CP Violation and Mixing in Multi-body $D$ decays

We present recent LHCb results and future prospects for CP violation and mixing measurements in multi-body charm decays. The complex amplitude structure of multi-body decays provides unique sensitivity to CP violation localised in certain phase space regions. A model-independent search in the phase space of $D \to \pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and $D \to K^{+}K^{-}\pi^{+}\pi^{-}$ decays showed no evidence for localised CP violation. If one assumes the no CP violation hypothesis, the probability of getting the observed results is $9.1\%$ and $41\%$, respectively. The model-independent determination of gamma from $B\to DK$ requires external input to account for the interference of $D^{0}$ and $\bar{D^{0}}$ amplitudes to the same final state. Previously this input could only be obtained at the charm threshold, but recently it has been proposed that $D$ mixing can provide complimentary information. For the example of $D \to K^{+}\pi^{-}\pi^{+}\pi^{-}$ decays, it is shown that charm mixing can be used to considerably improve current constraints on the coherence factor and average strong phase difference, with existing data.