2,567 research outputs found
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
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
Constructing exact Lagrangian immersions with few double points
We establish an -principle for exact Lagrangian immersions with transverse
self-intersections and the minimal, or near-minimal number of double points.
One corollary of our result is that any orientable closed 3-manifold admits an
exact Lagrangian immersion into standard symplectic 6-space \R^6_\st with
exactly one transverse double point. Our construction also yields a Lagrangian
embedding S^1\times S^2\to\R^6_\st with vanishing Maslov class.Comment: In the new version corrected some misprints, added clarifications and
filled a small gap in the proof of Lemma 3.
Coisotropic rigidity and C^0-symplectic geometry
We prove that symplectic homeomorphisms, in the sense of the celebrated
Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their
characteristic foliations. This result generalizes the Gromov-Eliashberg
Theorem and demonstrates that previous rigidity results (on Lagrangians by
Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are
manifestations of a single rigidity phenomenon. To prove the above, we
establish a C^0-dynamical property of coisotropic submanifolds which
generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of
generators for continuous analogs of Hamiltonian flows.Comment: 27 pages. v2. Significant reorganization of the paper, several typos
and inaccuracies corrected after the refeering process. A theorem (Theorem 5,
completing the study of C^0 dynamical properties of coisotropics) added. To
appear in Duke Mathematical Journa
Foliations of Isonergy Surfaces and Singularities of Curves
It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure
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