Quasi-states and symplectic intersections

By Michael Entov, L. Polterovich. 2004

158 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c39382c0c0d8294f99eb001d5a143f95c488edc

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Citations in the corpus

Lagrangian Floer theory on compact toric manifolds II: bulk deformations (2008, methodology): “0) in addition, then #(ψ(L(u0)) ∩L(u0)) ≥ 2n. (1.7) Theorem 1.5 will be proved in section 13. We would like to point out that (1.6) can be derived from a more general intersection result, Theorem 2.1 [EP1], obtained by Entov-Polterovich with a diﬀerent method using a very interesting notion of parti”
Conjugation-invariant norms on groups of geometric origin (2007, methodology): “Using methods of [12], one can show that frag U is unbounded on Ham( T 2 ) provided the subset U is displaceable by a Hamiltonian diﬀeomor-phism (e.g.”
Symplectic quasi-states and semi-simplicity of quantum homology (2007, unknown): “37 35 v3 [ m at h. SG ] 1”
Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, methodology): “Beginning with [EP03], work of Entov–Polterovich and others has shown that, on some symplectic manifolds, the asymptotic spectral invariant φ̃H 7→ limk→∞ ρ(e;H◊k)0 k (for H normalized and e a well-chosen idempotent with respect to the quantum multiplication ∗0) defines (after multiplication by a con”
Spectral numbers in Floer theories (2007, background): “(For instance, one can verify that the proofs in [1], when combined with the nondegenerate spectrality axiom, go through to show that there exists a “partial symplectic quasi-state” on any closed symplectic manifold (M, ω).”
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, methodology): “ilies of functions, µa: G→R, and ζa: C∞ c (T∗N) →R, where a∈H1(N;R). These functions possess properties analogous to those of partial quasi-morphisms and partial quasi-states of Entov and Polterovich [EP2].1) The precise properties are listed in theorems 1.3, 1.8 below. In the case N= Tn, the family”
Rigid subsets of symplectic manifolds (2007, methodology): “6) which form the axioms of a partial symplectic quasi-state introduced in [23].”
On the extrinsic topology of Lagrangian submanifolds (2005, background): “More generally, this holds true for the Clifford torus in CPn, see [ 6 ].”
Spectral invariants for monotone Lagrangians (2015, background): “ssociated to α1, α2. In case the ground ring Ris a ﬁeld, we have an equality. 1.4. An application to symplectic rigidity. We now present a sample application of this theory to symplectic rigidity. In [EP06] Entov–Polterovich introduced the notion of a symplectic quasistate. In the following deﬁnitio”