Rigidity in functional spaces, symplectic approximation and geometry on the group of Hamiltonian diffeomorphisms

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AbstractAward: DMS 1105813, Principal Investigator: Lev BuhovskiRigidity in functional spaces is a relatively recent discovery in symplectic geometry, and these projects will continue the principal investigator's study of rigidity of Poisson brackets and other multilinear differential operators on symplectic manifolds. An approximation problem will also be studied: Given two smooth functions on a symplectic manifold, is it possible to approximate them by another pair of functions whose Poisson bracket is small in the uniform norm? Another line of investigation concerns the group of Hamiltonian diffeomorphisms, in the Hofer metric. The behavior of geodesics under the Hofer metric and under perturbations of it is an example of the kinds of questions that remain open regarding these diffeomorphisms.Symplectic geometry is the study of the background structure for the Hamiltonian version of classical mechanics, and the Poisson bracket referred to above is an operation on pairs of functions that is used in mechanics to compare integrals of motion for a mechanical system and in passing from a classical mechanical system to a counterpart system in quantum theory. The subject has been studied since the nineteenth century, but the introduction of new tools into symplectic geometry from the 1980s onward has opened up many surprising phenomena, including the rigidity property cited above, which shows that small perturbations of the definition of a Poisson bracket operation have only small effects on the way it acts on pairs of functions.ˇ