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Humans interact with chaotic systems on everyday basis. Chaos plays a fundamental role on every length scale. It can be seen in the motion of asteroids, the formation of weather patterns, population growth, and even in the firing of neurons. When studying a chaotic system, one frequently faces both mathematical and experimental challenges. From mathematical point of view it is sometimes hard to even write down a complete set of equations describing the system behavior, while the experimental challenges arise from the fact that it is hard to isolate a chaotic system from random interactions with the environment. In recent years, atomic gasses have emerged as experimentally accessible systems for observing chaos under controlled conditions. In this talk I will present the study of geometric structures that arise in phase space of a Rydberg atom exposed to external electric field pulses. I will describe how these geometric structures govern the chaotic transport and how we can use them to both understand and to probe the ionization process. Finally, I will present a set of recent results which show that this approach is valid not only in the classical regime, but also for atoms whose energy levels are in the regime frequently thought of as requiring quantum computations. Host: Sebastian Deffner 