Michael Entov

Calabi quasimorphism and quantum homology

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

Rigid subsets of symplectic manifolds

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Quasi-states and symplectic intersections

We establish a link between symplectic topology and a recently emerg\-ed branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

CALABI QUASIMORPHISMS FOR THE SYMPLECTIC BALL

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition thes...

On Continuity of Quasimorphisms for Symplectic Maps

We discuss C0-continuous homogeneous quasimorphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasimorphisms extend to the C0-closure of this group inside the homeomorphism group. We show that for standard symplectic balls of any dimension, as well as for compact oriented surfaces other than the sphere, the space of such quasimorphisms is infinite-dimensional. In the case of surfaces, we give a user-friendly topological characterization of such quasimorphisms. We also present an application to Hofer’s geometry on the group of Hamiltonian diffeomorphisms of the ball.

An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics

Abstract We report on an “anti-Gleason” phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories—symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.

Lagrangian tetragons and instabilities in Hamiltonian dynamics

We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity of Lagrangian submanifolds. Applications include superconductivity channels in nearly integrable systems and dynamics near a perturbed unstable equilibrium.

Unobstructed symplectic packing for tori and hyper-Kähler manifolds

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by balls are unobstructed if any collection of disjoint symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994, McDuff and Polterovich proved that symplectic packings of Kahler manifolds by balls can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple), these Kahler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that for any Campana simple Kahler manifold, as well as for any manifold which is a limit of Campana simple manifolds in a smooth deformation, the symplectic packings by balls are unobstructed. This is used to show that the symplectic packings by balls of all even-dimensional tori equipped with Kahler symplectic forms and of all hyper-Kahler manifolds of maximal holonomy are unobstructed. This generalizes a previous result by Latschev–McDuff–Schlenk. We also consider symplectic packings by other shapes and show, using Ratner’s orbit closure theorem, that any even-dimensional torus equipped with a Kahler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).