Symplectic topology and ideal-valued measures.
Joint with Adi Dickstein, Leonid Polterovich and Frol Zapolsky. - Published in Selecta Mathematica.
Published Version (arXiv Preprint arXiv:2107.10012).
We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.
Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds.
Joint with Shira Tanny - - Published in Algebraic & Geometric Topology.
Published Version (arXiv Preprint: arXIv:2005.11096).
We study the Floer-theoretic interaction between disjointly supported Hamiltonians by comparing Floer-theoretic invariants of these Hamiltonians with the ones of their sum. These invariants include spectral invariants, boundary depth and Abbondandolo-Haug-Schlenk's action selector. Additionally, our method shows that in certain situations the spectral invariants of a Hamiltonian supported in an open subset of a symplectic manifold are independent of the ambient manifold.
Lagrangian isotopies and symplectic function theory.
Joint with Michael Entov and Cedric Membrez - Published in Commentarii Mathematici Helvetici.
Published Version. (arXiv Preprint: arXiv:1610.09631).
We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.
A homotopical viewpoint at the Poisson bracket invariants for tuples of sets - Published in Journal of Symplectic Geometry.
Published Version. (arXiv Preprint: arXiv:1806.06356).
We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data.
Enumeration of plane unicuspidal curves of any genus via tropical geometry.
Joint with Eugenii Shustin - Published in International Mathematics Research Notices.
Published Version. (open access link) (Preprint: arXiv:1807.11443).
We enumerate complex plane curves of any given degree and genus having one cusp and nodes as their singularities and matching appropriately many point constraints. The solution is obtained via the tropical enumerative geometry: We establish a correspondence between the algebraic curves in count and certain plane tropical curves and compute how many algebraic curves tropicalize to a given tropical curve. To enumerate these tropical curve, we provide a version of Mikhalkin's lattice path algorithm. The same approach enumerates unicuspidal curves of any divisor class and genus on many other toric surfaces. We also demonstrate how to enumerate real plane cuspidal curves.
Poisson Bracket Invariants and Wrapped Floer Homology.
(Preprint: arXiv:2410.04572).
The Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich and further studied by Entov and Polterovich, serve as invariants for quadruples of closed sets in symplectic manifolds. Their nonvanishing has significant implications for the existence of Hamiltonian chords between pairs of sets within the quadruple, with bounds on the time-length of these chords. In this work, we establish lower bounds on the Poisson bracket invariants for certain configurations arising in the completion of Liouville domains. These bounds are expressed in terms of the barcode of wrapped Floer homology. Our primary examples come from cotangent bundles of closed Riemannian manifolds, where the quadruple consists of two fibers over distinct points and two cosphere bundles of different radii, or a single cosphere bundle and the zero section.