Title: Determination of Dark Matter Mass at Lepton Colliders and Experimental Studies in 2D Turbulence Advisor: Professor Tao Han Abstract: We present a study of antler decay in determining dark matter mass. It was shown that the cusps and endpoints in some kinematic distributions of the antler decay at the LHC can probe the masses of the parity-odd missing energy particles as well as the intermediate particles. We extend this study into the high energy $e^+ e^-$ (or more generally lepton) linear collider, which will provide unambiguous center of mass frame and energy. We found new and more powerful cusp structures of new kinematic observables, possible only at the International Linear Collider (ILC). As a benchmark scenario, we study $e^+ e^-\to \tilde{l}^+\tilde{l}^- \to l^+ l^- \neuo\neuo$ and $e^+ e^-\to \tilde{\chi}_1^+\tilde{\chi}_1^- \to j \, j \, j \, j \, \neuo\neuo$ in the framework of the Minimal Supersymmetric Standard Model.