Proves existence of exact lagrangian cobordisms between legendrians.
problem Existence of exact lagrangian cobordisms between closed legendrians.
method Proves existence through mathematical proof.
result Existence of non-compact exact lagrangian cobordisms.
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.
problem Understanding Lagrangian cobordisms and their properties in cotangent bundles.
method Use microlocal theory of sheaves, sheaf quantization, and cone decompositions.
result Interleaving distance of sheaves is bounded by the shadow distance of the cobordism.
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.
problem Understanding sheaf equivalences and actions on Lagrangian cobordisms.
method Analyzing sheaf quantizations and Legendrian lifts, proving functorial properties.
result Lagrangian cobordism functor is action decreasing and Morita equivalent to sheaf categories of Legendrians.
New methods link Legendrian satellites to Lagrangian cobordisms.
problem Understanding relations between Legendrian and Lagrangian knots.
method Constructing Lagrangian concordances through satellite operations.
result Maximum Thurston-Bennequin number restricts Legendrian satellite Lagrangian sliceness.
Study Lagrangian cobordisms in Lefschetz fibrations, proving generation results.
problem Understanding relations among Lagrangian submanifolds in symplectic manifolds.
method Prove generation results for Lagrangian cobordisms in the Fukaya category.
result Unified treatment and generalization of relations among Lagrangian submanifolds.
Non-orientable Lagrangian cobordisms exist for Legendrian knots, differing from orientable ones.
problem Understanding non-orientable Lagrangian cobordisms for Legendrian knots.
method Analyzing symplectization of standard contact 3-space and using exact, non-orientable Lagrangian cobordisms.
result Existence of non-orientable Lagrangian endocobordisms with crosscap genus being a positive multiple of 4.
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
New obstructions found for Lagrangian cobordisms in knot Floer homology.
problem Existence of Lagrangian cobordisms in knot Floer homology.
method Functorial behavior of Legendrian invariants with respect to decomposable Lagrangian cobordisms.
result New computable obstructions to Lagrangian cobordisms.
Proves uniqueness of real Lagrangians up to cobordism.
problem Proving uniqueness of real Lagrangians in symplectic manifolds.
method Cobordism theory and Hamiltonian isotopy.
result Real Lagrangians are unique up to cobordism.
We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the nega…
Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prov…
GRID invariants block certain Lagrangian cobordisms in 3D.
problem Obstructing decomposable Lagrangian cobordisms in 3D.
method Filtered GRID invariants and link Floer homology.
result GRID invariants obstruct decomposable Lagrangian cobordisms.
The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.
problem Understanding how Lagrangian cobordisms impact DGAs of Legendrian ends.
method Adapting the map induced by cobordisms on DGAs to linearizations using augmentations, and showing invariance under Lagrangian isotopy.
result The induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses.
The study connects knot Floer homology and Lagrangian cobordisms.
problem Obstructing decomposable Lagrangian cobordisms in Weinstein cobordisms.
method Using knot Floer homology and transverse invariants, the study constructs maps between knot Floer homology groups.
result The invariant obstructs decomposable Lagrangian cobordisms of arbitrary genus.
Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X,L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L⊂X which agrees with cylinders over Legendrian links $…
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
New invariant stops certain types of geometric transformations.
problem Obstructing decomposable Lagrangian cobordisms.
method Defined an invariant for Legendrian links.
result Obstructs decomposable Lagrangian cobordisms.
Upper bounds for Legendrian links in tight contact 3-manifolds.
problem Bounding the complexity of Legendrian links in tight contact 3-manifolds.
method Constructing exact Lagrangian cobordisms and defining minimal Lagrangian genus.
result Established upper bounds for Legendrian links with a common rotation number.
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
Functorial properties of knot invariants under specific concordances.
problem Computing obstructions to regular Lagrangian concordances.
method Functoriality of EH class and LOSS invariant under Lagrangian concordances in Weinstein cobordisms.
result Computable obstructions to regular Lagrangian concordances.
