully geometrically
meaningful, but also in Alexandrov's spaces where some tricky regularity problems awaits us and in sub-riemannian geometry where one
would like a synthetic definition of curvature compatible with their homogeneous dimension and volume growth. We would also like to
investigate the Wasserstein space of some metric spaces, a new geometry worth studying on its own and intricately linked to the mass
transportation problem.

Project coordinator

Monsieur Constantin VERNICOS (UNIVERSITE DE MONTPELLIER II [SCIENCES TECHNIQUES DU LANGUEDOC])

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

I3M UNIVERSITE DE MONTPELLIER II [SCIENCES TECHNIQUES DU LANGUEDOC]

Help of the ANR 79,997 euros
Beginning and duration of the scientific project: December 2011 - 48 Months

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