Optimal Transport: continuous, discrete, and quantum
The problem of Optimal Transport is easily stated: how should one transfer mass from given initial locations to prescribed target locations, in such a way that the total transport cost is minimised?
This venerable optimisation problem plays an important role in recent development in mathematics, at the interface of areas as metric geometry, probability theory, and partial differential equations.
Moreover, optimal transport receives renewed interest due to applications in data analysis and machine learning.
In this talk we give a short introduction to the topic and present some recent contributions to the area: firstly, we discuss the discrete-to-continuous limit of dynamical optimal transport and show a homogenisation result in this context; secondly, we present a non-commutative extension of optimal transport that yields a variational structure in dissipative quantum systems.