Skip to content

Accueil > Les trimestres du LMB > Automne 2014 > Noncommutative Analysis month > Workshop on noncommutative geometry and optimal transport

Workshop on noncommutative geometry and optimal transport

Autumn 2014 Noncommutative month GDR meeting
Optimal Transport
Besançon, November 27, 2014


The distance formula in noncommutative geometry has been introduced by Connes at the end of the 80’s. Given an algebra A acting on a Hilbert space H and an operator D on H, one defines on the space of states of A the (possibly infinite) distance
d(φ, φ’) = sup { φ(a) -φ’(a) : ||[D,a]|| ≤ 1, a ∈ A }.
For A=C0∞(M), the algebra of smooth functions vanishing at infinity on a locally compact complete spin manifold M and D the usual Dirac operator, this distance computed between pure states gives back the geodesic distance on M. Later on, Rieffel noticed that it can be seen as a noncommutative version of the Wasserstein distance of order 1 in the theory of optimal transport. More exactly, this is a noncommutative generalization of Kantorovich dual formula of the Wasserstein distance.

In this perspective, the equation above may provide an interesting starting point for a theory of optimal transport in noncommutative geometry :

- what remains of the duality Wasserstein (minimizing a cost)/Kantorovich (maximizing a profit) in the noncommutative setting ? Is there some ``noncommutative cost’’ that one is minimizing while computing the supremum in the distance formula ? Hints may come from free probability, where there already exists a noncommutative version of the Wasserstein distance.

- what are the topological properties of this distance ?

- is it relevant for physics ?

These are some of the questions we would like to offer to the discussion in this short workshop.

The metric aspect of noncommutative geometry is a part of the theory that has been relatively little studied so far. Besides the questions listed above, several results - including explicit computations - have now been obtained, and links with other areas of geometry (like sub-riemannian geometry) have been discovered. This ``exploratory’’ workshop aims at pushing forward these ideas, taking advantage of the annual meeting of the GDR Non-commutative geometry to gather mathematicians and mathematical-physicists from various communities.

Contact : Pierre Martinetti (Università di Napoli)
ou Jean-Christophe Wallet (LPT Orsay, CNRS)

Back to the GDR meeting page