Symplectic reduction of quasi-morphisms and quasi-states

We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.Comment: 20 pages; v2: added remarks and updated references, to appear in Journal of Symplectic Geometr

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.Comment: 53 pages, 26 figures; v3: minor corrections and updated references. To appear in Algebraic & Geometric Topolog

Existence and classification of overtwisted contact structures in all dimensions

We establish a parametric extension h-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.National Science Foundation (U.S.) (Grant DMS-1510305

Euler Integration of Gaussian Random Fields and Persistent Homology

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.Comment: 21 pages, 1 figur

Quasi-states, quasi-morphisms, and the moment map

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.Comment: 22 pages, 7 figures; v3: minor corrections, added remarks, and altered numbering scheme to match published version. To appear in International Mathematics Research Notice

Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of W are non-degenerate. In this paper we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.Comment: 16 pages; corrected version, published in Electron. Res. Announc. Math. Sc