1,624 research outputs found
Fukaya categories of symmetric products and bordered Heegaard-Floer homology
The main goal of this paper is to discuss a symplectic interpretation of
Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of
Fukaya categories of symmetric products and Lagrangian correspondences. More
specifically, we give a description of the algebra A(F) which appears in the
work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer
homology for product Lagrangians in the symmetric product, and outline how
bordered Heegaard-Floer homology itself can conjecturally be understood in this
language.Comment: 54 pages, 11 figures; v3: minor revisions, to appear in J Gokova
Geometry Topolog
Cluster varieties from Legendrian knots
Many interesting spaces --- including all positroid strata and wild character
varieties --- are moduli of constructible sheaves on a surface with
microsupport in a Legendrian link. We show that the existence of cluster
structures on these spaces may be deduced in a uniform, systematic fashion by
constructing and taking the sheaf quantizations of a set of exact Lagrangian
fillings in correspondence with isotopy representatives whose front projections
have crossings with alternating orientations. It follows in turn that results
in cluster algebra may be used to construct and distinguish exact Lagrangian
fillings of Legendrian links in the standard contact three space.Comment: 47 page
Arnold diffusion for a complete family of perturbations with two independent harmonics
We prove that for any non-trivial perturbation depending on any two
independent harmonics of a pendulum and a rotor there is global instability.
The proof is based on the geometrical method and relies on the concrete
computation of several scattering maps. A complete description of the different
kinds of scattering maps taking place as well as the existence of piecewise
smooth global scattering maps is also provided.Comment: 23 pages, 14 figure
Intersections of Lagrangian submanifolds and the Mel'nikov 1-form
We make explicit the geometric content of Mel'nikov's method for detecting
heteroclinic points between transversally hyperbolic periodic orbits. After
developing the general theory of intersections for pairs of family of
Lagrangian submanifolds constrained to live in an auxiliary family of
submanifolds, we explain how the heteroclinic orbits are detected by the zeros
of the Mel'nikov 1 -form. This 1 -form admits an integral expression, which is
non-convergent in general. Finally, we discuss different solutions to this
convergence problem.Comment: Corrected typos, modified title, updated bibliograph
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