A Hofer-Type Norm of Hamiltonian Maps on Regular Poisson Manifold

We define a Hofer-type norm for the Hamiltonian map on regular Poisson manifold and prove that it is nondegenerate. We show that the L1,∞-norm and the L∞-norm coincide for the Hamiltonian map on closed regular Poisson manifold and give some sufficient conditions for a Hamiltonian path to be a geodesic. The norm between the Hamiltonian map and the induced Hamiltonian map on the quotient of Poisson manifold (M,{·,·}) by a compact Lie group Hamiltonian action is also compared

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

Spectral killers and Poisson bracket invariants

We find optimal upper bounds for spectral invariants of a Hamiltonian whose support is contained in a union of mutually disjoint displaceable balls. This gives a partial answer to a question posed by Leonid Polterovich in connection with his recent work on Poisson bracket invariants of coverings.Comment: 16 pages, 2 figures. V2: to appear in Journal of Modern Dynamic

Lectures on Groups of Symplectomorphisms

These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srni, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om) from various points of view. Lectures 1 and 2 provide an overview of our current knowledge of their algebraic, geometric and homotopy theoretic properties. Lecture 3 sketches the arguments used by Gromov, Abreu and Abreu-McDuff to figure out the rational homotopy type of these groups in the cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve techniques. Much of the recent progress in understanding the geometry and topology of these groups has come from studying the properties of fibrations with the manifold M as fiber and structural group equal either to the symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base is S^2 has proved particularly important. Lecture 4 describes the geometry of Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten invariants via the Seidel representation. It ends by sketching Entov's explanation of the ABW inequalities for eigenvalues of products of special unitary matrices. Finally in Lecture 6 we apply the ideas developed in the previous two lectures to demonstrate the existence of (short) paths in Ham(M,\om) that minimize the Hofer norm over all paths with the given endpoints.Comment: significantly revised, 36 pages; notes from summer school in Paris 2001 and winter school in Srni 200

A comparison of symplectic homogenization and Calabi quasi-states

We compare two functionals defined on the space of continuous functions with compact support in an open neighborhood of the zero section of the cotangent bundle of a torus. One comes from Viterbo's symplectic homogenization, the other from the Calabi quasi-states due to Entov and Polterovich. In dimension 2 we are able to say when these two functionals are equal. A partial result in higher dimensions is presented. We also give a link to asymptotic Hofer geometry on T^*S^1. Proofs are based on the theory of quasi-integrals and topological measures on locally compact spaces