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 of Eliashberg. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.
Nick Sheridan's notes from my lecture series at IAS on overtwisted contact structures.
Quasi-morphisms on contactomorphism groups and contact rigidity, with F. Zapolsky. Geom. Topol., 19(1):365-411, 2015.
We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain
prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact
reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact
We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs.
Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.
Bounding Lagrangian widths via geodesic paths, with M. McLean. Compos. Math. , 150(12):2143-2183, 2014.
The width of a Lagrangian is the largest capacity of a ball that can be symplectically
embedded into the ambient manifold such that the ball intersects the Lagrangian exactly
along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction
to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg-Murphy.
In this paper we introduce a new method for bounding the width of a Lagrangian Q by
considering the Lagrangian Floer cohomology of an auxiliary Lagrangian L with respect
to a Hamiltonian whose chords correspond to geodesic paths in Q. This is formalized
as a wrapped version of the Floer-Hofer-Wysocki capacity and we establish an associated
energy-capacity inequality with the help of a closed-open map. For any orientable
Lagrangian Q admitting a metric of non-positive sectional curvature in a Liouville manifold,
we show the width of Q is bounded above by four times its displacement energy.
Spherical Lagrangians via ball packings and symplectic cutting, with T.-J. Li and W. Wu. Selecta Math., 20(1):261-283, 2014.
In this paper we prove the connectedness of symplectic ball packings
in the complement of a spherical Lagrangian, S^2 or RP^2,
in symplectic manifolds that are rational or ruled.
Via a symplectic cutting construction this is a natural extension of
McDuff's connectedness of ball packings in other settings
and this result has applications to several different
questions: smooth knotting and unknottedness results for
spherical Lagrangian, the transitivity of the action of the
symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretic
essential Lagrangian tori in the del Pezzo surfaces.
Displacing Lagrangian toric fibers by extended probes, with M. Abreu and D. McDuff. Algebr. Geom. Topol., 14(2):687-752, 2014.
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.
Abreu's slides from various talks
Quasi-states, quasi-morphisms, and the moment map. Int. Math. Res. Not. IMRN, 2013(11):2497-2533, 2013.
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
Symplectic reduction of quasi-morphisms and quasi-states. J. Symplectic Geom., 10(2):225-246, 2012.
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.
Euler integration of Gaussian random fields and persistent homology, with O. Bobrowski. J. Topol. Anal., 4(1):49-70, 2012.
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 computation of a quantitative descriptor for the persistent homology of a Gaussian random field.
Persistent homology for random fields and complexes, with R. J. Adler, O. Bobrowski, E. Subag, and S. Weinberger. Borrowing Strength: Theory Powering Applications, A Festschrift for Lawrence D. Brown, IMS Collections 6, 124-143, 2010.
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.