Noncommutative Geometry and Optimal Transport
About this Title
Pierre Martinetti, Università di Genova, Genova, Italy and Jean-Christophe Wallet, CNRS, Université Paris-Sud 11, Orsay, France, Editors
Publication: Contemporary Mathematics
Publication Year: 2016; Volume 676
ISBNs: 978-1-4704-2297-4 (print); 978-1-4704-3560-8 (online)
This volume contains the proceedings of the Workshop on Noncommutative Geometry and Optimal Transport, held on November 27, 2014, in Besançon, 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's observation 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.
Graduate students and research mathematicians interested in noncommutative geometry.
Table of Contents
- Pierre Martinetti – From Monge to Higgs: a survey of distance computations in noncommutative geometry
- Frédéric Latrémolière – Quantum Metric Spaces and the Gromov-Hausdorff Propinquity
- Michel Dubois-Violette – Lectures on the classical moment problem and its noncommutative generalization
- Nicolas Franco and Jean-Christophe Wallet – Metrics and causality on Moyal planes
- Francesco D’Andrea – Pythagoras Theorem in noncommutative geometry
- Mijail Guillemard – An Overview of Groupoid Crossed Products in Dynamical Systems