Noncommutative geometry and optimal transport - Workshop on Noncommutative Geometry and Optimal Transport (Besancon, France 27 Nov 2014); Martinetti, Pierre; Wallet, Jean-Christophe (Providence : American Mathematical Society, 2016)
This volume contains the proceedings of the Workshop on Noncommutative Geometry and Optimal Transport, held on November 27, 2014, in Besancon, France. The distance formula in noncommutative geometry was introduced by Connes at the end of the 1980s. It is a generalization of Riemannian geodesic distance that makes sense in a noncommutative setting, and provides an original tool to study the geometry of the space of states on an algebra. It also has an intriguing echo in physics, for it yields a metric interpretation for the Higgs field. In the 1990s, Rieffel noticed that this distance is 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. Connes distance thus offers an unexpected connection between an ancient mathematical problem and the most recent discovery in high energy physics. The meaning of this connection is far from clear. Yet, Rieffel'sobservation suggests that Connes distance may provide an interesting starting point for a theory of optimal transport in noncommutative geometry. This volume contains several review papers that will give the reader an extensive introduction to the metric aspect of noncommutative geometry and its possible interpretation as a Wasserstein distance on a quantum space, as well as several topic papers.
From Monge to Higgs : a survey of distance computations in noncommutative geometry / Pierre Martinetti -- Quantum metric spaces and the Gromov-Hausdor propinquity / Frederic Latremoliere -- Lectures on the classical moment problem and its noncommutative generalization / Michel Dubois-Violette -- Metrics and causality on moyal planes / Nicolas Franco and Jean-Christophe Wallet -- Pythagoras theorem in noncommutative geometry / Frances D'Andrea -- An overview of groupoid crossed products in dynamical systems / Mijail Guillemard.
Includes bibliographical references.
Keywords: Noncommutative differential geometry -- Congresses; Mathematical optimization -- Congresses