Measurements of time-dependent $C\!P$ violation and mixing in charm at LHCb

Abstract: The LHCb experiment has opened the possibility to test mixing and $C!P$ violation in the charm sector with unprecedented precision thanks to the huge number of charm hadron decays collected, $\mathcal{O}(10^8)$. The first observation of $C!P$ violation in the decay of charm quarks in March 2019 has been a fundamental achievement. The latest LHCb measurements in the complementary sectors of mixing and time-dependent $C!P$ violation are illustrated in these proceedings. In particular, a new measurement of the $C!P$ violation parameter $A_\Gamma$ with 2015--2016 data that was presented for the first time at this conference is described. In the last section, prospects are given for the improvements in precision expected in the next few years.