Searches for Supersymmetry in multi-leptonic and $\tau$ pair final states with the ATLAS experiment at the LHC at $\sqrt{s}=13$~TeV

Supersymmetry (SUSY) is an attractive framework for extension of the Standard Model of particle physics providing answers for several open questions such as the nature of Dark Matter and the explanation of the value of the Higgs boson mass at the electroweak scale. It postulates for each Standard Model particle a superpartner with the same quantum numbers but differing in spin by half. None of these particles have been observed yet, leading to the conclusion that the superpartners are much heavier and SUSY must be broken. At the Large Hadron Collider~(LHC), proton beams collide at a centre-of-mass energy of actually $\sqrt{s}=13$~TeV in order to test the Standard Model and to search for new physics processes including supersymmetric particles. In this thesis, two searches for Supersymmetry have been performed using data of the ATLAS experiment at the LHC at $\sqrt{s}=13$~TeV with an integrated luminosity of 139~\ifb . If R-parity is violated due to additional lepton number violating interactions, the lightest supersymmetric particle (LSP) which is assumed to be produced in pairs and electrically neutral, can decay into two charged leptons and a neutrino. A search for such SUSY scenarios exploits final states wit four or more charged leptons which are experimentally very clean with low background and high reconstruction efficiency. The low background contributions also allow for probing supersymmetric models with gauge mediated SUSY breaking where the LSP is a gravitino originating from higgsino decays together with a Z or Higgs boson. The measurements were in good agreement with the SM predictions and improved lower limits were set on the supersymmetric particle masses and their decay modes into SM particles in the framework of simplified signal models. An essential prerequisite for searches in multi-leptonic final states is the precise knowledge of the lepton reconstruction efficiencies and of additional selection criteria. In this thesis, the muon reconstruction and identification efficiency as well as the efficiencies of a muon to originate from the primary collision vertex or to be isolated from hadronic activity in the detector have been estimated from data of $Z\to\mu\mu$ and $J/\psi\to\mu\mu$ decays with unprecedented precision of better than 0.1\% using the so-called tag-and-probe method. The second search presented in this thesis concerns the pair production of the supersymmetric partner of the $\tau$ lepton, the stau slepton. Light staus could reconcile the theoretical predictions with the measurements of the anomalous magnetic moment of the muon and also provide a mechanism for generating the Dark Matter relic density observed in the universe. The signature of such models consists of two $\tau$ leptons and missing transverse momentum from the two escaping LSPs and the neutrinos from $\tau$ decays. This final state is challenging at the LHC due to the large hadronic background. Hence, the limits on the stau mass from the LEP experiments prevailed until the end of the $\sqrt{s}=13$~TeV data taking. Events in which both $\tau$ leptons decay hadronically are used to search for direct stau production. No excess above the SM expectation has been observed and lower limits are placed on the stau masses which significantly improve the previous results. Prospect studies for combination of these results using di-tau events where one of the $\tau$ leptons decay leptonically are also presented.