"The objective of this proposal is to study ""continuous"" (or C^0) objects, as well as C^0 properties of smooth objects, in the field of symplectic geometry and topology. C^0 symplectic geometry has seen spectacular progress in recent years, drawing attention of mathematicians from various background. The proposed study aims to discover new fascinating C^0 phenomena in symplectic geometry.
One circle of questions concerns symplectic and Hamiltonian homeomorphisms. Recent studies indicate that these objects possess both rigidity and flexibility, appearing in surprising and counter-intuitive ways. Our understanding of symplectic and Hamiltonian homeomorphisms is far from being satisfactory, and here we intend to study questions related to action of symplectic homeomorphisms on submanifolds. Some other questions are about Hamiltonian homeomorphisms in relation to the celebrated Arnold conjecture. The PI suggests to study spectral invariants of continuous Hamiltonian flows, which allow to formulate the C^0 Arnold conjecture in higher dimensions. Another central problem that the PI will work on is the C^0 flux conjecture.
A second circle of questions is about the Poisson bracket operator, and its functional-theoretic properties. The first question concerns the lower bound for the Poisson bracket invariant of a cover, conjectured by L. Polterovich who indicated relations between this problem and quantum mechanics. Another direction aims to study the C^0 rigidity versus flexibility of the L_p norm of the Poisson bracket. Despite a recent progress in dimension two showing rigidity, very little is known in higher dimensions. The PI proposes to use combination of tools from topology and from hard analysis in order to address this question, whose solution will be a big step towards understanding functional-theoretic properties of the Poisson bracket operator."
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