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The talk is a journey on the topic of dynamical phase transitions in out-of-equilibrium quantum many body systems. I will first present results regarding a collective Heisenberg ferromagnet displaying a long-lived, stable dynamical ferromagnetic phase and discuss its realisation with ultra-cold fermions. The seminar will branch then into the possibility to engineer a dynamical chaotic ferromagnetic phase when quantum fluctuations reshape the out-o-equilibrium mean field phase diagram of the Lipkin-Meshkov-Glick model. Finally, I will present results, based on Keldysh field theory and renormalization group analysis, concerning the onset of critical aging in isolated interacting quantum systems. Host: Yigit Subasi |