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M i…
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…
We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over the real numbers. We define an index two subcategory (the even subcategory) of a 3-dimensional cobordism category. The objects of the category are surfaces are equipped with Lagrangian subspac…
The study finds an obstruction for decomposable exact Lagrangian fillings of certain Legendrian knots.
problem Properties of decomposable exact Lagrangian cobordisms between Legendrian links.
method Analysis of normal rulings associated with decomposable exact Lagrangian fillings.
result A Legendrian knot has a decomposable exact Lagrangian filling if and only if it has an even number of normal rulings.
Paper proves triple linking form vanishes under specific conditions.
problem Analyzing rational homology cobordism and linking forms.
method Proves vanishing of triple torsion linking form under specific conditions.
result Triple torsion linking form vanishes on a specific Lagrangian.
New approach solves problems in gauge theory and symplectic geometry.
problem Instanton Floer homology and Lagrangian Floer theory.
method Combination of cobordism type argument and homological algebra.
result Various problems solved and proofs simplified.
We prove that any Legendrian knot in (S3,ξstd) bounds an exact Lagrangian surface in R4∖B4 after a sufficient number of stabilizations. In order to show this, we construct a family combinatorial moves on knot projections with some additional data that correspond to Lagrangian cobordisms betw…
Regular sliceness implies once-stably decomposable sliceness in symplectizations.
problem Relationship between regular and decomposable Lagrangian cobordisms in symplectizations.
method Stabilization-free strategy and satellite operations.
result Regular sliceness implies once-stably decomposable sliceness.
Study proves a space of manifolds is homology equivalent to a loop space.
problem Understanding the stable moduli space of certain high-dimensional manifolds.
method Incorporates surgery data via Lagrangian subspaces in a cobordism category.
result The stable moduli space is homology equivalent to an infinite loopspace.
The study explores Legendrian fillings and augmentations, providing methods to compute induced augmentations.
problem Understanding and computing Legendrian isotopy invariants through augmentations and fillings.
method Developed methods to compute induced augmentations based on Morse complex families and Legendrian cobordisms.
result Established methods to compute Legendrian isotopy invariants using augmentations and fillings.
We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and ce…
We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator Dt on each regular level set Ct of a fixed Morse function defining this cobordism. We show that as we approac…
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
A new functor extends Magnus representation to 3D cobordisms.
problem Extending Magnus representation to 3D cobordisms.
method Constructing a monoidal functor from Cob_G to pLagr_R.
result Relates the Magnus functor to the Alexander functor.
This paper interprets topologically the tree reduction of the LMO functor using Johnson-Levine homomorphisms.
problem Interpreting the LMO functor's tree reduction topologically.
method Topological interpretation of the LMO functor's tree reduction using Johnson-Levine homomorphisms.
result Topological interpretation of the LMO functor's tree reduction.
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.
New examples of Legendrian links with infinitely many fillings.
problem Understanding the structure of Legendrian links and their fillings.
method New combinatorial formula for Legendrian contact DGAs and Floer-theoretic techniques.
result Construction of the first families of Legendrian links with infinitely many Lagrangian fillings.
The paper constructs TQFTs and Schrödinger representations for Heisenberg group.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
Study tangle invariants using pillowcase Fukaya category and character varieties.
problem Tangle invariants and their relationships with Khovanov cohomology.
method Define a twisted complex of compact Lagrangians in the pillowcase's Fukaya category, show functoriality, and relate to Khovanov cohomology.
result Hom set with a specific Lagrangian is naturally isomorphic to reduced Khovanov chain complex.
We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…
A new inequality linking submanifold's topology to its Reeb chords count.
problem Establishing a relationship between Legendrian submanifolds' topology and their Reeb chords count.
method Using a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by neck-stretching and splashing, under a small contact isotopy.
result The number of Reeb chords is at least the sum of the Z2-Betti numbers of the submanifold. The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.
problem Developing a new invariant for cobordism classes of manifolds.
method Introducing an invariant ϰR for null-cobordant n-manifolds and constructing cobordism groups ΩnR. result The invariant ϰR is a complete invariant of R-cobordism classes of null-cobordant n-manifolds.