Deformations and Rigidity in Poisson Geometry

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AbstractAward: DMS 1405671, Principal Investigator: Ioan-Tiberiu MarcutThe Hamiltonian description of classical mechanics describes the state of a moving particle by recording its position and momentum as two coordinates, and by introducing a relationship between those coordinates through the Hamiltonian function H, a form of total energy that originated as the sum of kinetic energy and potential energy. The physical property of the conservation energy is then reported mathematically as the property that the value of the Hamiltonian function H does not change as the system evolves, but H determines that evolution through a system of differential equations that is equivalent to Newton's law that F = ma. The Poisson geometry of this proposal's title is a version of the geometry underlying Hamiltonian mechanics that is well-adapted to the needs of quantum mechanics.Projects supported by this award are focused on deformation and rigidity properties of Poisson structures. A rigidity result established by the principal investigator in earlier work is one of the ingredients for new work; another ingredient is an explicit construction of local groupoids which can be applied to prove results such as a local normal form around Poisson transversals.