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We will start with classical mechanics and introduce the Lagrangian and Hamiltonian points of view, explaining how symplectic (and contact) geometry provides a natural framework for studying these systems. I will then give some of the background needed to understand a class of examples related to the so-called contact-type conjecture. If time permits, I will also sketch how these examples are constructed.
Part of this talk is based on joint work with Levin Maier and Tom Stalljohann. Some familiarity with differential geometry may be helpful, but it is not required.