In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Ha... more In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) 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 quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).
In this work we bring together tools and ideology from two different fields, Symplectic Geometry ... more 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 by its volume radius.
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and ... more In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.
In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and... more In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.
In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body... more In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal symplectic capacity. We present a proof of this fact up to a logarithmic factor in the dimension, and many classes of bodies for which this holds up to a universal constant.
In this work we bring together tools and ideology from two different fields, Symplectic Geometry ... more 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 by its volume radius.
In this work, we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry an... more In this work, we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.
We study certain algebraic properties of the small quantum homology algebra for the class of symp... more We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.
Let M be a closed symplectic manifold, and let · be a norm on the space of all smooth functions o... more 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 · ≤ C · ∞ , 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 · ∞ , then the induced pseudo-distance function on the group Ham(M, ω) of Hamiltonian diffeomorphisms of M vanishes identically. These results provide partial answers to questions raised by Eliashberg and Polterovich in [4]. Both results rely on an extension of · to the space of essentially bounded measurable functions, which is invariant under all measure preserving bijections.
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Papers by Yaron Ostrover