Search for Lepton Flavour violating decay $B^0 \to K^{∗0}τ^\pm e^\mp$ with the LHCb experiment, contribution in Rare Decay trigger configuration and implementation of isolation tools for the Upgrade of the detector

In the thesis, I present my two main contributions to the LHCb experiment.The first one concerns the development and configuration of the LHCb software trigger for the LHCb Upgrade. Multiple accomplishments were achieved. First, I developed several software algorithms and tools, named ThOr functors, oriented to fast and parallel programming logic. Part of this work is functional in implementing within the software framework a powerful tool called "isolation", which is used for background suppression in many physics analyses. Additionally, I performed rate and bandwidth studies needed to define the trigger selections for the physics analysis of the LHCb Rare Decays working group, in particular for Lepton Flavour Violation searches and Lepton Flavour Universality tests.The second main contribution is the first-ever search for the Lepton Flavour Violating decay $B^0 → K^{∗0}τ^{\pm}e^{\mp}$ using data collected at LHCb during Run 2 for a total integrated luminosity of $5.6 fb^-1$. I performed all the steps of the analysis: the definition of the selection, the evaluation of its efficiency, the determination of the systematic uncertainties, the fit to the data; the evaluation of the upper limit on the branching fractions of these decays, which correspond to $\sim 3 \times 10^-6$.