K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Given a closed symplectic manifold $(M,\omega)$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ${\hbox{\it Ham}} (M,\omega)$ by means of the Hofer metric on ${\hbox{\it Ham}} (M,\omega)$. We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold $(M,\omega)$ to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L.Polterovich's work on Hamiltonian fibrations over $S^2$.Comment: Corrected final version, accepted for publication in Inventiones Mathematica

Symplectic quasi-states and semi-simplicity of quantum homology

We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.Comment: A minor change: clarified the recipe for computing the quantum homology of a symplectic toric Fano manifol

C^0-rigidity of the double Poisson bracket

The paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe an hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau-Hadamard-Kolmogorov inequalities on the function-theoretic side.Comment: Minor corrections, to appear in IMR

Hofer's metric on the space of diameters

The present paper considers Hofer's distance between diameters in the unit disk. We prove that this distance is unbounded and show its relation to Lagrangian intersections.Comment: 11 pages, 4 figure