Dynamics and Quantum Optimal Transport: Three lectures on quantum entropy and quantum Markov semigroups

This document presents the contents of three lectures delivered by the author at the Erdős Center School ``Optimal Transport on Quantum Structures’', Septemer 19-23, 2022 in Budapest, Hungary. It presents a fairly self contained account of an active topic of current research, and this account should be accessible to most graduate students, as befits lectures for a school. The main results are known, but there a number of new proofs and some new results.