### Hamiltonian Dynamics on Pseudodifferential Symbols

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We survey two problems illustrating geometric-topological and Hamiltonian methods in fluid mechanics: energy relaxation of a magnetic field and conservation laws for ideal fluid motion. More details and results, as well as a guide to the literature on these topics can be found in [].

We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d\u20131$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...

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