Workshop on noncommutative geometry and optimal transport

Introduction

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)

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