Yaron Ostrover

Calabi quasi-morphisms for some non-monotone symplectic manifolds

In this work we construct Calabi quasi-morphisms on the universal cover e Ham.M/ of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasimorphisms descend to non-trivial homomorphisms on the fundamental group of Ham.M/.

A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

We compare Hofers geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham(M) into ℒ, f ↦ graph(f) is not an isometric embedding, although it preserves Hofers length of smooth paths.

From symplectic measurements to the Mahler conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahlers conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

The M-ellipsoid, symplectic capacities and volume

In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,

On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.

On the Uniqueness of Hofer’s Geometry

We study the class of pseudo-norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such pseudo-norm that is continuous with respect to the C∞-topology, is dominated from above by the L∞-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant pseudonorm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer’s metric.

Asymptotic Equivalence of Symplectic Capacities

A long-standing conjecture states that all normalized symplectic capacities coincide on the class of convex subsets of

REMARKS ABOUT MIXED DISCRIMINANTS AND VOLUMES

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A Remark on Projections of the Rotated Cube to Complex Lines

. In this note we focus on an asymptotic (in the dimension) version of this conjecture, and show that when restricted to the class of centrally symmetric convex bodies in

The existence of a billiard orbit in the regular hyperbolic simplex

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