Quantum Wasserstein topologies and applications

After having exhibited some lack of pertinence of standard Hilbert-Schmidt or trace class (or more general $L^p$-Schatten class) topologies usually used for linear PDEs, I will present a quantum notion of the Wasserstein-Monge-Kantorovich distance of order two canonically obtained through a simple dictionary between classical and quantum mathematical paradigms. This will lead to a quantum definition of optimal transport, actually shown to be ¨cheaper¨ than the classical one e.g. for the bi-partite problem and to make sense in situations where the standard classical (Brenier's) one fails to be true. As a bi-product I will show how quantization can be seen as a kind of Wasserstein geodesic path between a classical function and a quantum operator, thanks to a ¨semiquantum¨ Legendre transform.

No quantum mechanics prerequisites will be necessary for following the lecture.